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Consider an agricultural household that produces and also consumes part of its production.

Let y₁ = quantity of outputs produced by the household (e.g., food),

z₁ = quantity of the household outputs consumed by the household,

(y₁ - z₁) = "marketed surplus" = quantity of outputs sold on the market place at price p₁,

y₃ is purchased input used in household production,

L = (L₁, L₂, L₃), where L₁ denotes leisure, L₂ is labor used in household production, and L₃ is wage labor (with the wage rate = w),

y₁ = f(y₃, L₂, K) be the household production function, where K = household capital,

z₂ = quantity of purchased consumer goods (e.g., non-food) at a price p₂.

The household budget constraint is

I+wL₃ = -p₁(y₁-z₁)+p₂z₂+p₃y₃. (1)

The household time constraint is

T = L₁+L₂+L₃. (2)

Letting z = (z₁, z₂), assume that the household preferences are given by the utility function U(z, L₁). Letting y = (y₁, y₃), rational household decisions are then made in a way consistent with the household production model:

Max_z,L,y [U(z, L₁): I+wL₃ = -p₁(y₁-z₁)+p₂z₂+p₃y₃; T = L₁+L₂+L₃; y₁ = f(y₃, L₂, K)]

= Max_z,L,y [U(z, L₁): I+wT+p₁y₁-p₃y₃-wL₂ = p₁z₁+p₂z₂+wL₁; y₁ = f(y₃, L₂, K)], (3)

which has for solution z^*(I,p,w,K), L^*(I,p,w,K) and y^*(I,p,w,K).

Using a two-stage decomposition, we can choose (L₂, Y) in a first stage (conditional on (z, L₁), and then choose (z, L₁) in a second stage.

Given positive marginal utility of income, the first stage problem becomes the profit maximization problem:

p (p₁, p₃, w, K) = Max_L2,y [p₁y₁ - p₃y₃ - wL₂: y₁ = f(y₃, L₂, K)],

which has for solution the profit maximizing supply/demand functions: L₂⁺(p₁,p₃,w,K) and y⁺(p₁,p₃,w,K). This shows that profit maximizing behavior with respect to household labor L₂, marketed surplus Y₁, and purchased inputs y₃ is fully consistent with household behavior. It also shows that the production decisions are separable from the consumption decisions since they can be evaluated without any information on household preferences.

The second stage problem then becomes:

Max_z,L1 [U(z, L₁): I+wT+p (p₁,p₃,w,K) = p₁z₁+p₂z₂+wL₁],

which has for solution z^*(I,p,w,K) and L₁^*(I,p,w,K).

And combining the two stages gives the identities:

L₂⁺(p₁,p₃,w,K) = L₂^*(I,p,w,K), (4a)

and

y⁺(p₁,p₃,w,K) = y^*(I,p,w,K), (4b)

where y⁺ = (y₁⁺, y₃⁺). Differentiating these identities with respect to I and p₂ yields:

¶(L₂^*, y^*)/¶(I, p₂) = 0, (5)

which implies no effect of income (I) or consumer prices (p₂) on production decisions (y = (y₁, y₃) and L₂). This is an empirically testable implication of the separability of production decisions from the consumption decisions.

Note that, treating p as given and equal to some level p ₀, the above problem can be alternatively written as:

Max_z,L1 [U(z, L₁): I+wT+p ₀ = p₁z₁+p₂z₂+wL₁],

which has for solution z^a(I+wT+p ₀,p₁,p₂,w) and L₁^a(I+wT+p ₀,p₁, p₂,w). This is a traditional model of consumer behavior where "income" (I+wT+p ₀) is treated as "exogenous". But letting p ₀ = p (p₁,p₃,w,K), the following identities must necessarily hold:

z^*(I,p,w,K) = z^a(I+wT+p (p₁,p₃,w,K),p₁,p₂,w), (6a)

and

L₁^*(I,p,w,K) = L₁^a(I+wT+p (p₁,p₃,w,K),p₁,p₂,w). (6b)

