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Lecture 13: Production Economy

Advanced Microeconomics I

Yosuke YASUDA

National Graduate Institute for Policy Studies

November 26, 2013

From Pure Exchange to Production Economy

Previous lecture considered the general equilibrium in a pure exchange economy where all agents are consumers.

Now we expand our description of the economy to include production as well as consumption.

✞

✝

☎

Rm We will find that most of the important properties of✆ competitive market systems uncovered earlier continue to hold. In a general equilibrium model with production:

◮ Consumers, 1, 2, ..., I, act to maximize utility subject to their budget constraints.

◮ Firms, 1, 2, ..., J, seek to maximize profit.

◮ Both consumers and firms are price takers.

◮ Firms’ profits are shared among individual consumers.

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Firm Behavior (1)

Def Let Y denote the aggregate production possibilities set, defined as the sum of the individual production possibility sets:

Y =^X

j∈J

Y_j = {y | y =^X

j∈J

y_j where y_j ∈ Y_j}.

Let y(p) be the aggregate net supply function defined as the sum of the individual net supply functions:

y(p) =^X

j∈J

y_j(p).

where y_j(p) associates to each vector p the profit-maximizing net output vector at those prices.

✞

✝

☎

Rm Y represents all production plans that can be achieved by✆ some distribution of production among J individual firms.

Firm Behavior (2)

The next theorem says that if each firm maximizes profits, then aggregate profits must be maximized. Conversely, if aggregate profits are maximized, then each firm’s profits must be maximized.

Thm An aggregate production plan y maximizes aggregate profit, if and only if each firm’s production plan y_j maximizes its individual profit for all j ∈ J.

The theorem implies that there are two equivalent ways to construct the aggregate net supply function:

1. Add up the individual firms’ net supply functions. 2. Add up the individual firms’ production sets and then

determine the net supply function that maximizes profits on this aggregate production set.

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Firm Behavior (3)

Proof We only show “⇐”, since the converse is straightforward. Let {yj}j∈J be a set of profit-maximizing production plans for the individual firms. Suppose that y =^P_j∈Jy_j is not (aggregate) profit-maximizing at prices p. This implies that there is some other production plan y^′=^P_j∈Jy^′_j with y_j^′ in Y_j that has higher profits:

X

j∈J

py^′_j = p^X

j∈J

y^′_j > p^X

j∈J

y_j =^X

j∈J

py_j.

By inspecting the sums on each side of this inequality, we see that some individual firm j must have higher profits at y_j^′ than at y_j.

✞

✝

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Rm In a capitalist economy or private ownership economy,✆ consumers own firms and are entitled to a share of the profits.

Consumer Behavior (1)

Def Consumer i’s share of the profits of firm j is represented by 0 ≤ θ_ij ≤ 1 for all i ∈ I and j ∈ J

where X

i∈I

θ_ij = 1 for all j ∈ J.

That is, each firm is completely owned by individual consumers. Then, the budget constraint of each consumer i becomes:

px_i ≤ pe_i+^X

j∈J

θ_ijpy_j(p).

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✝

☎

Rm Given e✆ _i and θ_ij, the budget set is characterized by p alone. Hence, consumer i’s demand function can be written as a function of p, denoted by x_i(p).

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Consumer Behavior (2)

Def Let x(p) =^P_i∈Ix_i(p) be the aggregate (consumer) demand function, the sum of the individual demand functions.

Def The aggregate excess demand function is defined by z(p) = x(p) − y(p) − e

where e is the aggregate supply from consumers, e =^P_i∈Ie_i.

◮ Walras’ law holds in the production economy for the same reason that it holds in the pure exchange economy.

◮ Each consumer satisfies her budget constraint, so the economy as a whole has to satisfy an aggregate budget constraint.

Thm (Walras’ law) If u_i is strictly increasing for all i ∈ I, then pz(p) = 0 must hold for all p.

The Proof of Walras’ Law

Proof We expand z(p) according to its definition. pz(p) = p(x(p) − y(p) − e)

= p



 X

i∈I

x_i(p) −^X

j∈J

y_j(p) −^X

i∈I

e_i





=^X

i∈I

px_i(p) −^X

j∈J

py_j(p) −^X

i∈I

pe_i.

Since the budget constraint of every consumer holds with equality, px_i(p) = pe_i+^X

j∈J

θ_ijpy_j(p).

