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The Global Integrated Monetary and Fiscal Model (GIMF) - Theoretical Structure

Author(s):
Douglas Laxton, Susanna Mursula, Michael Kumhof, and Dirk Muir
Published Date:
February 2010
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I. Introduction

This paper presents a comprehensive overview of the theoretical structure of the International Monetary Fund’s Global Integrated Monetary and Fiscal Model (GIMF). GIMF is a multicountry dynamic general equilibrium model that is used extensively inside the IMF, and also at a small number of central banks, for policy and risk analysis. A significant number of GIMF-based IMF Working Papers and Special Issues Papers has been produced by country desks dealing with policy questions relating to their respective countries. GIMF simulations have been used to produce World Economic Outlook scenario analyses, and also a variety of internal risk assessment analyses, since 2008. Future uses of GIMF will include forecasting exercises based on filtering historical data.

The traditional strength of GIMF since it was first made available to staff has been its ability to analyze fiscal policy questions, due to incorporation of a variety of non-Ricardian features that make not only spending-based but also revenue-based fiscal measures non-neutral. These features have been deployed intensively in the analyses of the likely short-run effectiveness of recent fiscal stimulus packages. But, given its focus on the savings-investment balance, GIMF will also be very useful when the focus turns from short-run stimulus to long-run sustainability, that is to questions of the link between fiscal deficits and real interest rates, crowding out, and current account deficits.

Recent events have however also suggested some important extensions to GIMF, most but not all of which have been completed and are therefore described in this document. The first extension concerns macro-financial linkages. GIMF now has a financial accelerator mechanism for the non-financial corporate sector that gives a role to corporate net worth, corporate leverage, external finance premia and bankruptcies. This is described at length in this document. Currently under development is a banking sector that intermediates funds between households and the non-financial corporate sector, and which also has its own net worth and leverage. The second extension is motivated by the oil-price shocks that hit the world economy in 2008. GIMF now has a generic raw-materials sector with inelastic supply and with a demand whose price elasticity can be calibrated according to the raw material under consideration, which will not always be oil. This sector is also described in this document.

The structure of this paper follows the sectorial breakdown of GIMF. Because GIMF is now a highly modular tool, many of these sectors can be turned on or off depending on the complexity needed for the respective application. This modular structure is fully operational in TROLL versions of GIMF. For DYNARE users a similar modular structure is currently being developed, but it is not available to end users at this time.

At the end of every section that describes a separate economic agent we add some comments on this “Modularity”. Specifically, we comment on whether it is advisable, depending on the question being addressed using GIMF, to turn off the respective feature. Additionally, because this working paper will also serve a reference for the paper “Fiscal Stimulus to the Rescue?”, we will comment on what features are present in the version of GIMF used for that paper.

II. Model Overview

The world consists of Ñ countries. The domestic economy is indexed by j = 1 and foreign economies by j = 2,…, Ñ. In our exposition we will ignore country indices except when interactions between multiple countries are concerned. It is understood that all parameters except gross population growth n and gross technology growth g can differ across countries. Figure 1 illustrates the flow of goods and factors for the two country case.

Countries are populated by two types of households, both of which consume final retailed output and supply labor to unions. First, there are overlapping generations households with finite planning horizons as in Blanchard (1985). Each of these agents faces a constant probability of death (1 – θ(j)) in each period, which implies an average planning horizon of 1/(1 — θ(j)).1 In each period, N(j)nt(1ψ(j)) (1θ(j)n) of such individuals are born, where N(j) indexes absolute population sizes in period 0 and ψ(j) is the share of liquidity-constrained agents. Second, there are liquidity-constrained households who do not have access to financial markets, and who consequently are forced to consume their after tax income in every period. The number of such agents born in each period is N(j)ntψ(j) (1θ(j)n).2 Aggregation over different cohorts of agents implies that the total numbers of agents in country j is N(j)nt. For computational reasons we do not normalize world population to one, especially when our set of countries includes a small open economy j = 1. In that case we assume N (1) = 1, and set N(j) such that N(1)/j=2N˜N(j) equals the share of country 1 agents in the world population. In addition to the probability of death households also experience labor productivity that declines at a constant rate over their lifetimes. This simplified treatment of lifecycle income profiles is justified by the absence of explicit demographics in our model, and adds another powerful channel through which fiscal policies can have non-Ricardian effects. Households of both types are subject to uniform labor income, consumption and lump-sum taxes. We will denote variables pertaining to these two groups of households by OLG and LIQ.

Firms are managed in accordance with the preferences of their owners, myopic OLG households, and they therefore also have finite planning horizons. Each country’s primary production is carried out by manufacturers producing tradable and nontradable goods. Manufacturers buy capital services from capital goods producers (in GIMF without Financial Accelerator) or from entrepreneurs (in GIMF with Financial Accelerator), labor from monopolistically competitive unions, and raw materials from the world raw-materials market. They are subject to nominal rigidities in price setting as well as real rigidities in labor hiring and in the use of raw materials. Capital goods producers are subject to investment adjustment costs. Entrepreneurs finance their capital holdings using a combination of external and internal financing. A capital income tax is levied on capital goods producers (in GIMF without Financial Accelerator) or on entrepreneurs (in GIMF with Financial Accelerator). Unions are subject to nominal wage rigidities and buy labor from households. Manufacturers’ domestic sales go to domestic distributors. Their foreign sales go to import agents that are domestically owned but located in each export destination country. Import agents in turn sell their output to foreign distributors. When the pricing-to-market assumption is made these import agents are subject to nominal rigidities in foreign currency. Distributors first assemble nontradable goods and domestic and foreign tradable goods, where changes in the volume of imported inputs are subject to an adjustment cost. This private sector output is then combined with a publicly provided capital stock (infrastructure) as an essential further input. This capital stock is maintained through government investment expenditure that is financed by tax revenue and the issuance of government debt. The combined final domestic output is then sold to consumption goods producers, investment goods producers, and import agents located abroad. Consumption and investment goods producers in turn combine domestic and foreign output to produce final consumption and investment goods. Foreign output is purchased through a second set of import agents that can price to the domestic market, and again changes in the volume of imported goods are subject to an adjustment cost. This second layer of trade at the level of final output is critical for allowing the model to produce the high trade-to-GDP ratios typically observed in small, highly open economies. Consumption goods output is sold to retailers and the government, while investment goods output is sold domestic capital goods producers and the government. Consumption and investment goods producers are subject to another layer of nominal rigidities in price setting. This cascading of nominal rigidities from upstream to downstream sectors has important consequences for the behavior of aggregate inflation. Retailers, who are also monopolistically competitive, face real instead of nominal rigidities. While their output prices are flexible they find it costly to rapidly adjust their sales volume. This feature contributes to generating inertial consumption dynamics.3

Figure 1:Goods and Factor Flows in GIMF

The world economy experiences a constant positive trend technology growth rate g = Tt/Tt−1, where Tt is the level of labor augmenting world technology, and a constant positive population growth rate n. When the model’s real variables, say xt, are rescaled, we divide by the level of technology Tt and by population, but for the latter we divide by nt only, meaning real figures are not in per capita terms but rather in absolute terms adjusted for technology and population growth. We use the notation xˇt=xt/(Ttnt), with the steady state of xˇt denoted by x¯. An exception to this is quantities of labor, which are only rescaled by nt.

Asset markets are incomplete. There is complete home bias in government debt, which takes the form of nominally non-contingent one-period bonds denominated in domestic currency. The only assets traded internationally are nominally non-contingent one-period bonds denominated in the currency of Ñ. There is also complete home bias in ownership of domestic firms. In addition equity is not traded in domestic financial markets, instead households receive lump-sum dividend payments. This assumption is required to support our assumption that firm and not just household preferences feature myopia.

III. Overlapping Generations Households

We first describe the optimization problem of OLG households. A representative member of this group and of age a derives utility at time t from consumption ca,tOLG, leisure (StLa,tOLG) (where StL is the stochastic time endowment, which has a mean of one but which can itself be a function of the business cycle, see below), and real balances (Ma,t/PtR) (where PtR is the retail price index). The lifetime expected utility of a representative household of age a at time t has the form

where Et is the expectations operator, θ < 1 is the degree of myopia, γ > 0 is the coefficient of relative risk aversion, 0 < ηOLG< 14, um > 0, and βt is the (stochastic) discount factor.

As for money demand, in the following analysis we will only consider the case of the cashless limit advocated by Woodford, where um → 0. This has one major advantage for GIMF. Note first that the combination of separable money in the utility function and monetary policy specified as an interest rate rule implies that the money demand equation becomes redundant and that inflation is not directly distortionary for the consumption-leisure decision. But money also has a fiscal role through the government budget constraint, and any reduction in inflation tax revenue must be accompanied by an offsetting increase in other forms of distortionary taxation.5 Because of this indirect distortionary effect, an increase in inflation in this model would then actually reduce overall distortions, which is not plausible. Adopting the cashless limit assumption avoids this problem, by ensuring that inflation causes no distortions in either direction. GIMF is therefore clearly not designed to quantify the costs of inflation, and should not be used for that purpose.

Consumption ca,tOLG is given by a CES aggregate over retailed consumption goods varieties ca,tOLG(i), with stochastic elasticity of substitution σRt:

This gives rise to a demand for individual varieties

where PtR(i) is the retail price of variety i, and the aggregate retail price level PtR is given by

A household can hold two types of bonds. The first bond type is domestic bonds denominated in domestic currency. Such bonds are issued by the domestic government Ba,t and, in the case of GIMF with a Financial Accelerator, by banks lending to the nontradables and tradables sectors, Ba,tN+Ba,tT. The second bond type is foreign bonds denominated in the currency of country Ñ, Fa,t. The nominal exchange rate vis-a-vis Ñ is denoted by εt, and εtFa,t are nominal net foreign asset holdings in terms of domestic currency. In each case the time subscript t denotes financial claims held from period t to period t + 1. Gross nominal interest rates on domestic and foreign currency denominated assets held from t to t + 1 are it/(1+ξtb) and it(N˜)(1+ξtf). For domestic bonds, it is the nominal interest rate paid by the domestic government and ξtb is a domestic risk premium, with ξtb<0 characterizing a situation where the private sector faces a larger marginal funding rate than the public sector. For foreign bonds, it(Ñ) is the nominal interest rate determined in Ñ, and ξtf is a foreign exchange risk premium. For country Ñ, it = it(Ñ) and ξtf=0.6 Both risk premia are external to the household’s asset accumulation decision, and are payable to a financial intermediary that redistributes the proceeds in a lump-sum fashion either to foreigners or to domestic households. The functional form of the foreign exchange risk premium is given by

where Stfx is a mean zero risk premium shock, y1y4 are parameters, y1 is constrained to generate a zero premium at a zero current account by the condition y1 = –y2/(–y4)y3, and cat/gdpt is the current account-to-GDP ratio. Especially for emerging markets we have found this functional form to be more suitable than conventional linear specifications because it is asymmetric, allowing for a steeply increasing risk premium at large current account deficits. But a linear option is also available in GIMF as

The functional form of the domestic risk premium can similarly be made to depend on the government-debt-to-GDP ratio when it is intended to highlight the effect of government borrowing levels on domestic interest rates. But it can also be treated as an exogenous stochastic process when the emphasis is on shocks to the interest rate margin between the policy rate and the rate at which the private sector can access the domestic capital market. For example, recent financial markets events may be partly characterized by a persistent negative shock to ξtb.

Participation by households in financial markets requires that they enter into an insurance contract with companies that pay a premium of (1θ)θ on a household’s financial wealth for each period in which that household is alive, and that encash the household’s entire financial wealth in the event of his death.7

Apart from returns on financial assets, households also receive labor and dividend income. Households sell their labor to “unions” that are competitive in their input market and monopolistically competitive in their output market, vis-à-vis manufacturing firms.

The productivity of a household’s labor declines throughout his lifetime, with productivity Φa,t = Φa of age group a given by

where χ < 1.8 The overall population’s average productivity is assumed without loss of generality to be equal to one. Household pre-tax nominal labor income is therefore WtΦa,ta,tOLG. Dividends are received in a lump-sum fashion from all firms in the nontradables (N) and tradables (T) manufacturing sectors, from the distribution (D), consumption goods producer (C) and investment goods producer (I) sectors, from the retail (R) sector and the import agent (M) sector, from all unions (U) in the labor market, from domestic (X) and foreign (F) raw-materials producers, from capital goods producers (K), and from entrepreneurs (EP) (only in GIMF with Financial Accelerator), with after-tax nominal dividends received from firm/union i denoted by Da,tj(i), j = N, T, D, C, I, R, U, M, X, F, K, EP. Furthermore, in GIMF with Financial Accelerator OLG households receive remuneration for their services in the bankruptcy monitoring of firms, which equal rbra,t=ptNrbra,tN+ptTHrbra,tT. OLG households are liable to pay lump-sum transfers τTa,tOLG representing a small share of their dividend income to the government, which in turn redistributes them to the relatively less well off LIQ agents. Household labor income is taxed at the rate τL,t, and consumption is taxed at the rate τc,t. In addition there are lump-sum taxes τa,tls,OLG and transfers ϒa,tOLG paid to/from the government.9 It is assumed that retailers face costs of rapidly adjusting their sales volume. To limit these costs they therefore offer incentives (or disincentives) that are incorporated into the effective retail purchase price PtR. The consumption tax τc,t is however assumed to be payable on the pre-incentive price PtC.10PtC is the marginal cost of retailers, who combine the output of consumption goods producers, with price level Pt, with raw materials used directly by consumers, with price level PtX. We choose Pt as our numeraire. We denote the real wage by wt = Wt/Pt, the relative price of any good x by ptx=ptx/Pt, gross inflation for any good x by πtx=Ptx/Pt1x, and gross nominal exchange rate depreciation by εt = εt/εt–1.11

The household’s budget constraint in nominal terms is

The OLG household maximizes (1) subject to (2), (7) and (8). The derivation of the first-order conditions for each generation, and aggregation across generations, is discussed in detail in Appendices 1-3.

