The Collective Model of Household : Theory and Calibration of an Equilibrium Model Eleonora Matteazzi, Martina Menon, and Federico Perali University of Verona University of Verona University of Verona and and and THEMA-University of Cergy-Pontoise CHILD CHILD Summer School in Development Economics (July 30 August 3, 2010)
Aim Our aim is to extend the collective approach to the farm-household models Collective Models Apps and Rees (1997), and Chiappori (1997). Farm-Household Models Lopez (1984), Sing, Squire, and Straus (1986), Benjamin (1992), De Janvry et al. (1991, 1992), Jacoby (1993), Skoufias (1994), and Saudolet et al. (1996, 1998). We analyse the farm-household as a collection of individual rather than as an undifferentiated decision unit. 2
Farm-Household Models Agricultural household models integrate production and consumption decisions in rural farm-households. Household behaviour is described as maximizing household utility subject to a cash income constraint: household total expenditure = i w i T + profit + exogenous income Farm household behaves as though it is a single individual. Complete and competitive markets: even if production and consumption decisions are taken simultaneously in time they can be modelled recursively. Separability between production and consumption decision. 3
Farm-Household Models Definition 1. (Separability between production and consumption) A model of household enterprise exhibits a separation property between production and consumption decisions if household production technology, input and output prices affect consumption decisions but the reverse is not true. Otherwise, preferences, exogenous income and prices of consumption goods do not affect production decisions. This implies that production and consumption decisions can be solve recursively, that is: i) at the first stage, household makes production decisions. ii) at the second stage, household makes consumption decisions, given that the household full income includes profits of household enterprise. When markets are incomplete, or do not exist, or are not competitive, endogenous prices may arise to clear household markets.
How to deal with the multiplicity of decision-makers in the household? Unitary approach The differences that may exist between single-person and multi-person households are simply ignored. This approach turns out to be unsatisfactory on both theoretical and empirical grounds. i) At the theoretical level, the utility theory is intended to study the choices of individuals and not of groups. (ii) At the empirical level: the property of symmetry of the Slutsky Matrix is not well supported by evidence when household is composed by several decision-makers (see for example Browning and Meghir, 1991) the income pooling hypothesis has almost always been rejected in empirical studies (see for example, Thomas 1990, Schultz 1990, Browning, Chiappori and Lechene 1993).
How to deal with the multiplicity of decision-makers in the household? Collective approach The collective approach models household as a group of individuals. Each individual: 1. is characterized by specific preferences, 2. participates to the household decision process, and 3. shares household resources. The collective approach makes no assumption about the decision process. It only assumes Pareto-efficiency. This allows one to generate testable restrictions on household behavior. 6
Our objectives are threefold The collective model of household enterprise We characterize the collective household enterprise accounting for both Marketable and non-marketable production with non-marketable inputs. Estimation of an household equilibrium model within a collective approach We estimate the collective model of household enterprise using Italian data assuming complete and competitive markets. The collective household enterprise as an equilibrium model: calibration and simulation We use estimated elasticities and an Household Social Accounting Matrix, constructed using the same Italian data, to calibrate a programming model that exactly reproduces the econometric model. We use the programming model to conduct simulations under several assumptions about the markets functioning. We also control if simulation results are in line with theory predictions. 7
The Collective Household Enterprise We consider a two-person household which members i = 1,2 are, respectively, the husband and the wife. Assumption 1. Individual utilities are characterized by egoistic preferences, U i (C i, l i, z i ) where U i is a well-behaved twice continuously differentiable concave function increasing in its arguments. Notation: C i is individual consumption of a market good; l i is individual consumption of leisure; z i is individual consumption of a domestic good produced by the household. 8
The Collective Household Enterprise Assumption 2. Household welfare function is a weighted sum of individual's utilities, μ U 1 (C 1, l 1, z 1 ) + (1- μ) U 2 (C 2, l 2, z 2 ) where the Pareto-weight μ is assumed to be a differentiable function of market wages (w 1, w 2 ), non labour incomes (y 1, y 2 ) and good prices (p 1, p 2 ). It can also depends on distribution factors (s). The weight µ represents the bargaining power that each member can exert on family resources. If µ = 1 the household welfare is entirely determined by the preferences of individual 1 and it implies that he has the total control over the household resources.