Differentiating the identity (6a) with respect to p = (p₁, p₂, p₃) gives:

¶z₁^*/¶p₁ = ¶z₁^a/¶p₁ + [¶z₁^a/¶I] [¶p /¶p₁]

= ¶z₁^a/¶p₁ + [¶z₁^a/¶I] Y₁^*, (7a)

¶z₁^*/¶p₂ = ¶z₁^a/¶p₂, (7b)

¶z₁^*/¶p₃ = ¶z₁^a/¶p₃ + [¶z₁^a/¶I] [¶p /¶p₃]

= -[¶z₁^a/¶I] y₃^* ¹ 0, given ¶z₁^a/¶p₃ = 0, (7c)

where we used the envelope theorem: ¶p /¶p₁ = Y₁⁺ and ¶p /¶p₃ = -y₃⁺. Expressions (7) present the effects of market prices on consumer demand for the commodities z₁ (which are both produced and consumed by the household). Expression (7b) indicate that the price effects of a change in p₂, the price of consumer goods, is the same as in the traditional model of consumer theory. Perhaps more importantly, expressions (5a) and (5c) decompose Marshallian price effects into two additive terms: 1/ the first term on the right hand side is the classical Marshallian price effect of traditional consumer theory where "full income" (I+wT+p ₀) is treated as given; 2/ the second term is a "profit effect" which takes into consideration the effect of prices (p₁, p₃) on farm profit which can then be spend (through the effect of a higher income) on consumer goods. This second term indicates the non-separability of the consumption decisions from the production decisions: production decisions affect consumption decisions through the income effect generated by the profit from the production activities. If Y₁^* > 0 and z₁ is a non-inferior good (¶z₁^a/¶I ³ 0), then equation (7a) implies:

¶z₁^*/¶p₁ ³ ¶z₁^a/¶p₁ < 0,

This shows that, as compared to the household production model, the traditional consumer demand model (which treats profit as "exogenous") would give a biased estimate of the effect of an increase in price p₁ on household demand for the commodity z₁.

In addition, differentiating the identity (6a) with respect to the wage rate w gives:

¶z^*/¶w = ¶z^a/¶w + [¶z^a/¶I] [¶p /¶w] + [¶z^a/¶I] T

= ¶z^a/¶w - [¶z^a/¶I] L₂^* + [¶z^a/¶I] T, (7d)

where we used the envelope theorem: ¶p /¶w = -L₂⁺ . Expression (7d) present the effects of the wage rate w on consumer demand for the commodities z. Expression (7d) indicate that the Marshallian price effects of the wage rate w can be decomposed into three additive terms. The first term on the right hand side is the classical Marshallian price effect of traditional consumer theory where "full income" (I+wT+p ₀) is treated as given. The second term is a "profit effect" which takes into consideration the effect of the wage rate w on farm profit. And the third term is an "time endowment effect", reflecting that [wT] is part of the full income. The second term indicates the non-separability of the consumption decisions from the production decisions: production decisions affect consumption decisions through the income effect generated by the profit from the production activities. The second and third terms are each proportional to the income effect [¶z^a/¶I]. When z are non-inferior goods (¶z^a/¶I > 0), the second term {-[¶z^a/¶I] L₂^*} is negative, the third term {[¶z^a/¶I] T} is positive, with the third term dominating the second term: [¶z^a/¶I][T - L₂^*] ³ 0 (since [T - L₂] ³ 0 from the time constraint).