Aggregation takes account of the size of each age cohort at the time of birth, and of the remaining size of each generation. Using the example of overlapping generations households’ consumption, we have

This also has implications for the intercept parameter k of the age-specific productivity distribution. Under the assumption of an average productivity of one, and for given parameters χ and θ, we obtain κ = (nθχ)/(n – θ).

Several of the optimality conditions that need to be aggregated are, or are derived from, nonlinear Euler equations. In such conditions, aggregation requires nonlinear transformations that are only valid under certainty equivalence. Tractable aggregate consumption optimality conditions therefore only exist for the cases of perfect foresight and of first-order approximations. For our purposes this is not problematic as all stochastic applications of GIMF will use linear approximations. However, for the purpose of exposition we find it preferable to present optimality conditions in nonlinear form. The expectations operator Et is therefore everywhere to be understood in this fashion.

The first-order conditions for the goods varieties and for the consumption/leisure choice are given by

The arbitrage condition for foreign currency bonds (the uncovered interest parity relation) is given by

The consumption Euler equation on the other hand cannot be directly aggregated across generations. For each generation we have

Here we have used the notation

where rt is the real interest rate in terms of final output payable by the government, while řt is the real interest rate payable by the private sector. We introduce some additional notation. The production based real exchange rate vis-a-vis.Ñ is et =(εtPt(Ñ))/Pt, where Pt(Ñ) is the price of final output in Ñ. We adopt the convention that each nominal asset is deflated by the final output price index of the currency of its denomination, so that real domestic bonds are bt = Bt/Pt and real foreign bonds are ft = Ft/Pt(Ñ).

The subjective and market nominal discount factors are given by

and the subjective and market real discount factors by

In each case the subjective discount factor incorporates an agent’s probability of death, which ceteris paribus makes him value near-term receipts more highly than receipts in the distant future.

We now discuss a key condition of GIMF, the optimal aggregate consumption rule of OLG households. The derivation of this condition is algebraically complex and is therefore presented in Appendix 3. The final result expresses current aggregate consumption of OLG households as a function of their real aggregate financial wealth fwt and human wealth hwt, with the marginal propensity to consume of out of wealth given by 1/Өt. Human wealth is in turn composed of hwtL, the expected present discounted value of households’ time endowments evaluated at the after-tax real wage, and hwtK, the expected present discounted value of capital or dividend income net of lump-sum transfer payments to the government. After rescaling by technology we have

where

The intuition of (20) is key to GIMF. Financial wealth (21) is equal to the domestic government’s and foreign households’ current financial liabilities. For the government debt portion, the government services these liabilities through different forms of taxation, and these future taxes are reflected in the different components of human wealth (22) as well as in the marginal propensity to consume (25). But unlike the government, which is infinitely lived, an individual household factors in that he might not be alive by the time higher future tax payments fall due. Hence a household discounts future tax liabilities by a rate of at least řt/θ, which is higher than the market rate řt, as reflected in the discount factors in (23), (24) and (25). The discount rate for the labor income component of human wealth is even higher at řt/θχ, due to the decline of labor incomes over individuals’ lifetimes.

A fiscal consolidation through higher taxes represents a tilting of the tax payment profile from the more distant future to the near future, so as to effect a reduction in the debt stock. The government has to respect its intertemporal budget constraint in effecting this tilting, and this means that the expected present discounted value of its future primary surpluses has to remain equal to the current debt it–1bt–1/πtwhen future surpluses are discounted at the market interest rate rt. But when individual households discount future taxes at a higher rate than the government, the same tilting of the tax profile represents a decrease in human wealth because it increases the expected value of future taxes for which the household expects to be responsible. This is true for the direct effects of lump-sum taxes and of labor-income taxes on labor-income receipts, and for the indirect effect of corporate taxes on dividend receipts. For a given marginal propensity to consume, these reductions in human wealth lead to a reduction in consumption. Note that with ξtb<0 this effect is not only due to myopia but also to the borrowing spread between the public and private sectors.

The marginal propensity to consume 1/Өt is, in the simplest case of logarithmic utility and exogenous labor supply, equal to (1 – βθ). For the case of endogenous labor supply, household wealth can be used to either enjoy leisure or to generate purchasing power to buy goods. The main determinant of the split between consumption and leisure is the consumption share parameter ηOLG, which explains its presence in the marginal propensity to consume (25). While other forms of taxation affect the different components of wealth, the time profile of consumption taxes affects the marginal propensity to consume, reducing it with a balanced-budget shift of such taxes from the future to the present. The intertemporal elasticity of substitution 1/γ is another key parameter for the marginal propensity to consume. For the conventional assumption of γ > 1 the income effect of an increase in the real interest rate r is stronger than the substitution effect and tends to increase the marginal propensity to consume, thereby partly offsetting the contractionary effects of a higher r on human wealth ȟwt. A larger γ therefore tends to give rise to larger interest rate changes in response to fiscal shocks.

Modularity: OLG households are a critical part of the core structure of GIMF, as they are partly responsible for the short-run effects of fiscal policies, and wholly responsible for the long-run effects. This sector can therefore not be removed, and is present in “Fiscal Stimulus to the Rescue?”.

IV. Liquidity Constrained Households

The objective function of liquidity-constrained (LIQ) households is assumed to be nearly identical to that of OLG households:12

These agents can consume at most their current income, which consists of their after tax wage income plus government transfers τTa,tLIQ. Their budget constraint is

The aggregated first-order conditions for this problem, after rescaling by technology, are

GIMF also allows for an alternative version where equation (31) is dropped and is replaced with an exogenous labor supply, the so-called “rule-of-thumb consumer”.

Modularity: The share of LIQ agents in the population is not strictly a part of the core of GIMF. But it is a critical determinant of the short-run effects of fiscal policies, especially revenue-based policies, as this sector exhibits a marginal propensity to consume out of current income of one. RESEM’s recent model comparison exercise for fiscal stimulus measures showed that there was virtual agreement on this question among modelers of several central banks and policy institutions. For all applications analyzing short-run fiscal measures this block should therefore remain part of the model, and it is present in “Fiscal Stimulus to the Rescue?”.

V. Aggregate Household Sector

To obtain aggregate consumption demand and labor supply we simply add the respective optimality quantities of the different consumers in the economy, OLG and LIQ households:

VI. Manufacturers

There is a continuum of manufacturing firms indexed by i ∈ [0,1] in two separate manufacturing sectors indexed by J ∈ {N, T}, where N represents nontradables and T tradables. For prices in these two sectors we introduce a slightly different index J˜{N,TH}, because the index T for prices is reserved for a different goods aggregate produced by distributors (see below). Manufacturers buy labor inputs from unions and capital inputs from capital goods producers (in GIMF without Financial Accelerator) or from entrepreneurs (in GIMF with Financial Accelerator). Sector N and T manufacturers sell to domestic distributors, and sector T manufacturers also sell to import agents in foreign countries, who in turn sell to distributors in those countries.13 Manufacturers are perfectly competitive in their input markets and monopolistically competitive in the market for their output. Their price setting is subject to nominal rigidities.

We first analyze the demands for their output, then turn to their technology, and finally describe their optimization problem.

Demands for manufacturers’ output varieties are given by

Where YtJ(z,i) and YtJ(z) are variety i and total demands from domestic distributor z in sector J, and YtTX(1,j,z,i) and YtTX(1,j,z) are variety i and total demands for exports from country 1 to import agent z in country j. Cost minimization by distributors and import agents generates demands for varieties

with price indices defined as

The aggregate demand for variety i produced by sector J can be derived by simply integrating over all distributors, import agents and all other sources of manufacturing output demand. We obtain

where ZtJ(i) and ZtJ remain to be specified by way of market-clearing conditions for manufacturing goods.

The technology of each manufacturing firm differs depending on whether the raw-materials sector is included. If it is included, the technology is given by a CES production function in utilized capital KtJ(i), union labor UtJ(i) and raw materials XtJ(i), with elasticities of substitution ξZJ between capital and labor, and ξXJ between raw materials and capital/labor. An adjustment cost GX,tJ(i) makes fast changes in raw-materials inputs costly. Labor augmenting productivity is TtAtJ, where AtJ is a country specific technology shock:1415

If the raw-materials sector is not included, the technology is given by a CES production function in capital KtJ(i) and union labor UtJ(i), with elasticity of substitution ξZJ between capital and labor:

We will from now on mostly ignore the version without raw-materials sector, for which the optimality conditions can be derived in the same fashion as below.

Manufacturing firms are subject to three types of adjustment costs. First, quadratic inflation adjustment costs GP,tJ(i) are real resource costs that represent a demand for the output of sector J. They are quadratic in changes in the rate of inflation rather than in price levels, which is important in order to generate realistic inflation dynamics. Compared to versions of the Calvo price setting assumption such adjustment costs have the advantage of greater analytical tractability. We have:

To allow a flexible choice of inflation adjustment costs we also allow for a version of Rotemberg sticky prices, whereby deviations of the actual inflation rate from the inflation target π¯t are costly. These may sometimes be preferable when working with a fixed exchange rates model, where sticky inflation can give rise to excessively large cycles. These costs are given by16

Second, adjustment costs on raw-materials inputs enter the production function rather than the budget constraint, and are given by17

where the term gn enters to ensure that adjustment costs are zero along the balanced growth path.

Third, adjustment costs on labor hiring are again resource costs that enter the budget constraint. They are given by

These costs are somewhat less common in the business cycle literature, and are only included as an option that can be switched off by setting ϕU = 0.

It is assumed that each firm pays out each period’s after tax nominal net cash flow as dividends DtJ(i). It maximizes the expected present discounted value of dividends. The discount rate it applies in this maximization includes the parameter θ so as to equate the discount factor of firms θ/řt with the pricing kernel for nonfinancial income streams of their owners, myopic households, which equals βθEt (λa+1,t+1/λa,t). This equality follows directly from OLG households’ first-order condition for government debt holdings λa,t=βEt (λa+1,t+1it/(πt+1(1+ξtb))).

Pre-tax net cash flow equals nominal revenue PtJ¯(i)ZtJ(i) minus nominal cash outflows. The latter include the wage bill VtUtJ(i), where Vt is the aggregate wage rate charged by unions, spending on raw materials PtXXtJ(i), where PtX is the price of raw materials, and the cost of capital services Rk,tJKtJ(i), where RtKJ is the nominal rental cost of capital in sector J, with the real cost denoted rk,tJ. Other components of pre-tax cash flow are price adjustment costs PtJ¯GP,tJ(i) that represent a demand for sectorial manufacturing output ZtJ, labor adjustment costs VtGu,t(i) that represent a demand for labor Lt, and a fixed cost PtJ¯TtωJ. The fixed resource cost arises as long as the firm chooses to produce positive output. Net output in sector J is therefore equal to max(0,ZtJ(i)TtωJ). The fixed cost is calibrated to make the steady-state shares of economic profits, labor and capital in GDP consistent with the data. This becomes necessary because the model counterpart of the aggregate income share of capital equals not only the return to capital but also the profits of monopolistically competitive firms. With several layers of such firms the profits share becomes significant, and the capital share parameter in the production function has to be reduced accordingly, unless fixed costs are assumed. More importantly, the introduction of an additional parameter determining fixed costs allows us to simultaneously calibrate not only capital income shares and depreciation rates but also the investment-to-GDP ratio. This would otherwise be impossible. We calibrate fixed costs by first noting that, in normalized form, steady-state monopoly profits equal Z¯J/σ¯J. We denote by the sπ share of these profits that remain after fixed costs have been paid, and we will calibrate this parameter to obtain the desired investment-to-GDP ratio. We assume that sπ is identical across the industries where fixed costs arise. Then fixed costs in manufacturing are given by

The total after tax net cash flow or dividend of the firm is

The optimization problem of each manufacturing firm is

subject to the definition of dividends (45), demands (37), production functions (38), and adjustment costs (40)-(43). The first-order conditions for this problem are derived in some detail in Appendix 4. A key step is to recognize that all firms behave identically in equilibrium, so that PtJ¯(i)=PtJ¯ and ZtJ(i)=ZtJ. Let λtJ denote the real marginal cost of producing an additional unit of manufacturing output. Also, rescale the optimality conditions by technology and population as discussed above, and denote stochastic markups by μJt=σJt/(σJt1). Then the condition for PtJ¯(i) under sticky inflation is

while under sticky prices we have

The first-order condition for labor demand UtJ(i) is

where FˇU,tJ is the marginal product of labor

The first-order condition for raw-materials demand XtJ(i) is

where FˇX,tJ is the marginal product of raw materials

The first-order condition for capital demand is

where FˇK,tJ is the marginal product of capital

For the sake of completeness we add here the marginal products of labor and capital for the version of GIMF without raw materials. They are

Rescaled aggregate dividends of firms in each sector are

We define aggregate capital and investment as

Finally, we turn to the market-clearing conditions for nontradables and tradables. They equate the output of each sector to the demands of distributors, of manufacturers themselves for fixed and adjustment costs, and in the case of tradables to the demands of foreign import agents. We have18

where rcˇutJ is the resource cost associated with variable capital utilization and SˇtJ,nwyshk is the net effect of entrepreneurs’ output destroying net worth shocks (in GIMF with Financial Accelerator). The term p˜texp in the second market-clearing condition refers to unit-root shocks to the relative price of exported goods. Specifically, tradables output is converted to exports YˇtTX using a technology that multiplies tradables output by Ttexp=1/p˜texp, where p˜texp is a unit-root shock with zero trend growth.

