Gender inequality is one of the biggest challenges facing Indian policy makers trying to achieve faster, sustainable, and more inclusive growth. In particular, division along gender lines in the labor market is one of the key concerns. For instance, India’s female labor force participation rate is the third lowest in the South Asian region at 27 percent in 2011-12; and less than one-third of male labor participation of 84 percent. In addition, they receive lower wages, are overrepresented in informal and unpaid domestic work, with gender gaps existing along several other dimensions including education, access to productive inputs, and bargaining power at home (Figure 1).
Figure 1:Gender inequality remains high in India
Note: The figure shows the female-to-male ratios for all variables, except for unpaid domestic work which is the male-to-female ratio.
Notwithstanding rising education levels amongst women in India, female labor participation has been declining and fell from 34 percent in 1999-00 to 27 percent in 2011-12, in both rural and urban areas (Figure 2). Lack of decent quality jobs in the formal economy discourage female participation. More than 80 percent of the workforce is employed informally in India, and among the ones that are employed in the formal sector, females constitute only 19-20 percent (Consensus 2011). Rigidities in the labor market due to strict regulations have been identified as the main drivers of this large informality1.
Figure 2:Falling female labor participation even in periods of economic growth
Source: World Bank Gender Data Portal
A considerably vast empirical literature finds a negative impact of gender inequality in employment and education on economic growth2. Indeed, researchers have attempted to model gender inequality within macroeconomic frameworks, to study the effect of a number of gender-specific public policies on both gender and the overall economy3. However, much of this literature has focused on how the relationship between gender and growth is mediated through changes in female labor force participation rate and productivity. Labor force participation rates only provide a partial analysis of women’s work, especially in India, where women are more likely to be engaged in vulnerable, low paid jobs in the informal sector. Hence, these frameworks are not rich enough to capture the impact of policies on the quality of female employment, wages, and unemployment, all of which impact economic growth.
Motivated by these observations, we ask the following questions: What is the interaction between informality and gender in the labor market? How do labor and gender-targeted policies affect female labor force participation, females’ access to formal employment, gender wage gaps, as well as aggregate economic activity? In this regard, we build an open economy dynamic stochastic general equilibrium (DSGE) model with both informality and gender inequality in the labor market. The model is estimated using Bayesian techniques and applied to quarterly data from India.
The key contribution of this paper is to link the issue of gender inequality to informality within a unified theoretical framework. In addition, in contrast to the existing literature, which has largely focused on long run outcomes alone, we are instead able to capture the short run effects of policy as well.
For our analysis, we consider an economy where two goods are produced, a market-good and a home-good. Firms in the formal and informal sector combine labor with capital to produce different varieties of market goods. There are labor market rigidities (modeled as hiring costs) in both sectors, where informality results from significantly higher rigidities in the formal sector, and wages are determined through Nash bargaining between workers and firms4. Home goods are instead produced by individuals working at home in home production5. Households consist of males and females, where the labor supply decision of each individual is an outcome of an optimal allocation among paid market-good production, unpaid home-good production, job search, and leisure; while also being dependent on their relative intra-household bargaining power6.
We use the findings from the empirical literature to model gender inequality by imposing several frictions on workers’ labor supply and demand7. Constraints on female labor demand are modeled as: (i) lower education (skill) level of female relative to male workers (World Bank (2013a)); (ii) lower bargaining power in the wage bargaining process (ILO (2011)); and (iii) firm-based gender bias against females in formal employment (ILO (2012), Campbell and Ahmed (2012), OECD (2008)). Whereas, constraints on female labor supply are modeled in the utility function as: (i) lower preference for working outside home in paid market-good production relative to staying at home, which corresponds to concerns regarding female safety and mobility in developing countries (World Bank (2011)); and (ii) higher preference for working in home-production relative to leisure, which is related to social norms and the lack of public provisions, including childcare support (Duflo (2012)).
Using this framework, we study and quantify the impact of a number of gender-based policies on aggregate economic activity (GDP, formality in the labor market, and unemployment), and on labor market indicators of gender inequality (female labor force participation, female formal employment, and gender wage gaps). Increase in female education and stronger enforcement of laws against gender discrimination which lower constraints on female labor demand, as well as increase in public provisions and improvements in female safety and mobility which lower constraints on female labor supply are considered as gender-specific policies. Finally, we also investigate the effects of increasing labor market flexibility in the formal sector, i.e. lowering labor regulations.
Policy analysis yields three key insights. First, gender-targeted policies boost female labor force participation, leading to gains in GDP in the long run. However, owing to the presence of labor market rigidities, these policies do not generate sufficient job creation in the formal sector, resulting in a large proportion of the increased female participants either being employed in low paying informal jobs or staying unemployed. This further widens gender gaps in wages and informal employment, thus worsening gender inequality, while also increasing aggregate informality and unemployment in the economy. For instance, an increase in public provisions that increases female participation by 1.5 percent would increase GDP by 1.25 percent, but it would also increase unemployment and informality by 1 and 0.1 percent, respectively. However, we do find one exception with the policy that increases female education, which by increasing female workers’ efficiency in employment, leads to an increase in female formal employment. This in turn increases the overall productivity in the formal sector, leading to creation of more formal jobs for both males and females. However, due to slow reallocation of workers to the formal sector, female formality falls and gender wage gaps are higher in the short run.
Second, lowering formal regulations that increase labor market flexibility, allows more women and men to be employed in the formal sector, which lowers aggregate informality and unemployment in the economy boosting GDP in the long run. However, male workers gain more, as unchanged constraints on female labor supply and demand along with a positive household income effect, both lower female labor force participation (as opposed to increasing in the short run and falling marginally in the long run for males) and lead to a smaller increase in female formality in comparison to males. For instance, lower labor regulations that decrease informality by 1.5 percent would increase GDP by 2 percent and lower unemployment by 1.5 percent, but it would also lower female participation by 0.5 percent.
Finally, we show that combining gender-targeted policies that lower constraints on female labor participation with reforms that boost formal job creation not only improves gender equality in the labor market but also leads to significantly larger gains in GDP, employment and formality; along with minimizing any short run losses.
The remainder of this paper is organized as follows. Section 2 presents a description of the previous literature. In Section 3 we outline the theoretical framework, and Section 4 discusses the data, calibration and method of estimation. In Section 5, we discuss results of the estimation and robustness checks. Section 6 presents an analysis of several policy experiments. Section 7 concludes the paper.
