Lecture 9: Firm Problem
Advanced Microeconomics I
Yosuke YASUDA
Osaka University, Department of Economics yasuda@econ.osaka-u.ac.jp
November 4, 2014
Production
Consumer theory imposes a strong structure on the choice sets (budget sets) but few constraints on the preferences or utility function. In contrast, classic producer theory assigns the producer a highly structured target function (profit function) but fewer constraints on the choice sets (technology or production functions).
Production is the process of transforming inputs into outputs. Let 1, ..., k be commodities. The producer’s choice will be made from subsets of the “grand set,” k-dimensional Euclidean space.
A vector z in this space is interpreted as a production combination; positive components in z are interpreted as outputs and negative components as inputs.
(Note that the firm’s profit can be expressed as pz=Pki=1pizi where p ∈ Rk++ is a vector of prices.)
2 / 14
Technology
Definition 1
A producer’s choice set is called a technology (or production possibility set), Z ⊂ Rk, which specifies the production constraints.
The following restrictions are usually placed on the technology Z:
1 0∈ Z: Producer can remain “idle.”
2 There is no z ∈ Z ∩ Rk+ besides the vector 0: No production with no resources.
3 If z ∈ Z and z′ ≤ z, then z′ ∈ Z: Free disposal.
4 Z is a convex set: Decreasing return to scale.
5 Z is a closed set.
✞
✝
☎
Rm Check each restriction by drawing a figure with k = 2.✆
Production Function (1)
Suppose that a commodity (denoted by 0) is produced from other commodities 1, ..., n, that is, for all z ∈ Z, z0 ≥ 0 and zi ≤ 0 for i= 1, 2, ..., n.
Another way to describe the firm’s technological constraints is a production function, f : Rn+→ R+, which specifies, for any positive vector of inputs x ∈ Rn+, the maximum amount of commodity 0 that can be produced:
f(x) = max{y|(−x, y) ∈ Z}.
Theorem 2
When a function f satisfies the assumptions of (i) f (0) = 0, (ii) increasing, (iii) continuity, and (iv) concavity, then Z(f ) satisfies the above assumptions 1-5, where Z(f ) is defined as
Z(f ) = {(−x, y)| y ≤ y′ and x ≥ x′ for some y′ = f (x′)}.
4 / 14
Production Function (2)
✞
✝
☎
Q A production function f (x) is said to be✆
1 Constant returns to scale (homogeneous of degree 1): if f (tx) = tf (x) for all t > 0 and all x.
2 Increasing returns to scale:
if f (tx) > tf (x) for all t > 1 and all x.
3 Decreasing returns to scale:
if f (tx) < tf (x) for all t > 1 and all x. Theorem 3
Suppose that the production function f (x) satisfies (i) f (0) = 0, (ii) increasing, (iii) continuity, and (iv’) quasi-concavity, and constant returns to scale. Then it is a (iv) concave function.
Cost function (1)
Let w = (w1, ..., wn) ≥ 0 be a vector of prevailing market prices at which the firm can buy inputs (x1, ..., xn).
Definition 4
The cost function is the minimum-value function of the following cost minimization problem,
c(w, y) =: min
x∈Rn+
wx s.t. f (x) ≥ y.
Mathematically speaking, the cost function is identical to the expenditure function.
e(p, u) =: min
x∈Rn+
px s.t. u(x) ≥ u.
So, the properties that the expenditure function e(·) possesses also hold for the cost function c(·).
6 / 14
Cost function (2)
Theorem 5
If f is continuous, strictly increasing and f (0) = 0, then the cost function c(w, y) : Rn+1+ → R+ is
1 Zero when y = 0.
2 Continuous on its domain.
3 For all w≫ 0, strictly increasing and unbounded above in y.
4 Increasing in w.
5 Homogeneous of degree one in w.
6 Concave in w.
Moreover, if f is strictly quasi-concave we have
7 Shephard’s lemma: c(w, y) is differentiable in w at (w0, y0) whenever w≫ 0, and
∂c(w0, y0)
∂wi
= xi(w0, y0), i = 1, ..., n.
