Department of Agricultural and Resource Economics ARE 251/Econ 270A, Fall Household Models

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1 Department of Agricultural and Resource Economics ARE 251/Econ 270A, Fall 2006 Department of Economics Elisabeth Sadoulet University of California at Berkeley Household Models I. The Basic Separable Household Model (Singh, I., Squire, L., and Strauss, J. (eds. Agricultural Household Models. Chapters 1 and 2. Baltimore, MD: The Johns Hopkins University Press, 1986 Two producer goods: food (a and cash crops (c Two factors of production: labor (l and other variable inputs (x Three consumer goods: food (a, manufactured goods (m, and leisure (l Definitions: q a production of food crop with price p a q c production of cash crop with price p c q l labor used in farm production with wage p l q x other variable inputs with price p x z q fixed factors in production and producer characteristics c a consumption of food product with price p a c m consumption of manufactured good with price p m c l consumption of leisure with price p l z h household characteristics in consumption l s time worked E total time endowment p l wage on labor market y income S exogenous cash transfers 1.1. The structural model Assume: perfect markets for all products and factors, including food and family labor. Household optimization problem: Max q a,q c,q l, q x,c a,c m,c l U( c a, c m, c l ;z h s.t. (1 g q a,q c,q l, q x ;z q ( = 0, production function ( + p c q c + p l ( l s q l + S, liquidity constraint (2 p x q x + p m c m = p a q a c a (3 l s + c l = E, time constraint Substituting l s in (2 for its value in (3 gives the full income constraint: p a c a + p m c m + p l c l = p a q a + p c q c p l q l p x q x ( + p l E + S = Π + p l E + S whereπ = p a q a + p c q c p x q x p l q l, restricted profit in agriculture. The household optimization problem can be rewritten as: Max W = U + φg + λ Π p c + p l E + S q a,q c,q l, q x,c a,c m,c l [ ] Hholds_handt_

2 Assume interior solution with q and c > 0. First order conditions: (4 (5 (6 (7 (8 : φ g q i = λp i, i = a,c i (producer goods : φ g q j = λp j, j = l, x j (factors φ : g = 0 (technology constraint : U c k = λp k, k = a, m, l k (consumption goods λ : ( Π + p le + S = 0 (full income constraint This indicates recursivity, called separability or separation, i.e.: Equations (4 (6 optimum levels of outputs, inputs, and maximum profit Π. Equations (7 and (8 identical to a pure consumer problem. Production decisions influence consumption only through profit Π Recursive solution: the reduced form First step: Solve the producer problem for maximum agricultural profit: Max Π = p a q a + p c q c p x q x p l q l, s.t. g q a,q c,q l, q x ;z q q a,q c,q l, q x ( = 0. This gives the reduced form: Supply functions q i = q i ( p a, p c, p l, p x ;z q, i = a,c Factor demands q j = q j ( p a, p c, p l, p x ;z q, j = l, x Maximum restricted profit Π = Π ( p a, p c, p l, p x ;z q Second step: Solve the consumer problem for maximum utility given the level of profit Π achieved in production Max c a, c m,c l U( c a,c m, c l ;z h s.t. p a c a + p m c m + p l c l = Π + p l E + S, full income constraint This gives the reduced form: Final demand functions: c k = c k ( p a, p m, p l, y ;z h, k = a, m, l ( + p l E + S. ( where y = Π p a, p c, p l, p x ;z q Hence: c k = c k p a, p c, p l, p x, p m ;z q, z h, E,S Note: under separability, the prices of consumption goods not produced at home ( p m and the z h, E, and S variables do not influence production decisions. This will provide a test of separability. Hholds_handt_

3 II. Household model with missing markets for food and labor (de Janvry, A., Fafchamps, M., and Sadoulet, E. "Peasant Household Behavior with Missing Markets: Some Paradoxes Explained." Economic Journal, Vol. 101, No. 409 (November, 1991, pp The structural model Market failures for food (a and labor (l: non-tradables Perfect markets for cash crops (c, other inputs (x, and manufactured goods (m: tradables with exogenous idiosyncratic prices: p c farm gate sale price of cash crop p x, p m farm gate purchase prices of other inputs and manufactured goods Max U(c a, c m, c l ; z h q a,q c,q l, q x,c a,c m,c l s.t. p x q x + p m c m = p c q c + S g q a,q c,q l, q x ;z q cash income constraint, ( = 0 production technology. p i = p i for i = c, x, m c a = q a c l = E q l exogenous effective prices for tradables equilibrium conditions for non-tradables 2.2. The first order conditions [ ( + φg + µ a ( q a c a + µ l ( E q l c l ] Max W = U + λ p c q c + S p x q x p m c m q a,q c,q l, q x,c a,c m,c l First-order conditions: : φ g q c = λp c ; : φ g c q x = λp x x (tradables : φ g a = µ a ; : φ g l = µ l q a q l (non-tradables : u c m = λp m m (tradables : u k = µ k, c k k = a, l (non-tradables : g = 0 φ (technology constraint λ : p xq x + p m c m = p c q c + S (cash income constraint : c µ a = q a a (equilibrium condition for food : c µ l = E q l l (equilibrium condition for labor. Define decision prices p as follows: p a * = µ a / λ, p l = µ l / λ shadow prices for the nontradables a and l p i * = p i effective market prices for the tradables c, x, and m. Hholds_handt_