2. Literature Survey
Now we turn to comparing our work with the existing theoretical policy literature on gender inequality. A considerably vast literature has investigated the effects of gender-specific policies with competitive labour markets. This literature can be broadly classified into talent allocation models (Cuberes and Teigner (2014)), occupational choice models (Esteve-Volart (2009), Hsieh et al. (2013)), overlapping generations (OLG) model (Galor and Weil (1996), Cavalcanti and Taveres (2008), Agenor and Canuto (2013), Agenor (2015)), and computable general equilibrium (CGE) models (Fontana and Wood (2000), Fontana (2004), Hendy and Zaki (2010)). Female labour supply in these models is often modeled using the framework of the time allocation model8, where women’s labour supply decision is based not only on the trade-off between leisure and labour, but also on home production modeled as investment in childcare. These studies analyze the impact of one or more of the following policies: increase in female education, increase in childcare provisions, better access to infrastructure, as well as fall in exogenously given gender wage discrimination. In sum, their results suggest that each of the above policies increase female labour participation that has a positive impact on their productivity, hence leading to higher growth. One or more of the following channels drive these results: (i) higher female labour participation increases female employment (under the assumption of flexible labour markets) and females’ labour income; (ii) this increase in females’ income improves the average human capital (skill) in the economy, as females are assumed to invest more in children’s education relative to males; and (iii) higher participation in turn has a direct effect on per capita GDP, as females move from unpaid home-good production (not accounted for in GDP) into market-good production.
However, this existing literature has not paid attention to how the presence and effects of labour market rigidities (i.e. regulations) vary by gender. In addition, much of the above analysis is based on small, illustrative models (with notable exceptions such as Agenor (2012, 2015)). Thus, they may not be adequate for policy analysis as important channels are ‘not modeled’ or ‘shut down’ - by imposing for instance exogenously given wages, exogenously given wage gaps in gender, in addition to assuming labour market flexibility. Hence, while this literature has significantly improved our understanding of the various links between gender equality and growth, the relevance of any policy-related study strongly depends on whether the specific model used to draw recommendations captures all the key transmission channels of policy or not. Moreover, the computable general equilibrium (CGE) modelling technique9 is commonly used to capture the general equilibrium effects of gender-specific policies. Compared with the more recent general equilibrium modelling strategies, CGE models are mainly non-stochastic and static. Thus, while they are useful for quantifying the long run effects of reforms, they do not take into account the dynamic impact and the interplay between macroeconomic policies and gender inequality.
We are only aware of one recent study by Albanesi and Patterson (2014) who model gender differences in labour force participation rates within a New Keynesian framework. However, they abstract from analysing gender-specific policies and instead focus on the impact of changes in female labour force participation on business cycles. Although the goal of their study is different to ours, they do highlight the relevance of using a DSGE framework for gender-related policy study.
Our model builds on their framework by adding a number of relevant frictions on the labour supply and demand of female workers, and by integrating this with the literature on labour market rigidities to model informality10.
3. The Model
This section presents the Baseline model. We provide a brief description before specifying the details of the model in the following subsections.
The small open economy consists of households, wholesale producers, retailers, capital producers, and a government. Two goods are produced in the economy: market-good and a home-good. Market-good consist of formal tradable goods (F), informal non-tradable goods (I), and imported goods (f*). The first two are produced domestically by formal and informal retailers in each sector sε (F, I), respectively, while the latter is produced in the foreign economy and sold domestically by import retailers in the formal sector. On the other hand, home goods (H0) are produced by individuals of the household who work at home, and is for household consumption only.
Households consist of male (m) and female (f) members who derive utility from consuming market goods, home goods, and leisure. Each member either supplies labor (i.e. participate in the labor market) to wholesale firms or instead stays at home. The ones that participate in the labor market are either employed in the formal sector, employed in the informal sector, or stay unemployed. The employed engage in paid market-good production, whereas the unemployed work in unpaid home-good production in the residual time when unoccupied by job search. On the other hand, the ones that stay at home, are either working in home-good production, or consuming leisure.
Formal and informal wholesale firms combine labor with capital to produce formal and informal wholesale goods, respectively. Unemployment exists as wholesalers in each sector pay a hiring cost when hiring new labor a laBlanchard and Gali (2006). Wages in each sector are determined through Nash bargaining between workers and firms.
Formal and informal retailers purchase wholesale goods from wholesalers, differentiate these into different varieties of market-goods, and set the retail price for each individual variety in an environment of monopolistic competition and price adjustment costs a laRotemberg (1982). A group of competitive capital producers combine formal market- and imported goods to produce final investment goods, which is then combined with the used capital goods rented from wholesalers to produce new capital. Government conducts monetary and fiscal policy: it sets the nominal interest rate using a Taylor-type rule, and receives tax wage income from households which is used to finance public spending and unemployment benefit payments.
Details regarding each agent’s behaviour are described below.
3.1. The Labor Market
There are a continuum of households (0,1), out of which pm proportion are males, and pf = 1 − pm proportion are females11. Households either supply their labor to wholesale firms, which determines the labor market participation rate, or stay at home forming the pool of non-participants.
Hence, there are two types of workers hε(m, f) in the labor market where m denotes male workers and f denotes female workers. They can either be employed in one of the two sectors sε(F,I), where F is the formal sector and I is the informal sector, or stay unemployed. The mass of male workers who are employed in the formal sector, employed in the informal sector, and unemployed, are denoted by
The pool of male and female workers who participate in the labor market is then given by:
The labor market dynamics closely follow the framework in Campolmi and Gnochhi (2014). The stock of employed labor varies because of the endogenous variation in hiring, and an exogenous probability of getting fired, σs, every period12. At the end of period t − 1, after all decisions have been taken and executed,
where male and female workers’ probability of getting hired,
The unemployed, the non-participants, and fired individuals,
Eq. 3.10 implies that in period t, total female workers employed in the formal sector increases with higher female labor participation,
Similarly, for the informal sector (s = I), we get:
Probability of getting hired in sector s is then given by the ratio of new hires to the pool of job searchers:
Ratio of total job searchers to the pool of individuals not employed at the end of period t − 1 determines the probability of searching for a job:
3.2. Wholesale Producer
We have a continuum of wholesalers (0,1) in each sector s producing different intermediate goods,
where Ψs is the capital intensity related to capital income share in sector s. They sell their goods to retailers in their respective sectors at a price of
Total labor in each sector is a constant elasticity of substitution (CES) aggregate of male and female workers, where
ωs,tε(0,1) is the firms’ relative preference for male workers over female workers in sector s19. Differences in skill level of male and female workers,
Labor Market Regulations
Wholesalers in each sector s face real hiring costs,
Capital and Labor Demand
Wholesalers in sector s choose
Capital and labor demand functions in sector s are obtained from the first order conditions as follows (see Technical Appendix for derivations):
The equation for capital demand (Eq. 3.25) is standard in the literature, however, the labor demand for males and females (Eq. 3.26 and Eq. 3.27) is now determined by equating marginal product to the marginal cost of employing labor, which includes the real wage plus the cost generated by hiring.