The Collective Household Enterprise The household is involved in production of two different goods The household runs a business activity, that may be an agricultural or a commercial activity. The good produced is termed household good. The household is also engaged in domestic activities like cleaning the house, cooking, caring The output of these activities is termed domestic good (Gronau 1977, Apps and Rees 1997, Chiappori 1997). Household Production The production technology of the household good is given by q = f (h 1,h 2,h 3,d h ) where: h 1 and h 2 are family members labor supply; h 3 are market production inputs and d h is a vector of demographic factors. p q is the market price of the household good q. 10
The Collective Household Enterprise Assumption 3. The function f is increasing and concave in each arguments and globally quasi concave. Domestic Production Domestic production was introduced in collective models (Chiappori 1997 and Apps and Rees 1997) after that several authors pointed out that omitting domestic production may distort welfare analysis. Why? Non market time Leisure The production technology of the domestic good is given by z = g(t 1,t 2, d h ) Assumption 4. The function g is increasing and concave in each arguments (t 1 and t 2 husband and wife s labor supply) and globally quasi concave. are
The Collective Household Enterprise Notice that domestic production is not observable. Is it marketable? Marketable Domestic Production Following Gronau (1977), we may assume that the same goods and services, such as cleaning or catering or caring, can be produced at home or bought on outside markets at a given exogenous price. The collective household enterprise model is separable. Identifiability of individual preferences and the sharing rule Chiappori (1997) shows that if we assume Pareto-efficiency, egoistic preferences, marketability of domestic production and if there is at least one distribution factor, the rule that describe how household resources are shared between husband and wife the sharing rule - can be identified (up to a constant) from the sole estimation of market and domestic labor supplies.
The Collective Household Enterprise Non-marketable domestic production The domestic good is entirely sold in the home market, that is z = z 1 + z 2 The price of the domestic good is endogenously determined and vary across households. CRS ensures separability between production and consumption decisions. Identifiability of individual preferences and the sharing rule Chiappori (1997) shows that if we assume Pareto-efficiency, egoistic preferences, constant return to scale in domestic production and if there is at least one distribution factor, the sharing rule can be identified (up to a function of wages) from the sole estimation of market and domestic labor supplies. See also Apps and Rees (1997) an Rapoport, Sofer and Solaz (2009).
The Collective Household Enterprise with Marketable and Non-Marketable Production The household centralized maximization program is Max μ U 1 (C 1, l 1, z 1 ) + (1- μ) U 2 (C 2, l 2, z 2 ) subject to: C i, l i, z i, h i, t i, L i, q Household BC: p 1 C 1 + p 2 C 2 = w 1 L 1 + w 2 L 2 + y 1 + y 2 + p q q w 3 h 3 Domestic technology constraint: z 1 + z 2 = z = g(t 1, t 2,d h ) Household technology constraint: q = f (h 1,h 2,h 3,d h ) Individual time constraints: T = l i + h i + t i + L i i = 1,2 where T is individual time endowment and L i is individual market labor supply. 14
Constant Return to Scale Because of the non-marketability of domestic good, the model looses the separable property. Assumption 5. The cost function of the domestic production is homothetic C(w1,w2, z) = c (w1,w2) z and the underlying domestic production technology g is linearly homogeneous and exhibits constant returns to scale. This allows us to determine the price of domestic good applying Shepard s lemma (Gronau 1973, Apps and Rees 1996, 2002), that is p z* = C/ z = c(w1,w2), corresponding to the unitary cost function. Notice that p z* is affected only by market wages.
The Collective Household Enterprise with Marketable and Non-Marketable Production The production side of the household economy Non-marketable Domestic Production Given that the price of the domestic good is not given exogenously by the market, we can say that optimal household behavior results in minimizing total production cost given the production function g. Min TC = w 1 t 1 + w 2 t 2 z = g(t 1, t 2,d h ) t 1, t 2 Optimal input demands and total cost are functions of (w 1, w 2, z, d h ). We derive p z* by applying Shepard s Lemma to the total cost function.