4 Combining the last three conditions gives the full income constraint: p x q x + p m c m + p a c a + p l c l = p c q c + p a q a + p l ( + S. E q l By analogy with the first-order conditions for the separable model in 1.1, the first order conditions for the non-separable model can be rewritten using decision prices p as: φ g i = λp i, i = c,a products φ g j = λp j, j = l, x g = 0 u k = λp * k, k = m, a,l p k c k = p i q i k =a,m, l i =a,c factors technology consumer goods p j q j + p l E + S full income constraint j =l, x c a = q a c l = E q l equilibrium conditions for non-tradables 2.3. The household's decision structure (semi-structural form Production decisions from profit maximization: supply and derived demand: q i = q i ( p a, p c, p l, p x ;z q, i = a, c,l, x. Profit and full income: Π * = Σ p * i q i Σ p * j q j i =a,c j =l, x y * = Π * + p l * E + S. Consumption from utility maximization (with prices p* and income y* c k = c k ( p a, p m, p l, y ;z h, k = a,m,l Equilibrium conditions c a ( p, y ;z h = q a ( p a, p c, p l, p x ;z q ( = E q l ( p a, p c, p l, p x ;z q c l p, y ;z h for non tradables Solving these equilibrium conditions for the shadow prices of non-tradables: p j = p j p c, p x, p m ;z q,z h,e,s (, j = a,l. The p for nontradables are function of the prices of tradable consumption goods and of z q, z h, E, and S. The semi-structural solution of the model is thus: q i = q i ( p a, p c, p l, p x ;z q, i = a,c,l,x c k = c k ( p a, p m, p l, y ;z h, k = a,m,l Hholds_handt_

5 and p j = p j ( p c, p x, p m ;z q,z h,e,s, j = a,l Hence, household characteristics in consumption, z h, E, and S and consumption prices, p m, affect production decisions, as opposed to the separable model. The system would be recursive if there were only tradables The reduced form Substituting the expression just derived for the shadow price p j into the production and consumption decisions give: q i = q i ( p c, p x, p m ;z q,z h,e,s, i = a,c,l,x c k = c k ( p c, p x, p m ;z q,z h,e,s, k = a,m,l 2.5. Price elasticities (E Supply response ( p j = E q i p j E G q i ( + E q i p a where E G is the global elasticity. Consumption ( p j = E H c k p j E G c k ( E p a p j ( + E H c k p a ( + E q i p l ( E p a p j ( E( p l p j, i = a,c; j = c. ( + E H c k p l ( E( p l p j, k = m,l; j = m where E H is the elasticity in the separable household model with endogenous income effects: ( p k = E c k p k E H c k ( + E c k y III. Examples of empirical analyses: ( E( y p k, k = a, m. Benjamin, D. "Household Composition, Labor Markets, and Labor Demand: Testing for Separation in Agricultural Household Models." Econometrica, Vol. 60, No. 2 (1992, pp Jacoby, Hanan. "Shadow Wages and Peasant Family Labour Supply: An Econometric Application to the Peruvian Sierra." Review of Economic Studies, Vol. 60 (1993, pp Bowlus, Audra J. and Terry Sicular, Moving toward markets? Labor allocation in rural China, Journal of Development Economics, Vol. 71, No. 2 (2003, pp Strauss, J Does better nutrition raise farm productivity. Journal of Political Economy 94(2, pp Key, Nigel, Elisabeth Sadoulet, and Alain de Janvry. Transactions Costs and Agricultural Household Supply Response, American Journal of Agricultural Economics, Vol. 82, No. 2, (May 2000, pp Hholds_handt_

6 Simulation Results 2.2. Impact of a 10 percent 2.4. Impact of a 10 percent 2.1. Impact of a 10 percent increase in the 2.3. Impact of a monetary increase in productivity increase in the price of cash crops price of manufactured goods head tax of food production Market failures Market failures Market failures Market failures Food Food Food Food and and and and labor Labor Food None labor Labor Food None labor Labor Food None labor Labor Food None Percentage changes over base Percentage changes over base Percentage changes over base Percentage changes over base Consumption Food Leisure Manufactured good Production Food crop Cash crop Fertilizer Labor Prices Food crop a Cash crop Fertilizer Labor Manufactured good Net labor supply b Marketed surplus of food b a Blanks indicate no change relative to base value. b Net labor supply in percent of household labor effort, and marketed surplus in percent of food production. Source: A. de Janvry, M. Fafchamps, and E. Sadoulet, "Peasant Household Behavior with Missing Markets: Some Paradox Explained", Economic Journal, Vol. 101, No. 409 (November 1991, pp Hholds_handt_