Note: Similar results would be obtained from differentiating the identity (6b) with respect to p = (p₁, p₂, p₃) and w, thus providing insights on the effects of changing economic conditions on household time allocation. In particular, taking the derivative of (6b) with respect to w gives

¶L₁^*/¶w = ¶L₁^a/¶w + [¶L₁^a/¶I] [¶p /¶w] + [¶L₁^a/¶I] T

= ¶L₁^a/¶w - [¶L₁^a/¶I] L₂^* + [¶L₁^a/¶I] T. (8)

Again, expression (8) indicate that the Marshallian price effects of the wage rate w can be decomposed into three additive terms. The first term on the right hand side is the classical Marshallian price effect of traditional consumer theory where "full income" (I+wT+p ₀) is treated as given. The second term is a "profit effect" which takes into consideration the effect of the wage rate w on farm profit. And the third term is an "time endowment effect", reflecting that [wT] is part of the full income. If leisure L₁ is a non-inferior good (¶ L₁^a/¶ w > 0), then the first term [¶L₁^a/¶w] is negative, the second term {-[¶L₁^a/¶I] L₂^*} is negative, and the third term {[¶L₁^a/¶I] T} is positive. And the sum of the second and third terms is {[¶L₁^a/¶I] [T - L₂^*]}, which is positive when leisure is non-inferior (since [T - L₂] ³ 0 from the time constraint).

If the first term dominates the other two terms, then the total effect [¶L₁^*/¶w] will be negative in (8): increases the wage rate would reduce leisure. From the time constraint, this means an increase in labor, corresponding to an upward sloping household labor supply function. However, the presence of the third term in (8) makes it "more likely" that the total effect [¶L₁^*/¶w] will be positive in (8). In such a situation, increasing the wage rate would stimulate the demand for leisure, and thus reduce the incentive to work. This would correspond to a household labor supply function that is downward sloping or "backward-bending"…

Finally, the above analysis provides some insights on the properties of the "marketed surplus" (y₁ - z₁). The (Marshallian) elasticity of marketed surplus with respect to price p₁ is:

¶ln(y₁^*-z₁^*)/¶lnp₁ = [¶(y₁^*-z₁^*)/¶p₁] [p₁/(y₁^*-z₁^*)]

= [(¶ln y₁^*/¶ln p₁)(y₁^*/p₁) - (¶ln z₁^*/¶ln p₁)(z₁^*/p₁)] [p₁/(y₁^*-z₁^*)]

= {(¶ln y₁^*/¶ln p₁) [y₁^*/(y₁^*-z₁^*)]} - {(¶ln z₁^*/¶ln p₁) [z₁^*/(y₁^*-z₁^*)]},

This shows that the (Marshallian) elasticity of marketed surplus with respect to price p₁ is a weighted difference between the elasticity of output (¶ln Y₁^*/¶ln p₁) minus the elasticity of consumption (¶ln Y₁^*/¶ln p₁), where the weights involve the ratios of production or consumption with respect to marketed surplus.

Simulated results obtained from running GAMS:

Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 Scenario 6 Scenario 7

(Base case) +1$ Inc +20% Pnf +20% Pinp +20%Pfood +20% Plab "Hobby-farm"