Wage setting follows a Nash bargaining process between workers and wholesalers where wage bargaining power of worker h in the formal and informal sector is denoted by
Similarly, we get the value of being employed in the informal sector, the only difference being that the worker does not pay wage income tax, τI = 0:
An unemployed worker receives social benefits today, WU,t, and spends τUε(0,1) proportion of her time in home-good production, while the remaining time (1 − τU) is spent in searching for jobs23. In the next period, there is a probability
A non-participant household member, either works in home-good production with probability
Value of working in home-good production,
An unemployed worker h has a utility gain of
Following the derivations in Blanchard and Gali (2006), sector s wholesalers’ value of hiring an additional worker h in period t,
and the equation determining wages,
We derive the expressions for wage rate of male and female workers in the Techinacal Appendix and define average male and female wages,
where ratio of average male wages to average female wages,
A continuum jF and jI of monopolistically competitive formal and informal retailers buy wholesale goods to produce different final market-good varieties, YF,t (jF) and YI,t (jI), and sell these at different prices, PF,t(jF) and PI,t(jI), respectively27.
Total composite output in each sector s, Ys,t, produced by retailers is a Dixit-Stiglitz (1977) CES aggregate of different varieties of goods produced by individual retailers, Ys,t (js).
εs stands for the elasticity of substitution between different varieties of goods. The corresponding price of the composite consumption good, Ps,t is:
The demand function facing each retailer can be written as:
Formal final good, YF,t, is exportable where it is consumed both domestically
Retailer js sets its price, Ps,t (js) that maximizes its expected discounted stream of future profits:
where the one-period profit in the formal sector,
Per-period profits of informal retailers are similar, except that the informal sector only sells its goods domestically:
The first order condition of the retailer optimization problem determines the price in each sector s (refer to Technical Appendix):
The households aggregate utility function is a weighted sum of male utility,
β is the nominal discount rate and BPtε (0, 1) is the endogenously determined intra-household bargaining power of males relative to females29. Following Klaveren (2009), BPt is an increasing function of male to female wage income ratio given by:
Bargaining power of male increases with an increase in his own steady state wage income, whereas it decreases with a rise in the steady state wage income of females.
Each member derives utility from consuming market-produced goods Ct, home-produced goods
Market and home consumption are public goods, and there is risk sharing within the household, so that all its members - males and females, consume the same amount of these goods. The disutility of working, on the other hand, accrues to each member individually. Therefore, males do not get any utility from female leisure and vice-versa. Ct denotes aggregate consumption at time t, while Ct−1 is the average level of consumption in t − 1, where hc ∈ [0, 1) is the external habit formation parameter.
Home goods are produced by males and females working in home production (home workers),
θHt is the exogenous AR(1) shock to home productivity31. Intra-household bargaining power, BPt, determines the weight on female relative to male workers in home-good production, where higher the bargaining power of males at home, i.e. higher BPt, lower is the weight on male workers in home production.
The corresponding partial derivatives with respect to male and female leisure and home-work are:
Eq. 3.47 suggests that females’ leisure choice influences the household utility directly through the utility function of the female,
Aggregate Market-good Consumption
Aggregate market-good consumption, Ct consists of domestically produced market goods, CD,t, and imported market goods, Cf*,t (in terms of domestic currency), and is given by the following Dixit-Stiglitz (1977) aggregator:
where α ∈ (0, 1) can be interpreted as a measure of domestic bias in consumption, and η > 1 is the elasticity of substitution between domestic and foreign goods.
Aggregate price level Pt can be expressed as a composite of domestic price PD,t and import price Pf*,t, and is given by the following CES form:
Domestic market-good consumption is a composite of formal market-good consumption, CF,t, and informal market-good consumption, CI,t expressed as:
where wε (0, 1) is the weight on formal sector market-good, and μ > 1 is the elasticity of substitution between the goods produced in the two sectors. Then, aggregate domestic market-good price, PD,t, is determined by:
By minimizing household expenditure on the total composite demand, we can derive the following optimal consumption demand functions for aggregate domestic and imported market goods:
Similarly, we derive the optimal consumption demand functions for domestically produced formal and informal market-goods:
The representative household enters period t with one period (real) foreign and domestic bonds,
The resulting first order conditions with respect to Ct, Bt, and Dt yield the standard Euler equation for consumption (see Technical Appendix):
The remaining first order conditions for
Finally, probability that a non-participant household member h consumes leisure,
3.5. Capital Producer
Capital producers combine the existing undepreciated capital stock, (1 − δK)Kt − 1, leased from wholesalers, with investment goods, It, to produce new capital Kt, using a linear technology. The capital-producing sector is perfectly competitive. Capital evolves according to the following equation:
Capital production is confined to the formal sector, and investment is thus a composite of domestic formal goods and foreign imports:
and the price of investment is:
We assume that it is in the same proportion as in the consumption basket (Eq. 3.52 and Eq. 3.53), except that now weight on formal good is w = 1. Hence, optimal demand for domestic and imported investment goods is:
The capital producer invests such that its profit is maximized, where Qt is the real price of capital:
The corresponding first order condition w.r.t. to the choice of It determines the capital supply equation (see Technical Appendix):
This is the Tobin’s (1969) Q equation relating the price of capital to marginal adjustment costs. In the absence of capital adjustment costs (κ = 0), the price of capital is constant and equal to one.
Demand for capital by wholesalers in sector s must satisfy the following condition:
3.6. Rest of the World
Foreign economy consumes domestic formal exports,
The demand for domestic exports by the foreign economy is assumed to have a similar structure to that of domestic consumption in Eq. 3.52:
Following Schmitt-Grohe and Uribe (2003), interest rate on foreign bond,
This is a standard assumption in the small open economy literature35.
3.7. Government Policy
Government consists of monetary and fiscal authorities. The monetary authority sets the nominal interest rate, it, based on a Taylor-type (1993) feedback rule. It responds to deviations in inflation and gross domestic product:
where αi captures interest rate smoothing, and the Taylor rule coefficients, απ and αY, are the relative weights on inflation and output stabilization respectively. i, π, and Y are the steady state values for nominal interest rate, inflation, and gross domestic product. εi,t is a monetary policy shock to capture unanticipated changes in the nominal interest rate.