Substituting it into the above equation, we obtain pz(p) =^X

i∈I

pe_i+^X

i∈I

X

j∈J

θ_ijpy_j(p) −^X

j∈J

py_j(p) −^X

i∈I

pe_i

=^X

j∈J

py_j(p) −^X

j∈J

py_j(p) = 0.

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Competitive Equilibrium

The production economy is represented by (u_i, e_i, θ_ij, Y_j)_i∈I,j∈J. Def An allocation-price pair (x, y, p) where p ≫ 0 is called a competitive (Walrasian) equilibrium, if z(p) = 0.

The next theorem guarantees the existence of equilibrium. Thm Consider a production economy (ui^{, e}i^{, θ}ij^{, Y}j)i∈I,j∈J. Suppose that the following conditions are satisfied:

◮ Utility function u_i is continuous, strongly increasing, and strictly quasiconcave for all i ∈ I.

◮ _{0∈ Yj ⊆ R}ⁿ_{, Yj} is closed, bounded and strongly convex.

◮ _y+^P

i∈I^ei ≫ 0 for some aggregate production vector y ∈ Y . Then, there exists at least one price vector p^∗ ≫ 0 such that z(p^∗) = 0. That is, the competitive equilibrium price exists.

Competitive Equilibrium: Example (1)

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☎

Ex Consider a Robinson Crusoe economy where a consumer has✆ the following Cobb-Douglas utility function for consumption x and leisure R and initial endowments e:

u(x, R) = x^aR^1−a

The consumer is endowed with one unit of labor/leisure and there is one firm that has a production function x = L^1/2. Let the price of x be normalized by 1. Then, solve a competitive equilibrium price of labor, denoted by w^∗.

Answer Let us first solve the profit maximization problem: maxL ^L

1_/2

− wL

From the first order condition, ¹₂L^−1/2− w = 0, we obtain L(w) = ¹

(2w)^{2, xs(w) =} 1

2w^{, π(w) =} 1 4w^.

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Competitive Equilibrium: Example (2)

Since the firm’s profits are distributed to the consumer, she solves: maxx,R ^x

a_R¹⁻a _{s.t. x+ wR = w +} ¹

4w

By the property of the Cobb-Douglas utility function, we obtain x_d(w) = a

 w+ ¹

4w



, R(w) = ^{1 − a} w

 w+ ¹

4w

 .

In a competitive equilibrium, the supply and demand for x coincide, x_s(w^∗) = x_d(w^∗) ⇔ ¹

2w^{∗ = a}



w^∗+ ¹ 4w



⇒ w^∗ =^{ 2 − a} 4a

1_/2

.

Note that, by Walras’ law, the labor market also clears.

First Welfare Theorem

Thm If each u_i is strictly increasing on Rⁿ+, then every competitive equilibrium is Pareto efficient.

Proof Suppose not, and let (x^′, y^′) be a Pareto dominating allocation. Then, since consumers are maximizing utility,

px^′_i > pe_i+^X

j∈J

θ_ijpy_j

must hold for all i ∈ I. Summing over consumers,

p^X

i∈I

x^′_i>^X

i∈I

pe_i+^X

j∈J

py_j. Feasibility of x^′, i.e., x^′ = e + y^′, implies

p



 X

i∈I

e_i+^X

j∈J

y^′_j



>^X

i∈I

pe_i+^X

i∈J

py_j ⇔^X

j∈J

py^′_j >^X

j∈J

py_j, which contradicts profit maximization by firms.

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Second Welfare Theorem

Thm Suppose the conditions stated in the existence theorem are satisfied. Let (x^∗, y^∗) be a feasible Pareto efficient allocation. Then, there are income transfers, T1, ..., T_I, satisfying^P_i∈IT_i = 0, and a price vector p such that for all j ∈ J and for all i ∈ I.

1. y^∗_j maximizes py_j s.t. yj ∈ Yj.

2. x^∗_i maximizes u_i(x_i) s.t. px_i ≤ pe_i+^P_j∈Jθ_ijpy^∗_j + T_i.

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✝

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Rm The transfer T✆ i must be set equal to T_i= px^∗_i −



pe_i+^X

j∈J

θ_ijpy^∗_j



 Note that by feasibility of (x^∗, y^∗),

X

i∈I

T_i = p^X

i∈I

(x^∗_i − ei) − p^X

j∈J

y^∗_j = p(x^∗− (e + y^∗)) = 0.