Modularity: The tradables manufacturing sector is part of the core of GIMF and cannot be removed. The nontradables sector can be removed. For many applications it can however have critically important effects on the real exchange rate that should not be overlooked. For example, for applications that aim at a realistic description of worldwide feedback effects of fiscal policies, including their effects on trade, it is advisable to keep the nontradables sector. Both tradables and nontradables sectors are therefore present in “Fiscal Stimulus to the Rescue?”.

VII. Capital Goods Producers

A. GIMF with Financial Accelerator

These agents produce the capital stock used by entrepreneurs in the nontradables and tradables sectors, indexed as before by J ∈ {N, T}. They are competitive price takers. Capital goods producers are owned by households, who receive their dividends as lump-sum transfers. They purchase previously installed capital K˜tJ from entrepreneurs and investment goods ItJ from investment goods producers to produce new installed capital K˜t+1J according to

where Stinυ is an investment demand shock. They are subject to investment adjustment costs

The nominal price level of previously installed capital is denoted by QtJ. Since the marginal rate of transformation from previously installed to newly installed capital is one, the price of new capital is also QtJ. The optimization problem is to maximize the present discounted value of dividends by choosing the level of new investment ItJ:19

The solution to this problem is

The stock of physical capital evolves as

We allow for shocks to the deprecation rate of capital, which in the context of the Financial Accelerator we will refer to as capital destroying net worth shocks:

Physical capital K¯tJ is different from the capital rented by manufacturers KtJ because the stock of physical capital is subject to variable capital utilization utJ. The normalized relationship between physical capital K¯tJ accumulated by the end of period t – 1 and capital KtJ used in manufacturing in period t is therefore given by

The real value of dividends is given by

We let dˇtK=dˇtKN+dˇtKT, and also Iˇt=IˇtN+IˇtT, K¯t=K¯tN+K¯tT

Modularity: This sector is part of the core of GIMF, as it determines the dynamics of investment. It is therefore also present in “Fiscal Stimulus to the Rescue?”.

B. GIMF without Financial Accelerator

Capital goods producers produce the physical capital stock K¯t+1J. They rent out capital K¯tJ inherited from period t – 1 to manufacturers in the nontradables and tradables sectors J ∈ {N,T}, after deciding on the rate of capital utilization utJ. They are competitive price takers and are subject to a capital income tax. Capital goods producers are owned by households, who receive their dividends as lump-sum transfers. The accumulation of the physical capital stock is given by

As before, we allow for shocks to the deprecation rate of capital

Investment goods ItJ are purchased from investment goods producers, and Stinυ is an investment demand shock. Investment is subject to investment adjustment costs

After observing the time t aggregate shocks the capital goods producer decides on the time t level of capital utilization utJ, and then rents out capital services KtJ(j)=utJK¯tJ(j). High capital utilization gives rise to high costs in terms of sector J goods, according to the convex function a(utJ)K¯tJ(j), where we specify the adjustment cost function as20

The optimization problem is to maximize the present discounted value of dividends by choosing the level of new investment ItJ the level of the physical capital stock K¯t+1J, and the rate of capital utilization utJ:

The first-order conditions for investment demand ItJ(i) and capital K¯t+1J(i) are

The first-order condition for capital utilization is

and the resource cost associated with variable capital utilization is

The real value of dividends is given by

We let dˇtK=dˇtKN+dˇtKT, and also Iˇt=IˇtN+IˇtT, Kˇ¯t=Kˇ¯tN+Kˇ¯tT.

Modularity: This sector is part of the core of GIMF, as it determines the dynamics of investment. It is therefore also present in “Fiscal Stimulus to the Rescue?”.

VIII. Entrepreneurs and Banks

This sector is based on the models of Bernanke and others (1999) and Christiano and others (2007). Entrepreneurs in sectors J ∈ {N, T} purchase a capital stock from capital goods producers and rent it to manufacturers. Each entrepreneur j finances his end of time t capital holdings (at current market prices) QtJK¯t+1J(j) with a combination of his end of time t net worth NtJ(j) and bank loans BtJ(j). His balance sheet constraint is therefore given by

or in real normalized terms by

After the capital purchase each entrepreneur draws an idiosyncratic shock which changes K¯t+1J(j) to ωt+1JK¯t+1J(j) at the beginning of period t + 1, where ωt+1J is a unit mean lognormal random variable distributed independently over time and across entrepreneurs. The standard deviation of ln(ωt+1J), σt+1J, is itself a stochastic process. While the realization of ωt+1J is not known at the time the entrepreneur makes his capital decision, the value of σt+1J is known. The cumulative distribution function of ωt+1J is given by Pr(ωt+1Jx)=Ft+1J(x).

After observing the time taggregate shocks the entrepreneur decides on the time t level of capital utilization utJ, and then rents out capital services KtJ(j)=utJK¯tJ(j) to entrepreneurs. High capital utilization gives rise to high costs in terms of sector J goods, according to the convex function a(utJ)ωtJK¯tJ(j), where we specify the adjustment cost function as

The entrepreneur chooses utJ to solve

which has the solution

The resource cost associated with variable capital utilization is given by

The entrepreneur’s real after tax return to utilized capital is given by

The nominal return to utilized capital is equal to

We assume that the entrepreneur receives a standard debt contract from the bank. This specifies a loan amount BtJ and a gross rate of interest iB,t+1J to be paid if ωt+1J is high enough. Entrepreneurs who draw ωt+1J below a cutoff level ω¯t+1J cannot pay this interest rate and go bankrupt. They must hand over everything they have to the bank, but the bank can only recover a time-varying fraction (1μt+1J) of the value of such firms. The cutoff ω¯t+1J is given by the condition

The bank finances its loans to entrepreneurs by borrowing from households. We assume that the bank pays households a nominal rate of return iˇt=it/(1+ξtb) that is not contingent on the realization of time t +1 shocks. The parameters of the entrepreneur’s debt contract are chosen to maximize entrepreneurial profits, subject to zero bank profits in each state of nature and to the requirement that iˇt be non-contingent on time t + 1 shocks. This implies that iB,t+1J and ω¯t+1J are both functions of time t + 1 aggregate shocks, in other words the optimal contract specifies state-contingent schedules of interest rates and bankruptcy cutoffs.

The bank’s zero profit or participation constraint is given by:21

This states that the stochastic payoff to lending on the l.h.s. must equal the non-stochastic payment to depositors on the r.h.s. in each state of nature. The first term on the l.h.s. is the nominal interest income on loans for borrowers whose idiosyncratic shock exceeds the cutoff level, ωt+1Jω¯t+1J. The second term is the amount collected by the bank in case of the borrower’s bankruptcy, where ωt+1J<ω¯t+1J. This cash flow is based on the return Retk,tJω on capital investment QtJK¯t+1J(j), but multiplied by the factor (1μt+1J) to reflect a proportional bankruptcy cost μt+1J. Next we rewrite (91) by using (90) and (82):

We adopt a number of definitions that simplify the following derivations. First, note that capital earnings are given by Retk,tJQtJK¯t+1J(j). The lender’s gross share in capital earnings is defined as

while his monitoring costs share in capital earnings is given by μt+1JG(ω¯t+1J), where

The lender’s net share in capital earnings is therefore Γ(ω¯t+1J)μt+1JG(ω¯t+1J). The entrepreneur’s share in capital earnings on the other hand is given by

Using this notation and denoting the multiplier of the participation constraint by λt, the entrepreneur’s optimization problem can be written as

Note the expectations operator: The entrepreneur is risk-neutral and absorbs all aggregate risk, so that his realized profits depend on time t + 1 shocks, while the bank is guaranteed zero profits in each state of nature. Before deriving the optimality conditions we rewrite this expression by dividing through by iˇtNtJ(j), rewriting the resulting expression in terms of normalized variables, and finally replacing nominal returns by real returns:

We let Γt+1J=Γ(ω¯t+1J), Gt+1J=G(ω¯t+1J), ΓJ,t+1=Γt+1J/ω¯t+1J and GJ,t+1=Gt+1J/ω¯t+1J. We obtain the following first-order condition with respect to ω¯t+1J:

which implies

The condition for the optimal loan contract, that is the first-order condition with respect to Kˇ¯t+1J(j), can be written using (99) as

The normalized lender’s zero profit condition is

where we have replaced time t + 1 and t subscripts with time t and t – 1 subscripts everywhere because this condition has to hold for each state of nature, that is it has to hold exactly ex-post. Also, for correct timing we need to define ex-post realized returns for this expression as

rather than using reˇtk,t1J and řt–1. Notice that we have omitted entrepreneur specific indices j for capital and net worth and replaced them with the corresponding aggregate variables. This is because each entrepreneur faces the same returns reˇtk,tJ and řt, and the same risk environment characterizing the functions Γ and G. Aggregation of the model over entrepreneurs is then trivial because both borrowing and capital purchases are proportional to the entrepreneur’s level of net worth.

A key problem for coding the Financial Accelerator version of GIMF in a standard software such as TROLL and DYNARE consists of finding a closed-form representation for the terms ΓtJ, GtJ and their derivatives. In TROLL we can use the hard-wired (like e.g. LOG) PNORM function, which is the c.d.f. of the standard normal distribution.22 In Appendix 5 we therefore derive the relevant expressions in terms of PNORM, for which we use the notation Φ(.). We obtain the following set of equations, starting with an auxiliary variable z¯tJ:

As for the evolution of entrepreneurial net worth, we first note that banks make zero profits at all times. The difference between the aggregate returns to capital net of bankruptcy costs and the sum of deposit interest paid by banks to households therefore goes entirely to entrepreneurs and accumulates. To rule out a situation where over time so much net worth accumulates that entrepreneurs no longer need any loans, we assume that they regularly pay out to households dividends which, in terms of sector J output, are given by diυtJ. Net worth is also subject to output-destroying shocks StJ,nwyshk. We assume that for an individual entrepreneur both dividends and output destroying shocks are proportional to his net worth, which given our above result concerning the proportionality of borrowing and capital purchases to net worth implies that the evolution of aggregate net worth is a straightforward aggregation of the evolution of entrepreneur specific net worth. Nominal aggregate net worth therefore evolves as

This can be combined with the aggregate version of the balance sheet constraint (82) and normalized to yield

Dividends in turn are given by the following expressions:

Regular dividends, given by expression (112), are a fraction StJ,nwd (with S¯J,nwd typically in a range between 0 and 0.05) of smoothed (moving average) gross returns on net worth invested in the previous period, as per equation (109). The dividend related net worth shock StJ,nwd can cause temporary losses or gains of net worth that are a pure redistribution between households and entrepreneurs, without direct resource implications. The second determinant of dividends in (111) consists of a dividend response to deviations of net worth from its long-run value, the latter proxied by a moving average of past and future values of net worth. This allows us to model dividend policy as a tool to rebuild net worth more quickly following a negative shock. The parameter θnwJ (typically in a range between 0 and 0.05) measures the increase/decrease in dividends if net worth rises/falls below its long-run value. The relative price ptJ enters because dividends are in units of sector J output while net worth is in units of final output.

To parameterize moving averages we use a general formula, as in (113) and (114), that minimizes the number of leads or lags needed. This is critical for computational economy in GIMF. The same type of formula will be used throughout for all moving average terms, with one exception. This is that while for dividend income and net worth terms we have found it useful to employ a moving average of future terms, all other moving average terms contain only lags, in other words the t +1 in the formula becomes t – 1. The terms kincJ and knw index the degree to which the moving average moves with actual values of income and net worth, with high values (typically around 10) representing a very slow-moving average and low values allowing for a quicker adjustment. For backward-looking moving averages, we have found that a value for this coefficient of around 3 generates reasonable dynamics for quantities like potential output.

Output-destroying and capital-destroying net worth shocks are easier to calibrate if they are expressed as fractions of steady-state net worth.23 We therefore adopt the definitions

and express the shock processes as autocorrelated shocks to ŠtJ,nwy and ŠtJ,nwk.

We define the real sector J bankruptcy monitoring cost as

This is not a physical resource cost but a remuneration for monitoring work performed. We therefore assume that it is received by OLG households in the same lumps-sum fashion as dividends.