Firm’s Objectives
We assume the goal of the firm (producer) is to maximize profits.
✞
✝
☎
Q Why is the producer’s goal “profit maximization”?✆
(Only) Profit maximizing firms would survive in the long-run under competitive pressure.
Profit maximization is the single most robust and compelling assumption. (JR, pp.125)
Just for mathematical convenience? (Rubinstein, Lecture 7)
✞
✝
☎
Ex Alternative plausible(?) targets:✆
Increase market share subject to not incurring a loss. Increase salaries with less regard for the level of profits.
8 / 14
Cost minimization (1)
If the object of the firm is to maximize profits, it will necessarily choose the least costly production plan for every level of output. (Note this must be true for all profit-maximizing firms, whether monopolists, perfect competitors, or anything between.)
min
x∈Rn+
wx s.t. f (x) ≥ y.
Assuming x∗ ≫ 0 and solve this problem by Lagrange’s method. (Note f (x) = y must be satisfied at the optimal solution.)
L = wx − λ(f (x) − y).
∂L
∂xi = wi− λ
∂f(x∗)
∂xi = 0 for i = 1, ..., n.
∂L
∂λ = f (x) − y = 0.
Cost minimization (2)
From the first order conditions, we obtain
∂f(x∗)
∂xi
∂f(x∗)
∂xj
= wi
wj for any i, j.
Cost minimization implies that the marginal rate of technical substitution (MRTS) between any two inputs is equal to the ratio of their prices.
Definition 6
The solution x(w, y) is referred to as the firm’s conditional (contingent) input demand, because it is conditional on the level of output y, which at this point is arbitrary and so may or may not be a profit maximizing.
10 / 14
Profit Maximization (1)
Suppose the competitive firm can sell each unit of output at the market price, p. There are two different ways to solve the firm’s profit maximization problem.
One-step procedure: max
x∈Rn+
py− wx s.t. f (x) ≥ y
which is equivalent to (since f (x) = y must hold) P M1 : max
x∈Rn+
pf(x) − wx.
Two-step procedure: P M2 : max
y≥0 py− c(w, y).
Profit Maximization (2)
✞
✝
☎
Rm It can be shown that the P M✆ 1 and P M2 are equivalent. Assuming x∗ ≫ 0, P M1 is just an unconstrained problem with
“multiple” variables. So we can solve it by just taking partial derivatives.
∂[pf (x) − wx]
∂xi
= p∂f(x
∗)
∂xi
− wi = 0 for i = 1, ..., n.
From the first order conditions, we obtain
∂f(x∗)
∂xi
∂f(x∗)
∂xj
= wi wj
for any i, j.
This is precisely the same as the condition for cost minimization input choice. It therefore confirms our earlier intuition that profit maximization “requires” cost minimization in production.
12 / 14
Profit Maximization (3)
Definition 7
The optimal choice of inputs x∗= x(p, w) gives the vector of input demand functions. (Note these input demands are no longer “conditional” since they are independent of output.) Assuming y∗ >0, P M2 is just an unconstrained problem with
“one” variable: d[py − c(w, y)]
dy = p −
dc(w, y∗)
dy = 0.
Therefore, output is chosen so that “price equals marginal cost”. Definition 8
The optimal choice of output y∗ = y(p, w) is called the firm’s (output) supply function. The maximum-value function, denoted by π(p, w), is called the profit function.
π(p, w) =: maxnpf(x) − wx.
Profit Maximization (4)
Theorem 9
Suppose f is continuous, strictly increasing, strictly quasi-concave, and f (0) = 0. Then, for p ≥ 0 and w ≥ 0, π(p, w), where
well-defined, is continuous and
1 Increasing in p.
2 Decreasing in w.
3 Homogeneous of degree one in (p, w).
4 Convex in (p, w).
5 Differentiable in (p, w) ≫ 0. Moreover, ( Hotelling’s lemma),
∂π(p, w)
∂p = y(p, w) and −∂π(p, w)
∂wi
= xi(p, w), i = 1, ..., n.
✞
✝
☎
Rm Note that profit maximization problem is mathematically not✆ identical to utility maximization problem.
14 / 14