In addition, the fiscal authority finances its consumption, Gt, and unemployment benefit payments by taxing wage income in the formal sector36. The government budget constraint every period is:
We assume that exogenously given government consumption basket, Gt, analogous to the investment basket in Eq. 3.60, consists of domestic formal market goods, GF,t, along with foreign imports, Gf*,t (in domestic currency):
Optimal demand for domestic formal, GF,t, and imported government consumption, Gf*,t, is given by:
3.8. Market Clearing and Aggregation
Sum of employment in the formal, LF,t, and in the sector, LI,t, is equal to aggregate employment Lt in the economy: LF,t + LI,t = Lt. Aggregate labor force participation in the economy (i.e. aggregate labor supply in the economy), Pt is a sum of the male and female labor participation:
Equilibrium in the labor market for males and females is given by equating aggregate supply of male and female labor,
Male and female unemployment is given by the ones searching for a job minus the ones that get hired:
Equilibrium in the asset market implies that the total number of bonds issued is equal to the cost of desired capital in the economy:
The resource constraint for the formal sector is:
where total demand for formal good, YF,t, is the sum of its domestic demand by households, capital producers and government,
Similarly, the resource constraint for the informal sector is:
where informal-market good is only consumed by domestic households,
Total foreign imports is given by the sum of imports by households, capital producers, and the government,
3.9. Shock Processes
We include fourteen exogenously given shocks in the economy: thirteen domestic, and two determined in the rest of the world. Domestic shocks include the following gender-specific shocks which form the basis of our policy experiments relating to gender-targeted policies: shock to male gender bias in formal employment (ωF,t), productivity of home production (θH,t), skill efficiency of female workers (
4. Estimation Methodology
This section describes our data, calibration approach, and presents details regarding the main estimation procedure for India. In order to evaluate the performance of the model, we use a combination of calibrated and estimated parameters. We choose to calibrate some parameters, as these are more important in matching the first moments of the Indian data, and estimate the remaining using Bayesian approach in Dynare.
To estimate the model, we use information on nine key macroeconomic variables for India: GDP, private consumption expenditure, investment, government consumption expenditure, exports, imports (all expressed in constant prices), the real exchange rate, the wholesale price inflation (WPI), and the nominal interest rate. The 3-month Treasury bill rate is used as a proxy for the nominal interest rate, and the real effective exchange rate (REER) is used as a proxy for the real exchange rate. The sample runs from 1996Q1 to 2012Q1, which gives us 65 observations for each of the time series. Prior to estimation, GDP, exports, imports, consumption, investment, and government spending are transformed into real per capita measures. This is done to align the scale of our data, with the steady state of our Baseline model. We remove a time trend in the data using the Hodrick-Prescott (HP) filter to obtain the stationary series, and measure these in terms of the percent deviation from the steady state (i.e. the HP trends corresponding to each)38. In addition, we remove seasonal effects in the series using the X12 arima filter (except the real exchange rate, and the nominal interest rate). All data is taken from the CEIC database.
Table 1 and Table 2 summarizes the calibrated values of parameter in our model for India, where we calibrate a set of parameters, and the steady state values for some endogenous variables, which characterise the model economy.
|δK||0.025||capital depreciation rate|
|α||0.8||share of home-good in consumption|
|η||1.2||substitutability between domestic and foreign goods|
|π||4.5||gross inflation in the steady state (% annually)|
|π*||2.5||gross foreign inflation in the steady state (% annually)|
|0.11||government spending-to-GDP ratio in the steady state|
|WU/Y||0.014||social spending-to-GDP ratio in the steady state|
|0.19||export-to-GDP ratio in the steady state|
|0.21||import-to-GDP ratio in the steady state|
|μ||1.5||substitutability between formal and informal goods|
|w||0.39||share of formal goods in consumption|
|4.5||price elasticity of exports|
|ψF||0.34||capital share in formal production function|
|ψI||0.34||capital share in informal production function|
|1.2||price mark-up in formal sector|
|1.09||price mark-up in informal sector|
|1.5||relative formal-to-informal productivity|
|3||share of formal hiring costs in formal wages|
|0.5||share of informal hiring costs in informal wages|
|σF||0.1||formal worker firing rate in steady state|
|σI||0.75||informal worker firing rate in steady state|
|1.2||male’s Frisch elasticity of labor supply|
|3.61||female’s Frisch elasticity of labor supply|
|2.5||substitutability btw male & female formal workers|
|5||substitutability btw male & female informal workers|
|1.7||male-to-female skill ratio in employment|
|0.67||bargaining power of male formal worker|
|0.27||bargaining power of male informal worker|
|0.46||bargaining power of female formal worker|
|0.02||bargaining power of female informal worker|
|ωF||0.62||male gender bias in formal employment|
|ωI||0.5||male gender bias in informal employment|
|0.7||male utility weight on leisure|
|0.5||female utility weight on leisure|
|ϕm||0.7||male utility weight on staying at home|
|ϕm||1||female utility weight on staying at home|
As in much of the literature, the depreciation rate of capital, δK, is set at 10 percent per annum, implying a quarterly value of 0.025. Steady state inflation, π, is 4.5 percent which corresponds to the average seasonally adjusted quarterly WPI over this period on an annualized basis. The discount rate β is set at 0.994 which corresponds to an annual nominal interest rate, i of 7 percent, matching the mean of the sample. Foreign inflation, π*, is 2.5 percent annually, which corresponds to an annual foreign interest rate, i*, of 5 percent39. The depreciation rate of the nominal exchange rate, dep is calculated at 2 percent on an annual basis.
The share of government expenditure in GDP is calibrated at 11 percent, as in the data. In 2005, the Government of India spent 1.4 percent of its GDP on social protection, which forms the basis of our calibration for unemployment benefits to GDP ratio40.
The substitution elasticity between imported and domestically produced goods, η, is set at 1.2, close to the value estimated by Medina and Soto (2005) for Chile, and Castillo et al. (2006) who obtain values close to 1. This combined with the share of domestically produced goods in the market consumption basket, α, at 0.8, corresponds to a steady state import to GDP ratio of 21 percent, as in the data. Elasticity of substitution of exports,
Matching Informality Statistics
Next, we turn to parameters relating to the formal and informal sector. Because of the scarce empirical evidence on informality, our calibration strategy aims to match, as accurately as possible, the empirical evidence, and available data on key statistics relating to the two sectors in India.
Using industry level panel data for the period 1980-2007, Pal and Rathore (2013) estimate the size of the firms’ mark-up in India to have a long run average of 1.19 during 2000-07. Thus, the elasticity of substitution among different retail varieties, εF and εI, are calibrated at 6 and 12, so that the retail firms’ desired mark-up is pinned down at
Based on the estimates of share of compensation of employees in Chandrasekhar and Ghosh (2015), we calibrate the cost share of capital in the wholesalers’ production function, ψF and ψI, at 0.34, for both sectors. As in Ulyssea (2009), the productivity of informal wholesalers, θI is normalised to 1, whereas the productivity of the formal firms, θF, is 1.5 capturing a productivity differential of 50 percent between the two sectors42. Productivity and labor intensity of home production is assumed to be the same as the informal sector, i.e, θH = θI, and (1 − ψH) is 0.6743.
According to the The Global Competitiveness Report published by the World Economic Forum (2014), the redundancy costs of workers in India is estimated to be equivalent to 55.9 weeks of annual salary since 2006, which is equivalent to 4.53 times the quarterly wage rate. Since in our model the hiring costs also reflect the difficulty in firing workers, we calibrate the hiring cost to wage ratio in the formal sector for both male and female workers at 3, which corresponds to 38 weeks of annual salary. Since only the formal sector is regulated, for the corresponding informal sector ratio, we assume it to be much lower at 0.5.44.