Prod 6.225 6.225 6.225 6.202 6.287 6.161 6.383

Inpu 1.016 1.016 1.016 0.948 1.085 1.001 1.101

Hlab 0.794 0.794 0.794 0.785 0.899 0.675 1.478

Wlab 2.250 2.232 2.268 2.272 1.862 2.576 1.615

Leis 1.956 1.974 1.939 1.942 2.238 1.748 1.907

Mark 2.766 2.741 2.791 2.762 2.910 2.559 2.993

Food Cons. 3.459 3.484 3.434 3.439 3.378 3.601 3.390

Nfood Cons. 5.418 5.468 4.557 5.379 6.207 5.703 5.280

Utility 0.963 0.972 0.882 0.957 1.028 0.990 0.980

GInc 41.495 41.120 41.870 41.433 52.372 38.389 44.899

NInc1 37.432 37.057 37.807 36.881 48.034 34.385 40.493

NInc2 31.876 31.501 32.251 31.385 41.737 28.714 30.147

HInc 54.181 54.681 54.681 53.788 62.070 57.027 52.798

Scenario 1 (Base case): The household produces 6.225 units of food. This is allocated as follows: 2.744 (44.43 %) is marketed, and 3.459 (55.57 %) is consumed by the household. Household labor (T = 5) is allocated as follows: 0.794 (15.88 %) to household activities, 2.250 (45 %) to wage labor, and 1.956 (39.12 %) to leisure. Household income is Hinc = 54.181, which includes exogenous income Inc = 1 (1.85 %), net farm income Ninc1 = 37.432 (69.08 %), and wage income 15.75 (29.07 %).

Changes in the base model:

Changes in Inc or in Pnf have no impact on production decisions (Prod, Inpu, Hlab). This corresponds to equation (5). This is because production decisions are consistent with profit maximization, and are thus independent of the consumption environment (as reflected by Inc and Pnf). This reflects the separability of production decisions from other decisions.

Scenario 2: $1 rise in exogenous income, Inc.

A $1 rise in Inc tends to increase Leis, Food and Nfood consumption. Thus all consumer goods (including leisure) are non-inferior. The increase in leisure implies that Wlab must decrease (from the time constraint). In other words, increasing income tends to reduce the household wage-labor supply. Also, the increase in food consumption implies that marketed surplus (Mark) must decline. This means that farm income declines, implying that household income Hinc increases but by less than $1. The household is better off.

Scenario 3: 20 % rise in the price of nonfood, Pnf.

A 20% rise in Pnf tends to decrease Nfood consumption. This is expected since Nfood is a non-inferior good (implying that it cannot be a Giffen good). It also decreases the demand for leisure and food. This corresponds to equation (7b), where the price effect involves only a "consumption effect". Decreasing leisure implies that household wage-labor supply Wlab goes up, which contributes to increasing household income. And the decrease in food demand stimulates marketed surplus, which also contributes to increasing household income Hinc. But even with a rise in household income, the household is made worse off.

Scenario 4: 20 % rise in the price of inputs, Pinp.

A 20% rise in Pinp has negative effects on production decisions. This is consistent with profit maximization. Increasing the price of an input tends to decrease its demand (as a profit-maximizing input demand is always downward sloping). And the input being non-inferior, it also decreases output supply. The impacts on consumption/leisure decisions are also negative. This corresponds to equation (7c), where Pinp affects consumption only through a "profit effect" (which is necessarily negative for non-inferior goods). The decrease in leisure and in Hlab means that household wage-labor supply must go up (from the time constraint). Also, while both food production and food consumption decline, the former effect is stronger, generating a slight decline in marketed surplus and in gross farm income (Ginc). Because of the increased production costs, net farm income declines, and so does household income. The household becomes worse off.

Scenario 5: A 20 % rise in the price of food, Pfood.

A 20% rise in Pfood tends to stimulate production. This is consistent with profit maximization. Increasing output price tends to increase output supply (as profit-maximizing output supply functions are always upward sloping). And inputs being non-inferior, it also stimulates input demands (Inpu and Hlab). The impact on food consumption is negative. This corresponds to equation (7a), which decomposes the price effect into a consumption effect (negative for non-inferior goods), plus a "profit effect" (positive for non-inferior goods). Thus, as far as food consumption is concerned, the consumption effect dominates the profit effect. The impacts on leisure and non-food consumption are positive. The large increase in leisure means a large decrease in the household wage-labor supply (Wlab). Higher food production and lower food consumption generates a large increase in marketed surplus (Mark). This has a large positive impact on farm income, which greatly increases household income (in spite of a lower wage income). The household is much better off.

Scenario 6: A 20 % rise in the wage rate, Plab.