Modularity: The Financial Accelerator is a part of the core of GIMF, and is present in “Fiscal Stimulus to the Rescue?”.

IX. Raw-Materials Producers

A. Raw-Materials Output and Storage

The GIMF raw-materials sector has been constructed primarily with oil in mind. The modeling team’s priors are that this sector is characterized by extremely low demand and supply elasticities. This is the main reason, apart from analytical tractability, why the output of raw materials has been specified as having a zero price elasticity. But there is one drawback to this approach - without some escape valve on the demand or supply side the simulation of shocks to this sector can present serious numerical problems. We have therefore added such an escape valve, and one which is in addition quite plausible. This is that firms in the raw-materials sector can choose how much of their exogenous endowment they sell in any given period, by adding to or drawing down from a storage facility.

Specifically, in each period each country receives an endowment flow of raw materials Xtexog that is, in the absence of exogenous shocks, constant in normalized terms (i.e. it grows at the rate g). Its raw-materials producers decide on the size of a stored stock of raw materials given by Ot. Furthermore, storing some raw materials has both benefits and costs in terms of the amount that becomes available for sales. For simplicity, and because the dynamics of raw-materials storage are not central to the intended uses of GIMF, these benefits and costs are specified such that the steady-state stored stock equals zero, specifically as

The optimization problem of a raw-materials producer is therefore given by

where PtX is the nominal market price of the raw material. The first-order condition of this problem is

Finally, the actual sales of raw materials Xtsup are given, in normalized form, by

B. Raw-Materials Sales

The available supply of raw materials Xˇtsup is sold to manufacturers worldwide, with total demand for each country given by Xˇtdem. The value of a country’s normalized raw-materials exports is therefore given by

The world market for raw materials is perfectly competitive, with flexible prices that are arbitraged worldwide. A constant share sdx of steady-state (after normalization) raw-materials revenue is paid out to domestic factors of production as dividends d¯X. The rest is divided in fixed shares (1sfx) and sfx=Σj=2N˜sfx(1,j) between payments to the government gˇtX, for the case of publicly-owned producers, and dividends to foreign owners in all other countries fˇtX. This means that all benefits of favorable raw-materials price shocks accrue exclusively to the government and foreigners, and vice versa for unfavorable shocks. This corresponds more closely to the situation of many countries’ raw-materials sectors than the polar opposite assumption of assuming equal shares between the three recipients at all times. But it is straightforward to modify the code to allow for all three factors receiving variable revenue shares. We have where by international arbitrage we have

where by international arbitrage we have

The dividends received by country 1 households from ownership of country j raw-materials producers are then given by

and aggregate dividends are

The raw-materials sector is subject to shocks to domestic supply Xˇtexog and to foreign demand, the latter via the raw-materials share parameter in the manufacturing (αJtX) and retail (αCtX) sectors. Total demand for each country is given by

where XˇtC is demand from the retail sector, that is from direct household consumption. The market-clearing condition for the raw-materials sector is worldwide, and given by

Modularity: This sector is not part of the core of GIMF. It is typically omitted in applications that do not focus on the role of raw materials. It is not present in “Fiscal Stimulus to the Rescue?”.

X. Unions

There is a continuum of unions indexed by i ∈ [0,1]. Unions buy labor from households and sell labor to manufacturers. They are perfectly competitive in their input market and monopolistically competitive in their output market. Their wage setting is subject to nominal rigidities. We first analyze the demands for union output and then describe their optimization problem.

Demand for unions’ labor output varieties comes from manufacturing firms z ∈ [0,1] in sectors J ∈ {N, T}. The demand for union labor by firm z in sector J is given by a CES production function with time-varying elasticity of substitution σUt,

where UtJ(z,i) is the demand by firm z for the labor variety supplied by union i. Given imperfect substitutability between the labor supplied by different unions, they have market power vis-à-vis manufacturing firms. Their demand functions are given by

where Vt(i) is the wage charged to employers by union i and Vt is the aggregate wage paid by employers, given by

The demand (133) can be aggregated over firms z and sectors J to obtain

where Ut is aggregate labor demand by all manufacturing firms.

GIMF allows for three types of wage rigidities. The first two are the conventional cases of nominal wage rigidities. Sticky wage inflation takes the form familiar from (40),

and sticky wages follow (41). The level of world technology enters as a scaling factor in (136), as otherwise these costs would become insignificant over time. The third type of wage rigidities is real wage rigidities, whereby unions resist rapid changes in the real wage Vt/Ptc. We define πtrw(i)=πtυ(i)/(gπtC). Then these adjustment costs are given by

The stochastic wage markup of union wages over household wages is given by μtU=σUt/(σUt1).

The optimization problem of a union consists of maximizing the expected present discounted value of nominal wages paid by firms Vt(i)Ut(i) minus nominal wages paid out to workers WtUt(i), minus nominal wage inflation adjustment costs PtGP,tU(i). Unlike manufacturers, this sector does not face fixed costs of operation. It is assumed that each union pays out each period’s nominal net cash flow as dividends DtU(i). The objective function of unions is

subject to labor demands (135) and adjustment costs (136) or (137). We obtain the first-order condition for this problem. As all unions face an identical problem, their solutions are identical and the index i can be dropped in all first-order conditions of the problem, with Vt(i) = Vt and Ut(i) = Ut. We let πtV=Vt/Vt1, the gross rate of wage inflation, and we rescale by technology. For sticky wage inflation we obtain the condition

For real wage rigidities we have

Real “dividends” from union organization, denominated in terms of final output, are distributed lump-sum to households in proportion to their share in aggregate labor supply. After rescaling they take the form

We also have υˇt/υˇt1=(Vt/PtTt)/(Vt1/Pt1Tt1), so that

Finally, the labor-market clearing condition equates the combined labor supply of OLG and LIQ households to the labor demands coming from nontradables and tradables manufacturers, including their respective labor adjustment costs if applicable, and from unions for wage adjustment costs. We have:

Modularity: This sector is not part of the core of GIMF. But it is required in order to have sticky wages in the model. Sticky wages and therefore wage adjustment costs at the household level are not feasible in GIMF due to aggregation problems associated with the OLG structure. In most applications this sector is not removed because the assumption of flexible wages is not realistic or empirically successful. This sector is present in “Fiscal Stimulus to the Rescue?”.

XI. Import Agents

Each country, in each of its export destination markets, owns two continua of import agents, one for manufactured intermediate tradable goods (T) and another for final goods (D), each indexed by i ∈ [0, 1] and by J ∈ {T, D}. Import agents buy intermediate goods (or final goods) from manufacturers (or distributors) in their owners’ country and sell these goods to distributors (intermediate goods) or consumption/investment goods producers (final goods) in the destination country. They are perfectly competitive in their input market and monopolistically competitive in their output market. Their price setting is subject to nominal rigidities. We first analyze the demands for their output and then describe their optimization problem.

Demand for the output varieties supplied by import agents comes from distributors (sector T) or consumption/investment goods producers (sectors D), in each case indexed by z ∈ [0, 1]. Recall that the domestic economy is indexed by 1 and foreign economies by j = 2,…,Ñ. Domestic distributors z require a separate CES imports aggregate YtJM(1,j,z) from the import agents of each country j. That aggregate consists of varieties supplied by different import agents i, YtJM(1,j,z,i), with respective prices PtJM(1,j,i), and is given by

This gives rise to demands for varieties of

and these demands can be aggregated over z to yield

Nominal rigidities in this sector take the form familiar from (40),

and the costs represent a demand for the underlying exports. Import agents’ cost minimizing solution for inputs of manufactured intermediate tradable goods (or final goods) varieties therefore follows equations (34) - (36) above (or similar conditions for demands of consumption/investment goods producers). We denote the price of inputs imported from country j at the border of country 1 by PtJM,cif(1,j), the cif (cost, insurance, freight) import price. By purchasing power parity this satisfies PtJM,cif(1,j)=p˜texpPtJH(j)t(1)/t(j), where p˜texp is an exogenous price shock that equals the inverse of a shock to the technology that converts foreign exports into domestic imports. In real terms we have

The optimization problem of import agents consists of maximizing the expected present discounted value of nominal revenue PtJM(1,j,i)YtJM(1,j,i) minus nominal costs of inputs PtJM,cif(1,j)YtJM(1,j,i), minus nominal inflation adjustment costs PtGP,tJM(1,j,i). The latter represent a demand for final output. This sector does not face fixed costs of operation. It is assumed that each import agent pays out each period’s nominal net cash flow as dividends DtJM(1,j,i). The objective function of import agents is

subject to demands (147) and adjustment costs (148). The first-order condition for this problem, after dropping firm specific subscripts, rescaling by technology, and letting µJM = σJM/(σJM — 1), has the form:

The rescaled real dividends of country j’s import agent in the domestic economy, which are paid out to OLG households in country j, are

The total dividends received by OLG households in country 1, expressed in terms of country 1 output, are

Finally, the market-clearing conditions for import agents equate the export volume received from abroad to the import volume used domestically plus adjustment costs:

Modularity: This sector is not part of the core of GIMF. It can be dropped when local currency pricing (pricing-to-market) is not an important concern of the application. It is therefore frequently dropped, including in “Fiscal Stimulus to the Rescue?”.

XII. Distributors

Distributors produce domestic final output. They buy domestic tradables and nontradables from domestic manufacturers, and foreign tradables from import agents. They also use the stock of public infrastructure free of a user charge. Distributors sell their final output composite to consumption goods producers, investment goods producers and final goods import agents in foreign countries. They are perfectly competitive in both their output and input markets.

We divide our description of the technology of distributors into a number of stages. In the first stage a foreign input composite is produced from intermediate manufactured inputs originating in all foreign economies and sold to distributors by import agents. In the second stage a tradables composite is produced by combining these foreign tradables with domestic tradables, subject to an adjustment cost that makes rapid changes in the share of foreign tradables costly. In the third stage a tradables-nontradables composite is produced. In the fourth stage the tradables-nontradables composite is combined with a publicly provided stock of infrastructure.

Foreign input composites YtJF(1), J ∈ {T, D}, are produced by combining imports YtJM(1,j) originating in different foreign economies j and purchased through import agents. A foreign input choice problem therefore only arises when there are more than 2 countries. Also, distributors use only the composite indexed by T, while the composite indexed by D is used by consumption and investment goods manufacturers. We present the problem here in its general form and then reapply the results when describing these other agents. The CES production function for YtJF(1) has an elasticity of substitution ξJM and share parameters ζJ(1, j) that are identical across firms and that add up to one, Σj=2N¯ζJ(1,j)=1. We also allow for an additional effect of technology shocks on the intermediates import share parameters. Specifically, we posit that an improvement in technology in a foreign country not only leads to a lower cost in that country, but also to a higher demand for the respective good in all foreign countries, reflecting quality improvements due to better technology. The import share parameter between countries 1 and j is therefore given by

where ϰ = 0 corresponds to the standard case while ϰ > 0 introduces positive foreign demand effects of technological progress. This feature means that technological progress in the tradables sector leads to a stronger real appreciation. By contrast, for investment and consumption goods producers we assume ζ˜D(1,j)=ζD(1,j). The local currency prices PtJM(1,j) of imports in country 1 are determined by import agents, and the overall cost of the bundle YtJF(1) is PtJF(1). In the calibration of the model the share parameters ζJ(1, j) will be parameterized using a multi-region trade matrix. We have the following sub-production function:

with demands

and an import price index, written in terms of relative prices, of

Equations (158) and (159) are rescaled by technology and population to generate aggregate foreign input demand of country 1, YˇtJF(1) and aggregate demands for individual country imports YˇtJM(1,j). Note that for final goods YˇtDF there is a market-clearing condition because the imported bundle is sold to both consumption and investment goods producers:

In the two country case equations (158)-(160) simplify, after aggregation, to YˇtJF(1)=YˇtJM(1,2) and ptJF=ptJM. In our notation we will now revert to the two-country case and drop the index 1 for Home.

The tradables compositeYtT is produced by combining foreign produced tradables YtTF with domestically produced tradables YtTH, in a CES technology with elasticity of substitution ξT. This technology is modified in three distinct ways that account for important features of international trade. First, short-term to medium-term trade spillovers from domestic demand shocks are typically very weak in DSGE models because, when long-run elasticities are realistically calibrated, the real exchange absorbs much of their effects. We therefore allow for a quantitative spillover effect whereby an increase in domestic demand for tradables YtT relative to longer-run or potential output of tradables YtT,pot leads to a more than proportional increase in demand for the imported component of those tradables, the logic being that foreign tradables are in more elastic supply in the short run. Second, at the previous level we allowed for the possibility ϰ > 0, meaning foreign technology shocks affect relative demands for goods from different countries. We allow for an identical effect, dependent on the same parameter, to affect relative demands for domestic and foreign tradable goods. Specifically, an improvement in average world technology increases the relative demand for foreign produced tradables. Third, to prevent an excessive responsiveness of international trade to real exchange rate movements in the very short term, the model introduces adjustment costs GF,tT that make it costly to vary the share of Foreign produced tradables in total tradables production YtTF/YtT relative to the value of that share in the aggregate distribution sector in the previous period Yt1TF/Yt1T.