The unorganized sector employs nearly 84 percent of the Indian workforce according to the Employment and Unemployment Survey (EUS) of the National Sample Survey Organization (NSSO, 2009-10). Setting the exogenous probability of getting fired, αF and αI, at 0.1 and 0.75, gives us the informal employment share,
For the substitution elasticity between formal and informal goods, μ, we have chosen a value of 1.5 which matches values commonly used in the literature for the substitution elasticity between traded and non-traded goods. Then the formal goods bias in consumption basket, w, is set at 0.39, such that the share of informal sector output in GDP is obtained at 44 percent, close to the value of 49 percent estimated by the NCEUS (2009).
Matching Gender Inequality Statistics
Table 2 summarizes the calibration of gender-related parameters, which are chosen so as to match the Indian statistics on female participation, Pf, male participation, Pm, male formality in the labor market,
Plausible estimates for the substitution elasticity between female and male workers in production function of market goods,
We calibrate the ratio of skill level of males to female worker in each sector,
According to the World Economic Forum (2010), females earn 62 percent of the male’s salary for equal work, which implies a value of 1.62 for
According to the Global Gender Gap Report published by the World Economic Forum (2010), 86 percent of female workers were employed in the informal sector,
Household care work,
According to the NSSO report in 2009-10, female labor force participation rate,
4.3. Bayesian Estimation
We estimate the model using Bayesian approach in Dynare. This choice is driven by the widely recognized advantages of the Bayesian-Maximum Likelihood methodology, which are as follows48. First, prior information about parameters available from empirical studies or previous macroeconomic studies, can be incorporated with the data in the estimation process. Second, it facilitates representing and taking fuller account of the uncertainties related to models and parameter values. Third, it allows for a formal comparison between different mis-specified models that are not necessarily encapsulated in the marginal likelihood of the model. In addition, there has been a growing trend among central banks to employ Bayesian methods for conducting policy analysis.
Table 3 summarizes the choice of prior distributions for the estimated parameters. The prior densities for the estimated parameters are chosen by considering the theoretical restrictions for the parameters, and empirical evidence. Due to scarce empirical evidence on India, we choose relatively diffuse priors that cover a wide range of parameter values. The use of a diffuse prior reduces the importance of the mean of the prior distribution on the outcome of the estimation.
5. Empirical Results
Bayesian estimates for the parameters are summarized in Table 3, along with the 95 percent posterior confidence band. Looking at price adjustment costs, consistent with the estimates in Gabriel et al. (2010), the estimation indicates that price re-setting is highest in the informal sector (
Estimation results suggest a strong response of the Reserve Bank of India (RBI) to inflation deviations in the economy (απ = 3.2) and significantly lesser to output deviations (αY = 0.32), along with considerable policy inertia (αi = 0.86). The estimates for αY and αi are in the range of previous studies49, whereas απ is estimated to be slightly higher than the previous values in the literature, that range from 1.5 to 2.9 across different studies50. This may be the result of the combined shock-absorbing role of the informal sector and of low skilled female workers, because of which the RBI needs to be more aggressive in order to stabilize prices.
The persistence of most of the shocks in the economy are estimated to be high ranging from 0.58 to 0.97, with the exception of a few51. Posterior estimated means for the standard deviation of shocks are significantly higher than the corresponding prior means, which is consistent with higher business cycle volatility associated with emerging economies.
Overall, we obtain reasonable estimates for the common parameters used in other studies, in the sense that all of them are statistically significant and most of them are in the range of estimates in the existing studies on emerging economies relying on Bayesian DSGE models. Some of our estimates turn out to be away from the prior means, such as for
6. Policy Results
6.1. Gender-Specific Policies
In this section, we study the dynamic impact of the following gender-specific policies: (i) increase in public provisions (including childcare support); (ii) improvements in female safety and mobility; (iii) increase in female education; and (iv) stronger enforcement of laws against employment based gender discrimination. The former two policies lower constraints on female labor supply, whereas the latter two are targeted towards lowering female labor demand constraints.
To see how the economy transitions from the initial to the new steady state post-reform, we consider a permanent shock to the relevant variables. Table 5 summarizes the short run and long run impact of reforms on aggregate economic activity (i.e. GDP, formality in the labor market, and unemployment), and on labor market indicators of gender inequality (female labor force participation, female formality, and gender wage gaps).
Female Labor Supply Reforms
The impact of policies that lower constraints on female labor supply is a combination of its direct impact on female labor force participation, combined with its indirect impact on the same through changes in wages and employment in each sector.
(i) Increase in Public Provisions (including childcare)
Home production consists of both childcare and other household activities, where an increase in public provisions53 lowers the burden of home-work for all individuals in the household, which is captured by an increase in the home-production technology, θH. Childcare provisions, on the other hand, are assumed to be targeted towards females, which is captured by a simultaneous fall in
Figure 3 shows the combined effect of an increase in θH and a fall in
Figure 3:Increase in Public Provisions (including childcare)
Note: The figure shows a percentage deviation from the initial steady state for GDP, while for the rest of the variables the levels (%) are shown.
Higher productivity of home production implies that males and females can consume the same amount of home goods with lower amount of home-work. Substitution effect leads to an increase in male and female labor force participation in the long run. Childcare support targeted towards females (
More males relative to females get employed and are able to find formal employment, leading to a larger fall in female formality in comparison to that for males, which further widens wage gaps;
Qualitatively, the short run impacts of policy are similar to the ones in the long run. However, due to relatively higher rigidities, formal wholesalers increase hiring only gradually over time which increases unemployment and lowers aggregate, female and male formality by more in the short run.
(ii) Increase in Female Safety and Mobility
Figure 4:Increase in Female Safety and Mobility
Note: The figure shows a percentage deviation from the initial steady state for GDP, while for the rest of the variables the levels (%) are shown.
Qualitatively, the impacts on GDP and labor market indicators of gender inequality are similar to the ones with the policy of increased public provisions. However, there is an increase in gender inequality in relative burden of female-to-male home-work, as opposed to a decrease with the public provisions reform. Quantitatively, the impacts are now smaller - an improvement in female safety and mobility which increases female labor participation by 1.5 percent would lead to a 1.15 percent gain in GDP, however, it would also increase unemployment and informality by 0.5 and 0.2 percent, respectively. Below we provide a detailed analysis of the exact transmission channels involved.
Higher female safety and mobility lowers their disutility from working outside home, leading to a rise in their labor supply, Pf. Female labor supply curve shifts out, leading to a fall in female wages,
Home-work burden of females relative to males
Female Labor Demand Reforms
The impact of policies that lower constraints on female labor demand is a combination of its direct impact on female employment, and its indirect impact on labor market participation rates, through changes in wages and employment creation in each sector.
(iii) Increase in Female Education
Figure 5 shows the effect of an increase in the skill level of female workers in both sectors,
Figure 5:Increase in Female Worker Skill
Note: The figure shows a percentage deviation from the initial steady state for GDP, while for the rest of the variables the levels (%) are shown.