The Collective Household Enterprise with Marketable and Non-Marketable Production Marketable Household Production The optimal household behavior results in maximizing profit given the production technology f. Max = p q q w 1 h 1 w 2 h 2 w 3 h 3 q = f (h 1, h 2, h 3,d h ) h 1, h 2, h 3, q Optimal input demands and profit are functions of (w 1, w 2, w 3, p q, d h ). Household resources sharing Following Chiappori (1997) we assume that spouses divide household income y 1 + y 2 + according to a sharing rule φ(w 1, w 2, p 1, p 2, y 1, y 2 ) where individual 1 gets φ 1 = φ and individual 2 gets φ 2 = y 1 + y 2 + - φ.
The Collective Household Enterprise with Marketable and Non-Marketable Production The consumption side of the household economy Each household member maximizes her utility under her own budget constraint: Max U i (C i, l i, z i ) p i C i + w i l i + p z* z i = w i T + φ i (w 1, w 2, p 1, p 2, y 1, y 2 ) C i, l i, z i Optimal good demands of individual i are functions of (p i, w i, p z*, φ i (.)). From the time constraint we derive individual optimal market labor supply L i.
The Collective Household Enterprise with Marketable and Non-Marketable Production and Non-Marketable input While household's members are certain of being employed in the household enterprise and in domestic activities, they are not certain to work in the labor market. If the husband and/or the wife do not participate to the labor market, his and/or her time-uses (that is, leisure and labor supplies) are evaluated at the shadow wage w i* that can be derived from labor productivity at the optimal point. 19
The Collective Household Enterprise with Marketable and Non-Marketable Production and Non-Marketable input Household budget set results non-linear in hours worked, but, at the optimum, the slope of the budget set is given by the vector (w 1*, w 2* ). Thus, as in Jacoby (1993) and Skoufias (1994), we substitute the non linear budget constraint with an artificial linear constraint, where profit and total cost functions depend on (w 1*, w 2* ). This artificial linear constraint would induce the household to arrive at the same optimal choices and allows us to solve household program in two stages. 20
The case of non-marketable input The production side of the household economy. Domestic production: Min TC = w 1* t 1 + w 2* t 2 z = f (t 1, t 2,d h ) t 1, t 2 Household production: Max = p q q w * 1 h 1 w 2* h 2 w 3 h 3 q = f (h 1, h 2, h 3,d h ) h 1, h 2, h 3, q The consumption side of the household economy. Max U i (C i, l i, z i ) p i C i + w i* l i + p z* z i = w i* T + φ i (w 1, w 2, p 1, p 2, y 1, y 2 ) C i, l i, z i Notice that the sharing rule depends on individual market wages. The idea is that market wages are a measure of individual bargaining power. 21
The equilibrium structure of the model The production and the consumption sides of the household economy illustrates the equilibrium structure of the model. PRODUCTION SIDE CONSUMPTION SIDE z = g(t 1, t 2,d h ) c i = c i (p i, p z*, w i*, φ i (w 1, w 2, p 1, p 2, y 1, y 2 ) ) t i = t i (w 1*, w 2*, z, d h ) l i = l i (p i, p z*, w i*, φ i (w 1, w 2, p 1, p 2, y 1, y 2 ) ) q = f (h 1, h 2, h 3,d h ) z i = z i (p i, p z*, w i*, φ i (w 1, w 2, p 1, p 2, y 1, y 2 ) ) h i = h i (w 1*, w 2*, w 3, p q, d h ) CLEARING CONDITIONS T = t i + h i + l i + L i p z* (z 1 + z 2 ) = w 1* t 1 + w 2* t 2 φ 1 + φ 2 = y 1 + y 2 + 22
The equilibrium structure of the model Labor market clearing condition. The individual choice to participate in labor market is modeled as a Mixed Complementarity Problem (Khun-Tucker conditions). where: If w i* > w i then L i = 0 L i (w i* - w i ) = 0 If w i* = w i then L i = T l i h i t i 0 Let us not note that endogenous wages are function of all exogenous variables. 23