The domestic and foreign tradables share parameters are therefore given by

The sub-production function for tradables then has the following form:24,25

After expressing prices in terms of the numeraire, and after rescaling by technology and population, we obtain the aggregate tradables sub-production function from (168) - (170). We also obtain the following first-order conditions for optimal input choice:

The tradables-nontradables compositeYtA is produced with another CES production function with elasticity of substitution ξA. We again allow for a relative demand effect, this time of nontradables productivity shocks, with input share parameters given by

The sub-production function for the tradables-nontradables composite then has the following form:

The real marginal cost of producing YtA is, with obvious notation for sectorial price levels,

After expressing prices in terms of the numeraire, and after rescaling by technology, we obtain the aggregate tradables-nontradables sub-production function from (177), and the following first-order conditions for optimal input choice:

For the case where the nontradables sector is excluded from GIMF, we simply have YˇtA=YˇtT and ptA=ptT.

The private-public compositeZtD, which we will refer to as domestic final output, is produced with the following production function:

The inputs are the tradables-nontradables composite YtA and the stocks of public capital KtG1 and KtG2, which are identical for all firms and provided free of charge to the end user (but not of course to the taxpayer). Note that this production function exhibits constant returns to scale in private inputs while the public capital stocks enter externally, in an analogous manner to exogenous technology. The term S is a technology scale factor that can be used to normalize steady-state technology to one, (K¯G1)αG1(K¯G2)αG2S=1.

The real marginal cost of ZtD is denoted as ptDH, while the real marginal cost of YtA is ptA. After expressing prices in terms of the numeraire, and after rescaling by technology and population, we obtain the normalized production function from (181), and the following first-order condition:

The rescaled aggregate dividends of distributors (equal to zero in equilibrium) are

Finally, the market-clearing conditions for this sector equates its output to the demands of consumption and investment goods producers and of foreign import agents:

Modularity: This sector is part of the core of GIMF. But some elements can be dropped. Nontradables were already mentioned, in which case (177) would be removed. Public capital stocks can also be dropped when the effects of public investment are not of interest for the application. These effects are critical for fiscal multipliers as in “Fiscal Stimulus to the Rescue?”, which is why they are not dropped in that paper.

XIII. Investment Goods Producers

Investment goods producers buy domestic final output directly from domestic distributors, and foreign final output indirectly via import agents. They sell the final composite ZtI to capital goods producers, to the government, and back to other investment goods producers for the purpose of fixed and adjustment costs. There is a continuum of investment goods producers indexed by i ∈ [0,1]. They are perfectly competitive in their input markets and monopolistically competitive in their output market. Their price setting is subject to nominal rigidities. We first analyze the demand for their output, then we turn to their technology, and finally we describe their profit maximization problem.

Demand for investment goods varieties comes from multiple sources. Let z be an individual purchaser of investment goods. Then his demand 𝒟tI(z) is for a CES composite of investment goods varieties i, with time-varying elasticity of substitution σIt

with associated demands

where PtI(i) is the price of variety i of investment goods output, and PtI is the aggregate investment goods price level given by

Furthermore, the total demand facing a producer of investment goods variety i can be obtained by aggregating over all sources of demand z. We obtain

where 𝒟tI(i) and 𝒟tI remain to be specified by way of a market-clearing condition for investment goods output. The exogenous and stochastic price markup is given by μtI=σIt/(σIt1).

The technology of investment goods producers consists of a CES production function that uses domestic final output YtIH(i) and foreign final output imported via import agents YtIF(i), with a share coefficient for domestic final output of αHtI and an elasticity of substitution ξI. In the same way as for intermediates trade, we allow for trade spillover effects, and we introduce an adjustment cost GF,tI that makes it costly to vary the share of foreign inputs YtIF(i)/ZtI(i) relative to the value of that share in the aggregate investment goods distribution sector in the previous period Yt1IF/Zt1I. We therefore have

After expressing prices in terms of the numeraire, and after rescaling by technology and population, we obtain the aggregate investment goods production function from (189) - (193). Letting the marginal cost of producing ZtI be denoted by ptII, we also obtain the following first-order conditions for optimal input choice:

We finally turn to the profit maximization problem. It consists of maximizing the expected present discounted value of nominal revenue PtZI(i)𝒟tI(i) minus nominal costs of production PtII𝒟tI(i), a fixed cost PtZITtωI, and inflation adjustment costs PtZIGP,tI(i). The latter are real resource costs that have to be paid out of investment goods output ZtI. Their functional form is by now familiar:

Fixed costs are given by

It is assumed that the producer pays out each period’s nominal net cash flow as dividends DtI(i). The objective function is

subject to product demands (188) and given marginal cost PtII. We obtain the first-order condition for this problem, again using the fact that all firms behave identically in equilibrium. Using the equilibrium condition 𝒟tI=ZtI we obtain

The rescaled aggregate dividends of investment goods producers are

Finally, we allow for unit-root and stationary shocks to the relative price of investment goods. Specifically, the net output of investment goods producers,

is converted to final output of investment goods YˇtI using the technology

where AtI is a stationary technology shock and TtI is a unit-root technology shock with zero trend growth. We define the relative price terms p˜tI=1/TtI and ptI=1/AtI. Competitive pricing means that the price of final investment goods equals

The market-clearing condition for investment goods therefore equates output to the demands of manufacturers (as investors) or capital producers, the government, and the investment goods producers themselves for fixed and adjustment costs:

Modularity: This sector is part of the core of GIMF. It was introduced mainly to distinguish investment and consumption goods in countries’ international trade flows. This is because their shares in overall trade can differ dramatically between countries or regions, and because investment and consumption goods imports exhibit very different sensitivity to the business cycle. This sector is therefore present in “Fiscal Stimulus to the Rescue?”.

XIV. Consumption Goods Producers

Consumption goods producers buy domestic final output directly from domestic distributors, and foreign final output indirectly via import agents. They sell the final composite ZtC to consumption goods retailers, to the government, and back to other consumption goods producers for the purpose of fixed and adjustment costs. There is a continuum of consumption goods producers indexed by i ∈ [0,1]. They are perfectly competitive in their input markets and monopolistically competitive in their output market. Their price setting is subject to nominal rigidities. We first analyze the demand for consumption goods, then we turn to consumption goods producers’ technology, and finally we describe their profit maximization problem.

Demand for the consumption goods varieties comes from multiple sources. Let z be an individual purchaser of consumption goods. Then his demand 𝒟tC(z) is for a CES composite of final output varieties i, with time-varying elasticity of substitution σCt:

with associated demands

where Pt(i) is the price of variety i of consumption goods output, and Pt is the aggregate consumption goods price level given by

We choose this price level as the economy’s numeraire. The total demand facing a producer of consumption goods variety i can be obtained by aggregating over all sources of demand z. We obtain

where 𝒟tC(i) and 𝒟tC remain to be specified by way of a market-clearing condition for consumption goods output. The exogenous and stochastic price markup is given by μtC=σCt/(σCt1).

The technology of consumption goods producers consists of a CES production function that uses domestic final output YtCH(i) and foreign final output imported via import agents YtCF(i), with a share coefficient for domestic final output of αHtC and an elasticity of substitution ξC. In the same way as for intermediates trade, we allow for trade spillover effects, and we introduce an adjustment cost GF,tC that makes it costly to vary the share of foreign inputs YtCF(i)/ZtC(i) relative to the value of that share in the aggregate consumption goods distribution sector in the previous period Yt1CF/Zt1C. We therefore have

After expressing prices in terms of the numeraire, and after rescaling by technology and population, we obtain the aggregate consumption goods production function from (210) - (214). Letting the marginal cost of producing ZtC be denoted by ptCC, we also obtain the following first-order conditions for optimal input choice:

We finally turn to the profit maximization problem. It consists of maximizing the expected present discounted value of nominal revenue Pt(i)𝒟tC(i) minus nominal costs of production PtCC𝒟tC(i), a fixed cost PtTtωC, and inflation adjustment costs PtGP,tC(i). The latter are real resource costs that have to be paid out of consumption goods output ZtC. Their functional form is the familiar

Fixed costs are given by

It is assumed that the producer pays out each period’s nominal net cash flow as dividends 𝒟tC(i). The objective function is

subject to product demands (209) and given marginal cost PtCC. We obtain the first-order condition for this problem, again using the fact that all firms behave identically in equilibrium. Using the equilibrium condition 𝒟tC=ZtC we obtain

The rescaled aggregate dividends of consumption goods producers are

The market-clearing condition for consumption goods equates output to the demands of consumption goods retailers, the government, and the consumption goods producers themselves for fixed and adjustment costs:

Modularity: This sector is part of the core of GIMF. It was introduced mainly to distinguish investment and consumption goods in countries’ international trade flows. This is because their shares in overall trade can differ dramatically between countries, and because investment and consumption goods imports exhibit very different sensitivity to the business cycle. This sector is therefore present in “Fiscal Stimulus to the Rescue?”.

XV. Retailers

There is a continuum of retailers indexed by i ∈ [0,1]. Retailers combine final output purchased from consumption goods producers and raw materials purchased from raw-materials producers, where there are adjustment costs to rapid changes in raw-materials inputs. Retailers sell their output to households. They are perfectly competitive in their input market and monopolistically competitive in their output market. Their price setting is subject to real rigidities in that they find it costly to rapidly adjust their sales volume to changing demand conditions. We first analyze retailers’ technology, then the demands for their output, and finally their optimization problem.

The technology of each retailer is given by a CES production function in consumption goods Ctret(i) and directly consumed raw materials XtC(i), with elasticity of substitution ξXC. An adjustment cost GX,tC(i) makes fast changes in raw-materials inputs costly. We have

The optimal input choice for this problem, after normalizing by technology and population, and after dropping the agent specific index i, is given by

and marginal cost is

When the raw-materials sector is excluded from GIMF, the above simplifies to Cˇt=Cˇtret and ptC=1.

Demand for the output varieties Ct(i) supplied by retailers comes from households, and follows directly from (10) and (29) as

The optimization problem of retailers consists of maximizing the expected present discounted value of nominal revenue PtR(i)Ct(i) minus nominal costs of inputs PtCCt(i), minus nominal quantity adjustment costs PtGC,t(i), where the latter represent a demand for consumption goods output. This sector does not face fixed costs of operation. The quantity adjustment costs take the form26

It is assumed that each retailer pays out each period’s nominal net cash flow as dividends DtR(i). The objective function of retailers is

subject to demands (228) and adjustment costs (229). The first-order condition for this problem, after dropping firm specific subscripts, rescaling by technology and population, and letting μRt=σRt/(σRt1), has the form:

The real dividends and rescaled adjustment costs of this sector are given by

When the retail sector is excluded from GIMF the foregoing simplifies to ptR=ptC.

Modularity: This sector is not part of the core of GIMF. But it has never been dropped in applications, including “Fiscal Stimulus to the Rescue?”. The reason is that consumption dynamics without this sector becomes very implausible as it exhibits large jumps following shocks. There is no alternative to retailers to obtain inertial consumption dynamics. This is because the alternative of habit persistence is ruled out by the necessity of having a utility function consistent with both balanced growth and with aggregation across generations. In this class of utility functions habit persistence is only feasible in a form that generates minimal consumption inertia. The reason for having this sector is therefore similar to the reasons for the union sector (wage rigidities not feasible at the level of households) and for capital accumulation within firms rather than households (investment adjustment costs not feasible at the level of households).

XVI. Government

A. Government Production

The government uses consumption goods YtGC and investment goods YtGI to produce government output ZtG according to a CES production function with consumption goods share parameter αGC and an elasticity of substitution ξG:

Denoting the marginal cost of producing ZtG by ptZG, and normalizing by technology and population, we then obtain the normalized version of (234) and the following standard input demands:

We allow for unit-root shocks to the relative price of government output. Specifically, the output of government goods ZˇtG is converted to final output of government goods YˇtG using the technology

where TtG is a unit-root technology shock with zero trend growth. We define the exogenous and stochastic relative price as p̃tG=1/TtG. Then competitive pricing means that the final price of government output equals

Demand for government output Ğt comes from government consumption and investment:

and the market-clearing condition is given by Gˇt=YˇtG, and therefore by

Modularity: This technology is not part of the core of GIMF. It can be removed by setting the share parameters of consumption or investment goods to zero. The option is included to allow for a range of import contents of government output, between the often high content of investment goods and the often low content of consumption goods. For realism, it is also included in “Fiscal Stimulus to the Rescue?”.

B. Government Budget Constraint

Fiscal policy consists of a specification of public investment spending Gtinv, public consumption spending Gtcons, transfers from OLG agents to LIQ agents τT,t=τT,tOLG=τT,tLIQ, lump-sum taxes τls,t=τtls,OLG+τtls,LIQ, lump-sum transfers ϒt=ϒtOLG+ϒtLIQ, and three different distortionary taxes τL,t, τc,t and τk,t.