A 20% rise in Plab has negative effects on production decisions. This is consistent with profit maximization. Increasing labor (opportunity) cost tends to decrease the demand for household labor, Hlab (as profit maximizing input demand is always downward sloping). And Hlab being a non-inferior input, it also decreases output supply. The impacts on consumption decisions are positive, while the impact on leisure is strongly negative. This corresponds to equation (8), which decomposes the price effect into a consumption effect, a profit effect, and a time-endowment effect. The negative impact on leisure demand suggests that the consumption effect dominates. This means that a higher wage rate tends to reduce the demand for leisure, and to stimulate labor supply. In other words, this case does not provide evidence of a backward-bending labor supply function. The decrease in food production and increase in food consumption generate a decline in marketed surplus (Mark). As a result, farm income declines. But the increase in wage income dominates, implying an increase in household income Hinc. The household is better off.

Scenario 7: Case of a "hobby farm"

For a "hobby farm", household preferences now depend explicitly on household labor (Hlab). This provides some added incentives for labor to be used working in the household (as opposed to leisure or wage-labor). Thus, as expected, household labor (Hlab) increases, while leisure and wage labor decline.

In contrast with (3), rational household decisions are now made according to the following maximization problem (where L₂ = Hlab)

Max_z,L,y [U(z, L₁, L₂): I+wL₃ = -p₁(y₁-z₁)+p₂z₂+p₃y₃; T = L₁+L₂+L₃; y₁ = f(y₃, L₂, K)]

= Max_z,L,y [U(z, L₁, L₂): I+wT+p₁y₁-p₃y₃-wL₂ = p₁z₁+p₂z₂+wL₁; y₁ = f(y₃, L₂, K)], (3')

which has for solution z^*(I,p,w,K), L^*(I,p,w,K) and y^*(I,p,w,K). Since household labor now affects preferences, the two-stage decomposition presented above no longer holds. In particular, Hlab is no longer chosen in a way consistent with profit maximization (since household preferences clearly affect the choice of L₂ = Hlab). This suggests a different two-stage decomposition.

The relevant first stage profit maximization condition now becomes conditional on L₂ (= Hlab). It takes the form

p '(p₁, p₃, L₂, K) = Max_y [p₁y₁ - p₃y₃: y₁ = f(y₃, L₂, K)],

which has for solution the profit maximizing supply/demand functions: y'⁺(p₁,p₃, L₂, K), which are conditional on L₂. Similarly, p '(p₁, p₃, L₂, K) is now a profit function conditional on L₂. In this context, it remains true that y₃ (= Inpu) is still chosen in profit maximization fashion. However, since choosing L₂ is no longer consistent with profit maximization, the separability of production decisions from consumption decisions no longer holds.

The choice of household labor L₂ now becomes a second-stage decision. The second stage problem becomes:

Max_z,L1,L2 [U(z, L₁, L₂): I+wT+p '(p₁,p₃,L₂,K)-wL₂ = p₁z₁+p₂z₂+wL₁],

which has for solution z^*(I,p,w,K), L₁^*(I,p,w,K) and L₂^*(I,p,w,K).

And combining the two stages gives the identity:

y'⁺(p₁,p₃,L₂^*(I,p,w,K),K) = y^*(I,p,w,K), (4b')

where y'⁺ = (y₁'⁺, y₃'⁺). Differentiating this identity with respect to I and p₂ yields:

¶y^*/¶(I, p₂) = (¶ y'⁺/¶ L₂)(¶ L₂^*/¶ I, p₂). (5')

This clearly shows that, even if y₃ (= Inpu) is chosen in a profit maximization fashion, production decisions are no longer separable from the consumption environment.

This is illustrated by the simulation results (scenario 7). Enjoying work in the household means that household labor (Hlab) increases sharply, production rises, and leisure decreases. The increase in production induces a rise in marketed surplus and farm income. But wage labor declines strongly. This implies a sharp reduction in wage income, which generates a decline in household income (Hinc) and a decrease in consumption. Note that the household still appears to be "better off" compared to the one in the base scenario, even while facing a much lower household income.