Increase in females’ skill level improves their productivity in the labor market, leading to wholesalers substituting out male workers for more productive female workers in both sectors.
More females are hired in the formal relative to the informal sector, thus increasing female formality in labor employment,
(iv) Lower Gender Discrimination in Formal Employment
Figure 6 shows the effect of a decrease in formal wholesalers’ relative preference for male relative to female workers in formal employment, ωF59. This corresponds to stronger enforcement of laws against gender-based discrimination in employment. Results indicate that gender equality in the labor market and at home improve, i.e. gender gaps in labor participation rates, wages, formal employment, and relative home-work burden are reduced. However, unless accompanied by higher female education, we see a fall in aggregate economic activity, i.e. GDP and formality are lower in the long run.
Figure 6:Lower Gender Discrimination in Formal Employment
With this policy, formal wholesalers are forced to substitute out higher skilled male workers for relatively lower skilled female workers, increasing female formality. This reduces overall productivity and profits in the formal sector, which shrinks aggregate formality and worsens GDP60.
Higher probability of getting formal employment increases females’ returns to job search, which raises female labor supply, Pf.61 In contrast to the female education reform which leads to an increase in wages of both males and females in the new steady state, we find that female wages are higher, but male wage rates in both sectors are now lower. This is because demand for female relative to male workers is higher in the formal sector, increasing
To summarize, the above results indicate the following. First, policies targeted towards reducing constraints on female labor supply, directly increase female labor force participation, reducing gender gaps in participation rates. However, due to labor market rigidities, these policies are unable to simultaneously generate enough employment creation in the formal sector, leading to an increase in aggregate informality and unemployment. In addition, given gender-specific constraints on females’ access to decent formal jobs, it also worsens gender gaps in wages and informal employment. Regarding gender dynamics at home, increase in public provisions by directly reducing females’ childcare burden improves gender division in home-work and leisure, whereas it worsens with higher female safety and mobility. Second, effects of the two policies targeted towards reducing female labor demand constraints vary. Increase in the skill level of females along with reducing gender gaps in participation rates, also simultaneously leads to adequate formal job creation due to increase in overall formal sector productivity. Aggregate formality and employment increase, resulting in higher formality in male and female employment, while also lowering gender gaps in wages and informal employment. However, female formality in employment falls in the short run. On the other hand, unless accompanied by an increase in female education, strengthening firm-based discrimination laws forces firms to hire lower skilled female workers, thus lowering formal sector productivity, which shrinks formality and GDP in the economy.
Comparing our results to the ones in the gender-based policy literature, the impact on female labor force participation and GDP is consistent with these studies, both qualitatively and quantitatively. However, our framework is additionally able to capture the impact on the female quality of employment, female unemployment rate, and gender wage gaps, which even though shown to be empirically relevant, is largely ignored in this theoretical literature.
6.2. Labor Market Deregulation
Figure 7 shows the combined effect of a decrease in formal wholesalers’ labor hiring cost,
Figure 7:Labor Market Deregulation in the Formal Sector
There are two opposing effects on male and female participation rates, Pm and Pf: (i) substitution effect: increase in job-finding rate in the formal sector produces higher returns to job search, increasing Pf and Pm, and (ii) household income effect: as more household members are employed in higher paying formal jobs, this increases total household income, which decreases Pf and Pm. For males, substitution effect outweighs the income effect in the short run, increasing Pm, whereas it decreases in the long run due to stronger income effects. In contrast, due to a smaller increase in female’s job-finding rate in the formal sector relative to males (due to gender-related constraints, including education gaps and discrimination by firms), the household income effect outweighs the substitution effect for females in both the short run and long run, decreasing Pf. Therefore, gender gaps in participation rates widen;
Increase in male labor supply, Pm, has a negative impact on their wages, whereas decrease in female labor supply, Pf, has a positive impact on female wages, thus lowering gender wage gaps. At home, a larger increase in aggregate wage income of males relative to females, leads to a fall in the intra-household female bargaining power, worsening gender division in homework;
6.3. Reform Interactions
Given the above policy outcomes, in this section, we ask the following question: what is the optimal strategy for implementing gender-specific reforms, one that will lead to gains in aggregate economic outcomes, and simultaneously improve gender equality in labor force participation, wages, and formal employment? We study the impact of implementing a combined package of reforms. Specifically, we look at three cases: (i) combining a female labor supply reform with a female labor demand side reform, (ii) combining a female labor supply reform with a labor market deregulation reform, and (iii) combining two female labor supply side reforms.
Figure 8 shows the effects of a simultaneous reform package (black solid line) combining improvements in female safety and mobility (blue dashed line) with increase in female education (red dashed line). In contrast to when only the former reform is implemented, which leads to a fall in female and aggregate formality, higher unemployment, and wider gender wage gaps in the economy, combining it with the latter reform instead overturns these adverse outcomes. It helps mitigate the short run fall in female formality associated with the education reform, while also leading to significantly higher gains in GDP. Better safety and mobility outside home increases female labor supply, Pf, and due to a simultaneous increase in their skills, which increases female labor productivity, a larger proportion of these females get hired by formal relative to informal wholesalers, who also pay higher wages. Higher probability of getting employed and higher wages reinforces the positive impact on female participation, increasing it by more than when either policy is implemented on its own. Both substitution and income effect lower male participation, Pm. At home, increase in female wage income improves their intra-household bargaining power, thus reducing their home-work burden;
Figure 8:Improvements in Female Safety and Skill
Note: The blue dashed line corresponds to increase in female safety and mobility, the red dashed line corresponds to the policy of increase in female skill, and the black solid line is the policy impact with a combination of both policies. The figure shows a percentage deviation from the initial steady state for GDP, while for the rest of the variables the levels (%) are shown.
Figure 9 shows the effect of a simultaneous reform package (black solid line) combining the policy of increase in female safety and mobility (blue dashed line) with a labor market deregulation reform (red dashed line). We find that in contrast to when the former policy is implemented on its own, unemployment is lower, both female and male formality increases, and gender wage gaps are lower. Moreover, female labor force participation increases, as opposed to a decrease when labor market deregulation policy is implemented on its own. Better safety and mobility of females outside home, directly increases their labor participation rate, where these females are now able to find high paying formal jobs, as the formal sector expands due to lower formal labor market rigidities. This expansion of the formal sector also increases male formality in employment, which leads to higher male participation, as their return from job search is now higher. Thus, GDP increases by more now. Moreover, it also increases in the short run as opposed to when only the deregulation reform is implemented, however, there is a larger short run increase in unemployment65. The larger increase in unemployment is caused by two factors: increase in both male and female participation, and slower creation of jobs in the formal relative to the informal sector.