Data ISMEA Survey on Socio-Economic Characteristics of Italian Rural Household undertaken in 1996. The sample counts 1777 farm households. The questionnaire was designed on the basis of the collective household model presented. The survey combines information about: household and farm characteristics time use farm profits off-farm money income governmental and intra household transfers consumption the degree of autonomy in decision making by household members. 24
Empirical Specification and Estimation Method Empirical Specification: Translog specification for production technologies Gorman polar form for individual preferences linear in individual full income Estimation Technique: First stage - estimation of domestic technology and derivation of the price of the domestic good (under the assumption of CRS). Second stage - household technology and consumption are jointly estimate. The estimation is implemented using a Simulated ML technique. The assignability of clothing and leisure is the source of identification of the sharing rule. 25
Estimation results: Own and Cross-Price Elasticities of the Domestic Technology Husband wage Wife wage Others wage Price inputs goods Husband wage -0.147 0.420-0.911 2.580 Wife wage -0.558-0.958 1.350 0.089 Others wage -0.029-0.137-1.650 0.619 Price of Input Goods -0.060 0.278-0.670-0.939 26
Estimation results: Farm production Compensated Elasticities Crops Beef Milk Fruits Crops 0.942 0.439 0.167-0.912 Beef 0.435-0.09 0.339 Milk 0.129 1.146 Fruits 0.412 Hired labour Chemicals Materials Hired labour -0.375 0.875 0.999 Chemicals -0.075 1.051 Materials -0.096 27
Estimation results: Consumption Compensated Elasticities Husband Domestic good Food Clothing Other Leisure Domestic good -0.760 1.558 0.012-0.018-0.345 Food 0.899-1.158 0.021 0.057-0.470 Clothing 0.000 0.000-0.205 0.001 0.003 Other -0.056 0.067 0.062 0.027-0.018 Leisure -0.250-0.388 0.178 0.003 0.945 Wife Domestic good Food Clothing Other Leisure Domestic good -0.293 1.543-0.210-0.080-0.792 Food 0.639-1.166 0.035 0.076-0.423 Clothing 0.007-0.006-0.133-0.004-0.001 Other -0.081 0.100 0.065 0.067-0.011 Leisure -0.261-0.478 0.237-0.064 1.218 28
The Household Social Accounting Matrix The Household Social Accounting Matrix (HSAM) is an accounting scheme of the medium Italian farm-household economy. It describes the links among household members and between the farm household and the rest of the economy. It reveals household production decisions and individual consumption decisions. Notice that It is possible to construct the HSAM for each household or for different types of farm household. 29
The Household Social Accounting Matrix Market inputs Land Family work Agric. Prod. Man Woman Market work Dom. Prod. Leisure Rest of econ. Market inputs 3379.4 Land 1370.9 Family work Agric. Prod. 1780.4 Profit from agricultural production shared according to the estimated sharing rule Man 923.5 824.3 108 0 484.8 1007.5 1363.9 Woman 447.3 956 125.3 0 718.5 1132.5 1581.4 6764 Market work Dom. Land value and non labor 484.8 718.5 Prod. income are shared according Leisure to the estimated sharing rule 1007.5 1132.5 0 Rest of econ. 3379.4 3219.7 3110.6 30
The Household Social Accounting Matrix The farm-household produces four outputs (crops, beef, milk, and fruit, olives and grapes) that are sold on the market at an exogenous price. The farm household receives also transfer from Italian Government. The production factors are partly bought on the market (as hired labor, chemicals, materials, capital stock), and partly owned by household's members (as land and family labor fixed inputs). They are remunerated from the value added. Individual full income is given by the value of time endowment plus the share of household resources (profits + non labor income + land value). Individual spends this full income to a) purchase the market goods (food, cloth and other goods), b) consume the domestic good, and c) enjoy leisure. 31
The Household Social Accounting Matrix Disaggregate values for agricultural production (shares) Hired Labor Inputs Chemicals Materials Capital Family labor Land Agricultural production 0.10 0.08 0.31 0.02 0.21 0.28 Outputs Crop Beef Milk Fruit Agricultural production 0.40 0.22 0.24 0.14 32