Government investment and consumption spendingGt=Gtinv+Gtcons represents a demand for government output. Both types of government spending are exogenous and stochastic. Government investment spending has a critical function in this economy. It augments the stock of publicly provided infrastructure capital KtG1, the evolution of which is, after rescaling by technology and population, given by

where δG1 is the depreciation rate of public capital. Government consumption spending on the other hand can be modeled as either unproductive or productive by choosing the coefficient αG2 in the production function. For the case of a αG2 > 0 government consumption accumulates a second productive capital stock:

The government’s policy rule for transfers partly compensates for the lack of asset ownership of LIQ agents by redistributing a small fraction of OLG agents’s dividend income receipts to LIQ agents. Specifically, dividends of the retail and union sectors are redistributed in proportion to LIQ agents’ share in consumption and labor supply, while the redistributed share of dividends in the remaining sectors is ι, which we will typically calibrate as being smaller than the share ψ of LIQ agents in the population, ι = ψdshare with dshare < 1. Finally, in the baseline of GIMF government lump-sum transfers and taxes are received and paid by LIQ agents in proportion to their share in aggregate consumption, but this rule can easily be changed, for example to allow for transfers that are 100% targeted to LIQ agents. After rescaling by technology we therefore have the rule:

The sources of nominal tax revenue are labor income taxes τL,tWtLt, consumption taxes τc,tPtCCt, taxes on the return to capital τk,tΣj=N,T[Rk,tJδKtJPtqtJ]K¯tJ, and lump-sum taxes Ptτls,t. We define the rescaled aggregate real tax variable as

Furthermore, the government issues nominally non-contingent one-period nominal debt Bt at the gross nominal interest rate it. The rescaled real government budget constraint is therefore

Modularity: These equations are part of the core of GIMF, and are therefore included in “Fiscal Stimulus to the Rescue?”.

C. Fiscal Policy

The model makes two key assumptions about fiscal policy. The first concerns dynamic stability, and the second stabilization of the business cycle.

1. Dynamic Stability

Fiscal policy ensures a non-explosive government-debt-to-GDP ratio by adjusting tax rates to generate sufficient revenue, or by reducing expenditure, in order to stabilize the overall, interest inclusive government surplus-to-GDP ratio gstrat at a long-run level of gsstrat chosen by policy. The government surplus is given by

and its ratio to GDP (gdpt will be defined below) is

We allow for the possibility that gsstrat follows an exogenous stochastic process. We denote the current value and the long-run target for the government-debt-to-GDP ratio by bˇtrat and bˇsstrat, expressed as a share of annual GDP. We have the following relationship between long-run government balance and government-debt-to-GDP ratios:

Here π¯t is the inflation target of the central bank. In other words, for a given nominal growth rate, choosing a surplus target gsstrat implies a debt target bˇsstrat and therefore keeps debt from exploding.

2. Business Cycle Stabilization

Fiscal policy ensures that the government surplus-to-GDP ratio, while satisfying its long-run target of gsstrat, can also flexibly respond to the business cycle. Specifically, we have the following structural fiscal surplus rule:

The relationship (248) implies that even with ddebt = 0 the rule (249) automatically ensures a non-explosive government-debt-to-GDP ratio of bˇsstrat. But the long-run autoregressive coefficient on debt in that case, at 1/(π¯tgn), is very close to one. Setting ddebt > 0 ensures faster convergence of debt at the expense of more volatile government surpluses.

The remaining terms in (249) represent responses to the state of the business cycle. The first term, which follows dgdp, is an output gap. This uses current and potential Fisher-weighted GDP gdˇptfisher and gdˇptpot as the relevant output measures, and can be calibrated using OECD estimates of fiscal rules. As for potential GDP, our model allows for unit-root shocks to technology and to savings, where the latter have permanent real effects due to the non-Ricardian features of the model. Potential GDP is therefore subject to these nonstationary shocks, and the fiscal rule has to reflect these changes. This is why potential GDP is proxied by a moving average of past actual values of GDP.27 For the same reason a number of other model variables require moving average approximations of their moving “potential” or long-run values, including tax bases, long-run tradables composites in the formulas for spillovers (see above), and even some parameters (ϕaN, ϕaT, κo, ωN, ωT, ωc, ωI) that need to change when the model’s steady state changes permanently. The moving average expression for potential GDP is given by

The second term in the fiscal rule in (249), which follows dtax, is a tax revenue gap, where potential tax revenue τˇtpot is tax revenue at current tax rates but multiplied by the respective moving average tax bases:

For the moving average tax bases we have

The third term in the fiscal rule in (249), which follows drawmat, is a raw-materials revenue gap. Potential raw-materials revenue gˇXtpot is based on estimates of the potential or long-run international price and domestic output of the raw material, thereby yielding an estimate of potential dollar revenue. Changes in the real exchange rate are allowed to affect the estimate of potential revenue in terms of domestic currency. We have

where the two moving average terms are given by

Setting ddebt = dgdp = dtax = drawmat = 0 in (249) corresponds to a balanced budget rule, which is highly procyclical and therefore undesirable. Actual fiscal policy in individual countries can typically be characterized by the degree to which automatic stabilizers work in response to the business cycle. This idea has been quantified by the OECD, who have produced estimates of dgdp for a number of countries. But a small number of countries has instead implemented structural fiscal surplus rules that can be characterized by dgdp = 0 and dtax = 1.28 In this case during a boom, when tax revenue exceeds its long run value, the government uses the extra funds to pay off government debt by reducing the deficit below its long run value. The main effect is to minimize the variability of fiscal instruments, but it also reduces the variability of output relative to a balanced budget rule. A more explicitly counter-cyclical rule would set dtax > 1.

The rule (249) is not an instrument rule but rather a targeting rule. Any of the available tax and spending instruments can be used to make sure the rule holds. The default setting is that this instrument is the labor tax rate τL,t, because this is the most plausible choice. However, other instruments or combinations of multiple instruments are possible. For example, we can posit

With dctax = dktax = 1 this generates a perfect comovement between the three tax rates, while dctax = dktax = 0 means that only labor tax rates change.

Modularity: The fiscal rule is part of the core of GIMF, and are therefore included in “Fiscal Stimulus to the Rescue?”.

D. Monetary Policy

Monetary policy uses an interest rate rule that features interest rate smoothing and which responds to (i) deviations of one-year-ahead year-on-year inflation πt+129 from the (possibly time-varying) inflation target π¯t, (ii) the output gap, (iii) the year-on-year growth rate of Fisher-weighted GDP, and (iv) deviations of current exchange rate depreciation from its long run value ε¯=π¯t/π¯t(Ñ). Furthermore, we allow for autocorrelated monetary policy shocks Stint. The interest rate rule is very general and similar to conventional inflation-forecast-based rules, with one minor and one important exception. The minor exception is the presence of exchange rate depreciation, which we will however only use for the case of strict exchange rate targeting, which can be modeled as δi = 1 and δe→ ∞. The important exception is that the non-Ricardian nature of the model implies that there is no unchanging steady-state GDP or real interest rate. The former has already been discussed in the context of fiscal rules. As for the latter, the term proxying the nominal interest rate rteqπ̃t includes a geometric moving average of real interest rates, but this average is more complicated than in the case of GDP. Specifically, it contains separate moving averages of the underlying pre-risk-premium real interest rate, rtworld, and of the risk premium itself, ξtma. The former, in order to exclude excessive recent fluctuations in the domestic real interest rate from the proxy of the underlying equilibrium real interest rate, includes a smoothed measure of a worldwide GDP-weighted average real interest rate. The separate smoothing of the risk premium terms is done in the usual way and multiplies rtworld. We adopt the notation rtpreξ=rt/((1+ξtf)(1+ξtb)) and ξt=(1+ξtf)(1+ξtb). We also allow for the inflation rate targeted by monetary policy, π̃t, to be a weighted average of current and one-year-ahead inflation, where the weights δπ˜ and 1δπ˜ can be estimated along with the rest of the policy rule parameters. Then the complete monetary rule is given by

Modularity: The fiscal rule is part of the core of GIMF, and is therefore included in “Fiscal Stimulus to the Rescue?”.

XVII. Shocks

We assume that βt, αHtC, αHtI, αHtT, αCtX, αNtX, αTtX, Xtsup, σtN, σtT, μtN, μtT, SˇtN,nwd, SˇtT,nwd, SˇtN,nwy, SˇtT,nwy, SˇtN,nwk, SˇtT,nwk, Gˇtcons and Gˇtinv, and their foreign counterparts, are characterized by both transitory and unit-root components. Denoting any of these shocks by xt we have

For the two policy variables gsstrat and π¯t the transitory components are given by the endogenous responses of the fiscal and monetary rules, while the permanent components are specified as unit roots:

For the three relative price processes p̃ty, y ∈ {I, G, exp} we also assume unit roots:

Interest rate, investment, labor supply, foreign exchange risk premium, government risk premium and markup shocks are assumed to only have transitory components, and markup shocks in addition are assumed to be serially uncorrelated:30

For productivity shocks, we allow country specific technology to follow the U.S., in the following way:

The parameter catchup(j) can vary between 0 and 1, and ÃtJ can be subject to unit-root shocks. For the stationary shock to the price of investment goods we again allow for catchup growth with the U.S.:

Modularity: Shocks are part of the core of GIMF. But the catching up feature of technology shocks can be and often is turned off. It is turned off in “Fiscal Stimulus to the Rescue?”.

XVIII. Balance of Payments

Combining all market-clearing conditions with the budget constraints of households and the government and with the expressions for firm dividends we obtain an expression for the current account:

When we repeat the same exercise for all other countries we finally obtain the market-clearing condition for international bonds,

The current account balance is given by

The level of GDP is given by the following expression:

Modularity: This block is part of the core of GIMF, and is therefore included in “Fiscal Stimulus to the Rescue?”.

XIX. Calibration

We calibrate a five-region version of the model, with regions representing the United States (US), emerging Asia (AS), the euro area (EU), Japan (JA) and remaining countries (RC). The denomination of international bonds is in U.S. currency. The calibration described here is for an annual version of the model that excludes the raw-materials sector. The model has a large number of unit roots. Therefore, when we mention the calibration of the steady state of the model we refer to the initial baseline of the economy.

Table 1 lists our assumptions concerning real and nominal growth rates and the long-run real interest rate. We fix the steady-state world technology growth rate at 1.5% per annum or g = 1.015 and the world population growth rate at 1% per annum or n = 1.01. The steady-state inflation rates are 2.0% in US, AS, EU and RC, and 1% in JA. The long-run real interest rate is equalized across countries, and we assume a steady-state value of 3% per annum or r¯=1.03. We find the values of β̃t that are consistent with these and the following assumptions.

Table 1:Long Run Growth Rates and Interest Rates
USASEUJARC
World Technology Growth g1.0151.0151.0151.0151.015
World Population Growth n1.011.011.011.011.01
Steady State Inflation Rate π¯1.021.021.021.011.02
Long Run Real Interest Rate r¯1.031.031.031.031.03
Forex Risk Premium ξf00000
Government Risk Premium ξb00000

Table 2 shows the calibration of household utility functions, which are assumed equal across countries. The parameters θ and χ are critical for the non-Ricardian behavior of the model. We assume an average remaining time at work of 20 years, which corresponds to χ = 0.95. The degree of myopia is given by the planning horizon 1/(1 – θ), which we assume to equal 10 years, implying θ = 0.9. The main criterion used in choosing these parameters is the empirical evidence for the effect of government debt on real interest rates. Our model is calibrated so that a one percentage point increase in the government-debt-to-GDP ratio in the U.S. leads to an approximately three to four basis points increase in the U.S. (and world) real interest rate. This value is in the middle of the range of estimates provided by Laubach (2003), Engen and Hubbard (2004) and Gale and Orszag (2004). Household preferences are further characterized by an intertemporal elasticity of substitution of 0.25, or γ = 4. The elasticity of labor supply depends on the steady-state value of labor supply, which is in turn determined by the leisure share parameter η. We adjust this parameter to obtain an elasticity of 0.5, in line with a good part of the business cycle literature. Finally, the shares of liquidity-constrained agents in the population are 25% in US, EU and JA, and 50% in AS and RC. Their shares in dividend income are equal to half their shares in the population in all regions.

Table 2:Utility Functions
USASEUJARC
Average Planning Horizon in Years (θ = 0.9)1010101010
Average Remaining Working Life (χ = 0.95)2020202020
Intertemporal Elasticity of Substitution (γ = 4)0.250.250.250.250.25
Labor Supply Elasticity (endogenizes ηOLG, ηLIQ)0.50.50.50.50.5
Share of Liquidity Constrained Agents ψ0.250.500.250.250.50
Dividend Share of Liq. Constrained Agents i0.1250.250.1250.1250.25

Tables 3 and 4 turn to the calibration of technologies, specifically elasticities and markups. These are assumed to be equal across countries, while of course factor share parameters, especially trade shares, are not equal. Tables 5 and 6 deal with the calibration of the main expenditure and factor shares.

Table 3 shows elasticities of substitution. The elasticity of substitution between capital and labor in tradables and nontradables ξZN and ξZT equal 1. The elasticities of substitution between domestic and foreign goods, ξNM, ξTM, ξT, ξI and ξC equal 0.75.31 The elasticities of substitution between tradables and nontradables, ξA, and between government consumption goods and investment goods inputs, ξg, equal 0.5.