Figure 9:Increase in Female Safety and Labor Market Deregulation
Note: The blue dashed line corresponds to increase in female safety and mobility, the red dashed line corresponds to the formal labor market deregulation, and the black solid line is the policy impact with a combination of both policies. The figure shows a percentage deviation from the initial steady state for GDP, while for the rest of the variables the levels (%) are shown.
Figure 10 shows the effect of a simultaneous reform package combining the policy of increase in female safety and mobility with increase in public provisions. Since both reforms lower female labor supply constraints, there is a larger increase in female labor force participation rate, in contrast to when either policy is implemented on its own. Both policies are associated with inadequate formal job creation, which reinforces their individual impact resulting in an even larger increase in informality and unemployment.
Figure 10:Increase in Female Safety and Public Provisions
Note: The blue dashed line corresponds to increase in female safety and mobility, the red dashed line corresponds to increase in public provisions (including childcare), and the black solid line is the policy impact with a combination of both policies. The figure shows a percentage deviation from the initial steady state for GDP, while for the rest of the variables the levels (%) are shown.
Therefore, we find that not all combined policy packages lead to higher gains. In particular, we find that there is a complementarity in the impacts of policies that lower constraints on female labor participation and policies that boost formal job creation, leading to gains in both gender equality and overall economic activity. Based on the above findings, we suggest that policies should be designed to prioritize getting females into paid work outside home (i.e. lower constraints on female labor supply), while at the same time making sure that there are enough formal job opportunities (i.e. lower labor market rigidities in the formal sector), and that females have access to these formal jobs (i.e. lower constraints on female labor demand).
In this paper, we investigate the interaction between informality and gender inequality in the labor market. Specifically, we study the impact of a number of gender-specific policies on female labor force participation, their access to formal employment, gender wage gaps, as well as on aggregate economic outcomes. To achieve this goal, we build a small open economy DSGE model by incorporating both gender inequality and informality within our framework. The model is estimated using Bayesian techniques and applied to Indian data. Our framework integrates the literature on gender with the literature on labor market rigidities, and is detailed enough to provide a starting point for studying the impact of various public policies.
In our model, we have two sectors, formal and informal sector, where informality results from significantly higher rigidities in the formal sector. Households consist of males and females, where gender inequality is modeled as various frictions on their labor supply and demand, which are higher for females relative to males. Using this framework, we investigate the impact of the following gender-specific policies: (i) increase in female education; (ii) increase in public provisions (including childcare); (iii) increase in female safety and mobility to work; and (iv) fall in gender-based discrimination by firms in formal employment. In addition, the impact of lowering labor market rigidities (i.e. labor market deregulation) in the formal sector, is also considered.
Our findings carry both good and bad news. First, on the one hand, gender-targeted policies increase female labor force participation and GDP in both the short run and long run. On the other hand, however, due to labor market rigidities, gender-specific policies do not generate sufficient formal job creation. This results in a larger share of these increased female participants either being employed informally at low wages, or staying unemployed, which increases aggregate unemployment and informality and further widens gender gaps in wages and informal employment. The only exception to this is with the policy that increases female education, which by increasing female workers’ efficiency at work, leads to higher formal employment of females. In addition, this increase in female efficiency leads to an expansion of the formal sector engendering an increase in both female and male formal employment. However, slow reallocation of workers due to labor market rigidities leads to a fall in female formality and widens gender wage gaps in the short run.
Second, there is an increase in the formal employment of both males and females with a labor market deregulation reform. However, male workers gain more, as constraints on female labor supply and demand combined with the household income effect lowers female participation and lead to a smaller increase in female formality in comparison to males. Finally, we show that simultaneously implementing gender-based policies that lower constraints on female participation combined with policies that boost formal job creation, as opposed to a piecemeal approach, generates substantial gains in gender equality in participation, formal employment, and wages along with larger gains in GDP and formality.
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|Mean||Std. Dev.||Mean||95% CB|
|Price A.C. in F||G||50||20||64.546||[60.710, 68.270]|
|Price A.C. in I||G||50||20||24.1511||[20.240, 27.860]|
|Price A.C. in f*||G||20||20||42.4417||[38.750, 45.610]|
|κ||Cap. A.C.||IG||20||20||6.3579||[5.0900, 7.3700]|
|χ||Country premium||IG||0.002||0.002||0.0009||[0.0007, 0.0010]|
|αi||Coef. of it-1||B||0.75||0.20||0.862||[0.8249, 0.9000]|
|απ||Coef. of π||G||3||3||3.2349||[2.8987, 3.5581]|
|αY||Coef. of GDP||G||0.4||0.2||0.3246||[0.2821, 0.3675]|
|ρθF||Pers. of θF||B||0.75||0.20||0.9718||[0.9579, 0.9883 ]|
|ρθI||Pers. of θI||B||0.65||0.20||0.3301||[0.2731, 0.4147]|
|ρG||Pers. of G||B||0.75||0.20||0.7886||[0.7050, 0.8782]|
|ρθH||Pers. of θH||B||0.65||0.20||0.6989||[0.6493, 0.7559]|
|Pers. of ||B||0.75||0.20||0.778||[0.6911 0.8604]|
|Pers. of ||B||0.75||0.20||0.5823||[0.5387, 0.6232]|
|ρskillF||Pers. of ||B||0.75||0.20||0.3745||[0.2661, 0.4862 ]|
|Pers. of ϕf||B||0.75||0.20||0.7348||[0.6879, 0.7835]|
|Pers. of ||B||0.75||0.20||0.7782||[0.6618, 0.8775]|
|Pers. of ||B||0.75||0.20||0.1056||[0.0671, 0.1544]|
|ρπ*||Pers. of π*||B||0.75||0.20||0.6385||[0.5942, 0.6725]|
|ρi*||Pers. of i*||B||0.75||0.20||0.9492||[0.9232, 0.9751]|
|sd(εθF)||Std. Dev. of θD||IG||0.01||Inf||0.1271||[0.1061,0.1495]|
|sd(εθI)||Std. Dev. of θI||IG||0.01||Inf||0.0307||[0.0253, 0.0356]|
|sd(εG)||Std. Dev. of G||IG||0.01||Inf||0.1265||[0.1016, 0.1517]|
|sd(εθH)||Std. Dev. of θH||IG||0.01||Inf||0.0085||[0.0022, 0.0158]|
|sd(εHCF)||Std. Dev. of βHCF||IG||0.01||Inf||0.0103||[0.0023, 0.0208]|
|sd(||Std. Dev. of ||IG||0.01||Inf||0.1121||[0.0933, 0.1316]|
|sd(εskillF)||Std. Dev. of ||IG||0.01||Inf||0.0072||[0.0024, 0.0126]|
|sd (εskillF)||Std. Dev. of ||IG||0.01||Inf||1.1785||[0.9458, 1.4203]|
|sd(||Std. Dev. of ϕf||IG||0.01||Inf||0.0093||[0.0023, 0.0169]|
|sd(||Std. Dev. of ||IG||0.01||Inf||0.009||[0.0022, 0.0166]|
|sd(||Std. Dev. of ||IG||0.01||Inf||0.211||[0.1622, 0.2645]|
|sd (||Std. Dev. of π*||IG||0.01||Inf||0.02||[0.0024, 0.0208]|
|sd(||Std. Dev. of i*||IG||0.01||Inf||0.015||[0.0023, 0.0209]|
|sd(εi)||Std. Dev. of i||IG||0.01||Inf||0.0084||[0.0074, 0.0096]|
Robustness of the Result
We evaluate the robustness of our estimation result by re-estimating the Baseline model with alternative and less informative priors. In the alternative model, the uniform distribution is assigned to the parameters bounded between 0 and 1, such as
First of all, even though the posterior means of the Taylor rule coefficients αi, απ, and αY are smaller in the alternative model, our argument that there is considerable policy inertia, and that the RBI responds aggressively to inflation deviations and not much to output deviations, is robust to the change in priors. Second, and our argument that formal price rigidity is higher than informal price rigidity, and that import price rigidity falls within the range of formal and informal price rigidity, still holds. Third, the persistence and volatility of shock processes is estimated to take high values, and thus our third argument that shocks are more persistent and volatile in emerging economies could be maintained even under the looser prior.