The Household Social Accounting Matrix Disaggregate values for consumption (shares) Husband Wife Leisure 0.21 0.23 Domestic good 0.10 0.15 Clothing 0.01 0.01 Food 0.29 0.26 Time-uses (hours) Husband Wife Leisure 172 199 Domestic work 83 126 On-farm work 157 79 Market work - - Other market goods 0.39 0.35 Actual sharing rule Endogenous wages Husband Wife Husband Wife Sharing rule 0.463 0.537 Endogenous wage 5.6 5.9 33
The programming model The collective model presented is also programmed using the software GAMS (General Algebraic Modeling System). The programming model exactly reproduces the collective model underlying the econometric specification. It represent a powerful tool to describe the behavioral responses of both the farm and the household to economic and social policies and to evaluate their impact on individual welfare levels. Simulations may be conducted under several assumptions about the market functioning and degree of openness. 34
Micro-calibration The programming model in GAMS is calibrated using both the information coming from the estimated elasticities and the HSAM. The estimated shares and the shares derived from the HSAM differ slightly because of measurement errors. Definition of Micro-calibration We define micro-calibration the calibration of the intercept of demand and production shares to match the levels of the HSAM. 35
Micro-calibration An example: Y = F(x; α0, α) + ε = α0 + f(x; α) + ε (1) where: ε represents an optimization and measurement error α0 and α are estimated parameters (α0 is the constant) Y is a production or consumption share that has been estimated. Notice that Y and X are also the values reported in the HSAM. Thus, if we write (1) as Y = (α0 + ε) + f(x; α) the intercept micro calibrated is α 0 C = Y - f(x; α) = (α0 + ε) 36
Simulation results Increase of 10% in crop price PRODUCTION SIDE INPUT Hired labor -4.2 % Chemicals 27.3 % Materials 15.5 % Family labor -4.6 % OUTPUT Crop 12 % Beef -13.8 % Milk -2 % Fruit 8.3 % ENDOGENOUS PRICES HUSBAND WIFE Wage 18.2 % 19.6 % Domestic good 20 % 20 % 37
Simulation results Increase of 10% in crop price CONSUMPTION SIDE HUSBAND WIFE Full income 7.8 % 7.4 % Leisure -4.3 % -6.8 % Domestic good 8.9 % 0.4 % Clothing 5.2 % 7.2 % Food 4.3 % 4.3 % Other market goods 1.5 % 1.9 % TIME-USES HUSBAND WIFE Leisure -4.3 % -6.4 % Domestic work 16.9 % 14.1 % On-farm work -4.2 % -5.4 % Market work - - 38
Simulation results Increase of 10% in husband s market wage PRODUCTION SIDE INPUT Hired labor 0.6 % Chemicals 7.6 % Materials 3.6 % Family labor 0.4 % OUTPUT Crop -0.6 % Beef -4 % Milk -0.6 % Fruit 8.4 % ENDOGENOUS PRICES HUSBAND WIFE Wage 10 % - Domestic good 4.1 % 4.1 % 39
Simulation results Increase of 10% in husband s market wage CONSUMPTION SIDE HUSBAND WIFE Full income 2.8 % -1.5 % Leisure -2.1 % -3.7 % Domestic good 1.7 % -3.9 % Clothing 1.7 % -0.7 % Food 1.7 % -0.9 % Other market goods 0.1 % -1.2 % TIME-USES HUSBAND WIFE Leisure -2.1 % -3.7 % Domestic work 1.4 % 0.6 % On-farm work -5.8 % 3.6 % Market work 11 hours 4 hours SHARING RULE HUSBAND WIFE Before Sim. 0.463 0.537 After Sim. 0.458 0.542 40
Conclusions In this study: we present a collective model of household enterprise. We assume that household is involved in production of two different goods, a marketable and a non-marketable good, using marketable and non-marketable inputs. We show under which assumptions the model can be solved recursively. we estimate the collective model of the household enterprise under the assumption of complete and competitive markets. 41
Conclusions we reproduce in GAMS the collective model underlying the econometric specification. The programming model is micro-calibrate using the HSAM and estimated elasticities. The programming model is used to simulate the micro impact of macro policy changes under several assumptions about the market functioning. It is possible to evaluate their impact on individual welfare levels. 42