Table 3:Elasticities of Substitution
USASEUJARC
Nontradables: Capital-Labor ξZN11111
Tradables: Capital-Labor ξZT11111
Nontradables Import Agents: Different Countries ξNM0.750.750.750.750.75
Tradables Import Agents: Different Countries ξTM0.750.750.750.750.75
Distributors: Home-Foreign Tradables ξT0.750.750.750.750.75
Inv. Goods Producers: Home-Foreign Tradables ξI0.750.750.750.750.75
Cons. Goods Producers: Home-Foreign Tradables ξC0.750.750.750.750.75
Distributors: Tradables-Nontradables ξA0.50.50.50.50.5
Government: Consumption-Investment Goods ξG0.50.50.50.50.5
Table 4:Steady State Markups
USASEUJARC
Nontradables Manufacturing μ¯N1.11.11.11.11.1
Tradables Manufacturing μ¯T1.11.11.11.11.1
Union Wage Setting μ¯U1.11.11.11.11.1
Investment Goods Production μ¯I1.051.051.051.051.05
Consumption Goods Production μ¯C1.051.051.051.051.05
Retail Sector μ¯R1.051.051.051.051.05
Nontradables Import Agents μ¯NM1.0251.0251.0251.0251.025
Tradables Import Agents μ¯TM1.0251.0251.0251.0251.025
Table 5:Steady State Expenditure to GDP Ratios
USASEUJARC
Share in World GDP27.412.322.09.129.3
Consumption/GDP65.159.258.159.859.1
OLG Consumption/GDP51.334.345.846.934.0
LIQ Consumption/GDP13.824.912.312.925.1
Private Investment/GDP17.225.018.321.019.0
Government Spending/GDP17.516.023.519.522.0
Government Investment/GDP2.54.03.02.52.0
Government Consumption/GDP15.012.020.517.020.0
Government Transfers/GDP20.010.020.020.020.0
Trade Balance/GDP0.2-0.20.1-0.3-0.1
Exports/GDP11.726.817.510.821.9
Final Goods Exports/GDP8.320.513.78.09.6
Intermediate Goods Exports/GDP3.46.33.82.812.3
Imports/GDP11.527.017.411.021.9
Consumption Goods Imports/GDP5.25.76.83.89.1
Investment Goods Imports/GDP2.66.44.01.67.5
Intermediate Goods Imports/GDP3.714.96.65.65.3
Tradables Demand Effects of Technology ϰ11111
Nontradables Demand Effects of Technology ϰ˜11111
Table 6:Steady State Factor Shares and Depreciation Rates
USASEUJARC
Labor Income/GDP6054606060
Nontradables Labor Income/GDP6660666666
Tradables Labor Income/GDP5448545454
Depreciation Rate of Private Capital δK0.10.120.10.10.1
Nontradables Output/Manufacturing Output5050505050
Consumption Goods Input/Government Output5050505050

Table 4 shows steady-state markups. Steady state markups in tradables and nontradables manufacturing μ¯N and μ¯T, and in union wage setting μ¯U, equal 1.1. Steady state markups in investment and consumption goods production μ¯I and μ¯C, and in retailing μ¯R, equal 1.05. Finally, markups of import agents μNM and μTM, equal 1.025.

Table 5 shows the decomposition of steady-state GDP at consumer prices into its expenditure components. These numbers are based on recent historical averages. We note the very high investment-to-GDP ratio in AS. We make this a feature of the steady state by assuming a high capital share and a high depreciation rate for AS, see our discussion of Table 6. Figures 2-4 augment Table 5 with more detailed information on international trade flows between all regions. Finally, in relation to trade we allow for demand effects of technology shocks by setting all parameters ϰ and χ̃ equal to 1.

Figure 2:Trade Matrix: Intermediate Goods

Figure 3:Trade Matrix: Investment Goods

Figure 4:Trade Matrix: Consumption Goods

Table 6 shows the decomposition of steady-state GDP at producer prices into its factor components, depreciation rates of private capital, and factor shares of sectorial production functions. Except for AS, labor shares equal 60%, and the depreciation rate is 10% p.a. The nontradables labor share is 6 percentage points higher and the tradables labor share 6 percent lower than this average, reflecting the higher capital intensity of exports. The 50% share of the value of nontradables production in the overall value of manufacturing output is also a fairly common assumption supported by evidence. On the other hand, the 50% shares of consumption and investment goods inputs into the production of government output are arbitrary, but results are generally insensitive to all but very large deviations from this assumption.

Calibrating the depreciation rate of private capital would ordinarily present a problem given that we have already fixed the two capital income shares and the investment-to-GDP ratio. The only three free parameters available for to fix these four values would typically be the two labor share parameters and the depreciation rate. But in our model the income of capital consists not only of the return to capital in manufacturing, but also of economic profits due to market power. We have introduced fixed costs in distribution that partly or wholly eliminate these profits. The percentage of steady-state economic profits that is eliminated by fixed costs (1-pshare in the code) can therefore be specified as a third free parameter. This allows us to calibrate the annual depreciation rate of private capital at the conventional 10 percent for US, EU, JA and RC while maintaining the investment-to-GDP ratio and capital income shares stated above. For AS we calibrate a 12 percent depreciation rate. Together with the higher assumed capital share for AS this helps to reproduce that region’s very high investment-to-GDP ratio as a feature of the steady state.

Table 7 shows miscellaneous steady-state ratios and parameters. Calibrated government-debt-to-GDP ratios are roughly in line with the data, but will require some refinement in future work. Net foreign assets-to-GDP ratios on the other hand are based on detailed work by IMF staff. Table 7 also decomposes tax revenue into its four components. The assumed shares are used to infer the model’s steady-state tax rates. At this point the shares are not based on detailed data, but work is in progress to address this. The remaining two items in Table 7 deal with the specification of public capital stock accumulation and of its productivity. We adopt Kamps’ (2004) 4 percent per annum estimate of the depreciation rate of public capital, or δG = 0.04. Ligthart and Suárez (2005) estimate that the elasticity of aggregate output with respect to public capital equals 0.14, which based on model simulations requires αG = 0.1.

Table 7:Miscellaneous Steady State Ratios and Parameters
USASEUJARC
Government Debt/GDP5055607560
Net Foreign Assets/GDP-28.027.5-13.042.511.1
Labor Income Taxes/Total Taxes4040404040
Capital Income Taxes/Total Taxes1010101010
Consumption Taxes/Total Taxes2525252525
Lump-Sum Taxes/Total Taxes2525252525
Depreciation Rate of Public Capital δG0.040.040.040.040.04
Output Elasticity w.r.t. Public Capital (αG = 0.1)0.140.140.140.140.14

Table 8 shows our calibration of the financial accelerator sector. With some exceptions this calibration is generally based on those of Christiano and others (2007) and Bernanke and others (1999). We are in the process of updating this calibration with better data, and to allow us to differentiate better between regions. Our calibration fixes three key ratios. First, leverage, defined as the ratio of corporate debt to corporate equity, equals 100 in all sectors and regions. Second, the share of firms that goes bankrupt in any given year equals 8 percent. Third, the steady-state external finance premium equals 1.5 percent. Together these values endogenize the steady-state values of firm riskiness σ¯N and σ¯T, steady-state bankruptcy monitoring costs and μ¯N, and μ¯T the steady-state shares of net worth distributed as dividends, S¯N,nwd and S¯T,nwd.

Table 8:Financial Accelerator Sector
USASEUJARC
Leverage in Nontradables in %100100100100100
Leverage in Tradables in %100100100100100
Annual Bankruptcy Rate in Nontradables in %88888
Annual Bankruptcy Rate in Tradables in %88888
External Finance Premium in Nontradables in %1.51.51.51.51.5
External Finance Premium in Tradables in %1.51.51.51.51.5

Table 9 lists the monetary rule parameters in all five regions. For most regions this calibration is based on estimation results of reaction functions for an annual model. For AS we assume a fixed exchange rate.

Table 9:Monetary Rule Parameters
USASEUJARC
δi0.71510.3430.3920.715
δπ1.03401.4830.9131.034
δπ¯0.21610.2370.2160.216
δy00000
δygr0.250000
δe0106000

Table 10 lists the fiscal rule parameters in all five regions. The calibration assumes target surplus-to-GDP ratios consistent with the government-debt-to-GDP ratios calibrated above. Furthermore, it uses OECD estimates of countercyclical rule coefficients whereby the government surplus-to-GDP ratio responds to the output gap.

Table 10:Fiscal Rule Parameters
USASEUJARC
dgdp0.340.250.490.330.30
ddebt00000
dtax00000
drawmat00000
dptax00000
dktax00000

XX. Applications of GIMF

A. Central Banks Using GIMF

At the present time five central banks have started to work with GIMF at different levels of intensity.

The HongKong Monetary Authority (HKMA) started work in 2008Q3/4, supported by an APD Technical Assistance mission. They have produced a HKMA working paper on GIMF, and are using the model internally for area-wide policy simulations. HKMA staff have also presented GIMF-based policy work at workshops in Asia, with interest shown by other central banks in the region.

The Central Bank of Russia (CBR) started work in 2008Q4, supported by two RES Technical Assistance missions, and by visits of the Russian modeling team to IMF HQ. CBR started to use a production version of GIMF for policy simulations, and are actively involved in our new project of filtering GIMF, in this case with Russia — Rest of the World data. Russia is a test case for the oil sector in GIMF. Products so far include one finished working paper exploring fiscal and oil sector shocks for Russia.

Banco de Portugal (BdP) started work independently in 2007, after their staff attended several of our Modeling Workshops in Washington. A small team at BdP has adapted GIMF to the Portuguese setting, especially the monetary union and small open economy aspects but also a full specification of fiscal policy rules, with a very user friendly MATLAB front-end. This is now a production model at BdP, called PESSOA. Products so far include a working paper “Improving Domestic Competition, Fostering Growth in the Portuguese Economy”.

Banco Central de Reserva de Perú started work in late 2008, with technical support from RES and WHD. The first step was to convert TROLL-based GIMF into a DYNARE-based version calibrated to Peru. Projects are in the pipeline.

Banque de France (BdF) started work on GIMF in 2008Q4. This is part of our collaboration with Michel Juillard’s team at BdF, where GIMF is intended to be used for policy simulations. The main initial project is the translation of GIMF’s TROLL code into a DYNARE version that can handle extensive steady-state recalibrations, and that will therefore be much easier to use by other users in policy institutions. The DYNARE code is not finished yet, but very substantial progress has been made.

B. IMF Area Departments Using GIMF

The following is an incomplete listing of projects carried out by area departments that have used GIMF.

  • Allard, C., 2008, “Macroeconomic Effects of EU Transfers in New Member States”, SIP, Poland Article IV, IMF Country Report 08/131.
  • Allard, C. and S. Muñoz, March 2008, “Challenges to Monetary Policy in the Czech Republic — An Integrated Monetary and Fiscal Analysis”, IMF Working Paper 08/72.
  • Arslanalp, Serkan, 2008, “Bangladesh Tax Reform Scenarios”, SIP, Bangladesh Article IV.
  • Berkmen, P., 2009, “Macroeconomic Responses to Terms-of-Trade Shocks: A Framework For Policy Analysis For the Argentine Economy”, IMF Working Paper 09/117.
  • Canales Kriljenko, J., 2009, “Countercyclical Fiscal Policies under Alternative Monetary Policy Frameworks”, Box 2.5. in WHD’s Spring 2009 Regional Economic Outlook.
  • Clements, B., E. Flores and D. Leigh, 2009, “Monetary and Fiscal Policy Options for Dealing with External Shocks: Insights from the GIMF for Colombia”, IMF Working Paper 09/59.
  • Flanagan, M., 2008, “Resolving a Large Contingent Fiscal Liability: Eastern European Experience”, Study at the request of Ukrainian authorities, IMF Working Paper 08/159.
  • Freedman, C., M. Kumhof, D. Laxton and J. Lee, 2009, “The Case for Global Fiscal Stimulus”, IMF Staff Position Note 2009/3.
  • Gueorguiev, N., 2008, “Can Fiscal Policy Boost Growth and Employment in South Africa?”, SIP, South Africa Article IV, forthcoming.
  • Gueorguiev, N., 2009, “Between Scilla and Charybdis: Demand Management Policies to Support Growth and Maintain Stability in South Africa”, analytical note in support of the Article IV consultation with South Africa.
  • Hauner, D., 2008, “Macroeconomic Effects of Pension Reform in Russia”, IMF Working Paper 08/201.
  • Honjo, K., 2009, “Sweden - the Sustainability of Public Finances and Fiscal Rules”, in 2009 Article IV Report for Sweden, and IMF Working Paper (forthcoming).
  • Kinkaid, R., 2008, “Adjustment Dynamics in the Euro Area: A Fresh Look at the Role of Fiscal Policy Using a DSGE Approach”, DGECFIN/EC, Economic Papers Series #322.
  • Kumhof, M. and D. Laxton, August 2007, “A Party Without a Hangover: On the Effects of U.S. Government Deficits”, IMF Working Paper 07/202.
  • Kumhof, M. and D. Laxton, 2009, “Chile’s Structural Fiscal Surplus Rule: A Model-Based Evaluation”, IMF Working Paper 09/88.
  • Kumhof, M. and D. Laxton, 2009, “Simple, Implementable Fiscal Policy Rules”, IMF Working Paper 09/76.
  • Kumhof, M., D. Laxton, and D. Leigh, 2008, “To Starve or Not to Starve the Beast”, IMF Working Paper (forthcoming).
  • Leigh, Daniel, 2008, “Achieving a Smooth Landing: the Role of Fiscal Policy”, IMF Working Paper 08/69.
  • Leigh, Daniel, 2008, “The Scope for a Countercylical Fiscal Policy in Latin America”, Box in April 2008 WHD REO.
  • N’Diaye, P., P. Zhang and W. Zhang, 2008, “Structural Reform, Intra-Regional Trade, and Medium-Term Growth Prospects of East Asia and the Pacific”, HongKong Monetary Authority Working Paper 17/2008.
  • Rozhkov, D., and W. Schule. 2009, “Short-Term Benefits and Medium-Term Costs of Fiscal Stimulus”, IMF Selected Issues Paper (forthcoming).
  • Sgherri, S. and E. Zoli, 2009, “Fiscal Policy in Advanced European Countries: Effectiveness, Coordination, and Solvency Issues”, Chapter 2 in EUR’s Spring 2009 Regional Economic Outlook.