Overall, even though there are some quantitative differences for estimates of some parameters in the two model specifications, Bayesian estimates are similar across the models. Thus, our estimation analysis is robust to the prior specification and our arguments are strongly supported by the data.
|Param.||Alternative Model||Baseline Model|
This Appendix discusses the derivation of the model’s optimality conditions.
Partial Derivatives of Home Production
Here we derive the partial derivatives of home-good production with respect to male and female home-workers,
In the equation above, the total number of females unemployed can be written as:
and differencing the above with respect to
Partial derivatives with respect to female leisure consumption,
and similarly for males:
|Shock||Output||Formality||Unemp.||Female LFP||Female Formal||Wage gap|
|Y||LF/(LF + LI)||U/P||Pf/pf||wm/wf|
|Increase in public||SR||1||−0.3||1||2||−0.5||1.04|
|Increase in female safety & mobility||SR||0.62||−0.25||1.28||1.96||−0.48||1.07|
|Increase in female||SR||0.8||0.1||−1.3||2.4||−1.1||0.5|
|Lower firm-based gender||SR||−5.7||−2||−1.5||3.1||6||−34|
|Labor market deregulation||SR|
|Shock||Output||Formality||Unemp.||Female LFP||Female Formal||Wage gap|
|Y||LF/(LF + LI)||U/P||Pf/pf||wm/wf|
|Increase in female safety &||SR||1.5||−0.2||0.2||4.5||−1.5||1.5|
|Increase in female safety &||SR||1||−0.4||2||3||−0.7||1.6|
|Increase in female safety & labor||SR||2.2||0.2||1.5||2.2||0.1||−2|
Partial Derivatives and Marginal Rate of Substitution (MRS)
Here we obtain expressions for the partial derivatives of the household utility with respect to
Aggregate household utility is given by Eq. 3.42 in the text as:
Partial derivative of aggregate utility with respect to Ct is the marginal utility of household market-good consumption67:
Partial derivative with respect to female home production participation,
We have derived
Analogously, we obtain the partial derivative with respect to male home production labor:
Partial derivatives with respect to female home production participation,
By inserting the expressions for
Similarly, we obtain the partial derivative with respect to male leisure consumption:
The MRS between market-good and home-good consumption for females is given as:
MRS between market-good consumption and leisure for females is:
Solution to Households’ Utility Maximization
To solve for the households’ utility maximization problem, we first begin by inserting the following equations in the household budget constraint (Eq. 3.54) described in the text: (i) market clearing condition for unemployment of males and females,
In the household budget constraint the total wage income and unemployment benefits of females in period t is included as:
The total number of females unemployed can be written as:
Female labor force participation,
Analogously, total wage income plus the unemployment benefits of males in the budget constraint is included as:
which can then be written as:
To solve the households’ utility maximization problem described in the text, we insert Eq. B.10 and Eq. B.11 derived above in the household budget constraint, and establish the associated Lagrangian (L) as follows:
where λt is the shadow price for the budget constraint in period t, i.e. the value in terms of utility of relaxing the budget constraint at the margin. Differencing the above Lagrangian with respect to Ct, Bt, and Dt, yields the following first order conditions:
Eq. B.13 and Eq. B.14 imply the evolution of shadow price evaluated in domestic and foreign interest rate. Combining Eq. B.12 and Eq. B.13 is the Euler equation for domestic bonds given as Eq. 3.55 in the text, and combining Eq. B.12 and Eq. B.14 derives the Euler equation for foreign bond holdings given as Eq. 3.56 in the text.
Differencing the above Lagrangian with respect to
Combining these equations with Eq. B.12 yields the following first order conditions:
which corresponds to Eq. 3.57 for worker h described in the text.
Differencing the above Lagrangian with respect to
Combining these equations with Eq. B.12 yields the following first order conditions:
which corresponds to Eq. 3.58 for worker h described in the text.
Solution to the Wholesaler Profit Maximization Problem
To solve the wholesalers’ profit maximization problem described in the text, we establish the associated Lagrangian for the wholesalers in each sector s as follows where l = t + k:
The first condition is the wholesaler demand for capital Eq. 3.25, whereas the second and fourth conditions combined determine the demand for male labor Eq. 3.26 in the text. Similarly, the third and fifth conditions combined determine the demand for female labor Eq. 3.27 in the text.
We begin by deriving the wages of a female worker (h = f) in the formal sector (s = F), given as
Inserting the worker’s value of not participating in the labor market,
Add and subtract
Finally, expressions for wages of male and female workers in the formal sector are derived by plugging in Eq. B.15 and
Analogously, we derive expressions for informal wages, except that wage income taxes are nil in the informal sector (τI = 0), as follows:
Solution to Retailer Price Setting Problem
To solve the retailer’s profit maximization problem described in the text, we establish the associated Lagrangian for the retailer js in each sector s as follows where l = t + k:
Differencing the above equation with respect to PF,t (jF) yields the following first order condition:
As all firms are identical, i.e. PF,t (jF) = PF,t, we can write the above equation as:
Multiplying both sides by PF,t:
Dividing both sides by (εF– 1)(YF,t) results in the price setting equation in the formal sector given as:
Similarly, the informal price equation is obtained as:
The above equations correspond to Eq. 3.41 for sector s in the main text.
Solution to Capital Producer Profit Maximization
The capital producer invests It such that its profit is maximized, where Qt is the real price of capital, resulting in the following profit maximization problem described in the text:
Differencing the above equation with respect to It results in the following first order condition:
Rearranging terms gives us the supply of capital determined by:
which corresponds to the Tobin’s Q Eq. 3.62 given in the main text.