Appendices
APPENDIX 1

Population Growth

The population size at time 0 is assumed to equal N, with N(1 — ψ) OLG households and Nψ LIQ households. The size of a new cohort born at time t is given by Nnt(1θn), so that by time t + k this cohort will be of size Nnt(1θn)θk. When we sum over all cohorts at time t we obtain

This means that the overall population grows at the rate n. When we normalize real quantities, we divide by the level of technology Tt and by population, but for the latter we divide by nt only, meaning real figures are not in per capita terms but rather in absolute terms adjusted for population growth.

APPENDIX 2

Optimality Conditions for OLG Households

We have the following Lagrangian representation of the optimization problem of OLG households:32

where Λa,t is the marginal utility to the generation of age a at time t of an extra unit of domestic currency. Define the marginal utility of an extra unit of consumption goods output as

and let

Then we have the following first-order conditions for consumption and labor supply

which can be combined to yield

We can aggregate this as

and normalize it as

In this aggregation we have made use of the following assumptions about labor productivity:

Equation (293) is our specification of the profile of labor productivity over the lifetime. Equation (294) is the assumption that average labor productivity equals one. Equations (293) and (294), for a given productivity decline parameter χ, imply the initial productivity level κ in (295). Equation (296) is the definition of effective aggregate labor supply.

Next we have the first-order conditions for domestic and foreign bonds Ba,tN/Ba,tT33 and Fa,t:

Together these yield the uncovered interest parity condition

To write the marginal utility of consumption λa,t in terms of quantities that can be aggregated, specifically in terms of consumption, we use (287) and (290) in (288) to get

We use (300) in (297) to obtain the generation specific consumption Euler equations

APPENDIX 3

Consumption and Wealth

The key equation for OLG households is the one relating current consumption to current wealth. We start deriving this by reproducing the budget constraint:

We now derive an expression that decomposes human wealth into labor and dividend income. First, we note that after-tax wage income can be decomposed as follows:

The first expression on the right-hand side of (304) is the labor component of income, which equals the marginal value of the household’s entire endowment (one unit) of time. The second expression in (304), by (290), can be rewritten as

which can be combined with the consumption expression in (303) to obtain, on the left-hand side of (303), Ptca,tOLG(ptR+ptCτc,t)/ηOLG. The second component of income is dividend, remuneration for bankruptcy monitoring, and net transfer income net of redistribution to LIQ agents, the expression for which can be simplified by noting that in equilibrium all firms in a given sector pay equal dividends, so that we can drop the firm specific index and write 01Da,tj(i)di=Da,tj. We also assume that per capita dividends, remuneration payments for bankruptcy monitoring, and net transfers received by each OLG agent are identical. Finally, we incorporate the assumption that a share of dividend and net transfer income is redistributed to LIQ agents:

These assumptions imply

The preceding arguments imply that total nominal wage and dividend income of households of age a in period t is given by

We now rewrite the household budget constraint as follows:

We proceed to derive a condition relating current consumption to lifetime wealth through successive forward substitutions of (309). In doing so we use the arbitrage condition (298) to cancel terms relating to foreign bonds. After the first substitution we obtain

and successively substitute forward in the same fashion. We impose the following no-Ponzi condition on the household’s optimization problem:

Furthermore, we let

This expression denotes nominal financial wealth inherited from period t – 1. Next we define a variable HWa,t denoting lifetime human wealth, which equals the present discounted value of future incomes Inct. We have

Further forward substitutions on (310), and application of the transversality condition (311), then yields the following:

The left-hand side of this expression can be further evaluated by using (301) for all future consumption terms. We let

Then we can write

The infinite summation on the left-hand side is recursive and can be written as

so we finally obtain

We want to express this equation in real aggregate terms. We begin with real aggregate human wealth, denoted by hwt:

We break this down into its labor income and dividend income components hwtL and hwtK. For hwtL we have

where we have used (293) and (295). In recursive form, and scaling by technology, the last equation equals

For hwtK we have, using (307) and letting dtj=Dtj/Pt,

which has the recursive representation, again after scaling by technology, of

Finally, we have

Next we aggregate over the financial wealth of different age groups. We note here that aggregation cancels the 1/θ term in front of the bracket in (312). This is because the period by period budget constraint (303) from which (312) was derived is the budget constraint of the agents that have in fact survived from period t – 1 to t. Aggregation has to take account of the fact that (1 – θ) agents did not survive and their wealth passed, through the insurance company, to surviving agents. Noting that B−1, t−1 = 0, we therefore have34

For total nominal financial wealth, we therefore have

To express this in real terms, we define the real domestic currency asset stock as bt = Bt/Pt. We adopt the convention that each nominal asset is deflated by the consumption based price index of the currency of its denomination, so that ft = Ft/Pt(Ñ). With the real exchange rate in terms of final output denoted by et = εtPt(Ñ)/Pt, and after scaling by technology and population, we can then write

Finally, using (318)-(323) we arrive at our final expression for current period consumption:

The linearized form of the aggregate equation (324) can instead be derived by linearizing an individual age group’s budget constraint, using its linearized optimality conditions, and then aggregating over all generations. As mentioned above, it is therefore appropriate to use the expectations operator Et in nonlinear equations as long as it is understood that this is valid only up to first-order approximations of the system.

APPENDIX 4

Manufacturers

The objective function facing each manufacturing firm in sectors J ∈ {N, T} is

The price (and inflation) terms in the two sectors will be indexed with J˜{N,TH}. Then dividend terms are given by

Optimization is subject to the equality of output with demand

We also have the following adjustment costs:

We write out the profit maximization problem of a representative manufacturing firm in Lagrangian form. Terms pertaining to period t and t +1 are sufficient. We introduce a multiplier ΛtJ for the market-clearing condition F(KtJ(i),UtJ(i),XtJ(i))=(PtJ˜(i)PtJ˜)σJZtJ. The variable ΛtJ equals the nominal marginal cost of producing one more unit of good i in sector J. We have

We take the first-order condition with respect to PtJ̃(i) and then impose symmetry by setting PtJ̃(i)=PtJ̃ and ZtJ(i)=ZtJ because all firms face an identical problem. We let λtJ=ΛtJ/Pt and rescale by technology. Then we obtain

For UtJ(i), XtJ(i), ItJ(i), and KtJ(i) we have

where we have used

APPENDIX 5

Entrepreneur’s Problem - Lognormal Distribution

Basic Properties of Γ and G

We first repeat the expressions for Γ and G here for ease of reference:

Then we have

Basic Properties of the Lognormal Distribution

The assumption is that ωtJ is lognormally distributed with E(ωtJ)=1 and Var(ωtJ)=(σtJ)2. This implies the following:

Derivations

We will change integrands at various points in order to obtain solutions that can be expressed in terms of the cumulative distribution function Φ of the standard normal distribution. We begin by defining terms:

Manipulating the second expression in each case gives the following expressions:

Using (338)-(342) we can now evaluate the expressions determining Γ and G in terms of the c.d.f. Φ(.). We start with

Next we have

To summarize:

Final Equation System

The entrepreneur’s optimal loan contract condition (100) determines the equilibrium return to capital reˇtk,tJ, the lender’s zero profit condition (101) determines the lender’s gross profit share Γt+1J, and the net worth accumulation condition (109) determines the entrepreneur’s net worth nˇtJ. The conditions derived in this appendix close the system. To summarize, we have:

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1In general we allow for the possibility that agents may be more myopic than what would be suggested by a planning horizon based on a biological probability of death.
2We use the term liquidity-constrained agents, but this could also include agents that simply choose to consume all of their income. In the literature these agents are commonly referred to as rule-of-thumb consumers or hand-to-mouth consumers. This is important for interpreting the calibration of the model because we will be using higher estimates of the shares of these agents than is consistent with micro data on the share of agents in the economy that do not have any access to credit markets.
3The alternative of using habit persistence to generate consumption inertia is not available in our setup. This is because we face two constraints in our choice of household preferences. The first is that preferences must be consistent with balanced growth. The second is the necessity of being able to aggregate across generations of households. We are left with preferences that, while commonly used, do not allow for a powerful role of habit persistence.
4For flexible model calibration we allow for the possibility that OLG households attach a different weight ηOLG to consumption than liquidity constrained households. This allows us to model both groups as working during an equal share of their time endowment in steady state, while OLG households have much higher consumption due to their accumulated wealth.
5Except for the special case of lump-sum taxation.
6The most recent version of GIMF is symmetric in that it also allows for a nonzero foreign exchange risk premium payable by country Ñ.
7The turnover in the population is assumed to be large enough that the income receipts of the insurance companies exactly equal their payouts.
8Declining income profiles are necessary to eliminate excessive sensitivity of human wealth to changes in the real interest rate, see Faruqee and Laxton (2000). In models with exogenous labor supply and stationary population shares it can also be shown that declining productivity profiles can be calibrated to produce identical macro behavior as more plausible hump-shaped life-cycle productivity profiles.
9It is sometimes convenient to keep these two items separate when trying to account for a country’s overall fiscal structure, and when allowing for targeted transfers to LIQ agents.
10Without this assumption consumption tax revenue could become too volatile in the short run.
11We adopt the convention throughout the paper that all nominal price level variables are written in upper case letters, and that all relative price variables are written in lower case letters.
12The distinction of generations could be dropped as all agents must act identically.
13There are also some small sales of aggregate manufacturing output back to manufacturing firms, related to manufacturers’ need for resources to pay for adjustment costs.
14Note that, for the sake of clarity, we make a notational distinction between two types of elasticities of substitution. Elasticities between continua of goods varieties, which give rise to market and pricing power, are denoted by a σ subscripted by the respective sectorial indicator. Elasticities between factors of production, both in manufacturing and in final goods distribution, are denoted by a ξ subscripted by the respective sectorial indicator.
15The factor T is a constant that can be set different from one to obtain different levels of GDP per capita across countries.
16In all other instances of nominal rigidities that follow, GIMF offers this as one option. It will however not be mentioned again in this document.
17Note that, unlike other adjustment costs, this expression treats lagged inputs as external. This has proved more useful than the alternatives in our applied work.
18The tradables market clearing condition is reported for the example of country 1.
19Any value of capital is profit maximizing.
20This follows Christiano, Motto and Rostagno (2007), “Financial Factors in Business Cycles”. Papers where the model is linearized prior to solving it only require the elasticity σa of the function a(ut). Because for some applications GIMF is solved in nonlinear form we require a full functional form.
21Note the absence of expectations operators because this equation has to hold in each state of nature. Likewise for subsequent equations.
22In DYNARE this will have to be replaced by the complementary error function unless the Statistical Toolbox is available.
23Dividend related shocks are easier to calibrate as they are already in terms of a share of gross returns on net worth.
24Home bias in tradables use depends on the parameter αTH and on a similar parameter αDH at the level of final goods imports.
25For the ratio tT we assume as usual that the distributor takes the lagged denominator term as given in his optimization.
26The presence of the growth terms ensures that adjustment costs are zero along the balanced growth path.
27For applications of the model where unit root processes are not allowed for, potential GDP (and potential tax bases) can simply be evaluated at their non-stochastic steady state.
28Under many calibrations of GIMF such rules exhibit superior properties to automatic stabilizers.
29In quarterly versions of GIMF this is replaced by a one-year-ahead four-quarter geometric moving average of inflation.
30Inflation persistence in the model is therefore exclusively due to inflation adjustment costs.
31The trade spillover parameters spillT, spillI and spillC are calibrated at 0.175.
32For simplicity we ignore money given the cashless limit assumption. We also treat some stochastic parameters like βt as constants.
33With ξtb=0 the condition for government bonds is identical. When ξtb0 we assume that the private sector will absorb all government bonds irrespective of their return relative to private sector bonds. Recent versions of GIMF have not used shocks to ξtb because an external financing premium arises endogenously with a financial accelerator sector.
34Take the example of bonds held by those of age 0 at time t−1. Only θ of those agents survive into period t, but those that do survive obtain 1/θ units of currency for every unit they held in t − 1. Their weight in period t bonds aggregation is therefore θ1θ=1.

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