3 The Theory of Supply

3.1 Introduction

3.1.1 Module Meta Information

Reading material: MWG Ch. 5

3.2 Production plans

A production plan is a vector \(y \in \mathbb{R}^{L}\) that describes both inputs (negative) and outputs (positive) of a firm. The production set \(Y\) is the set of all feasible production plans.

Some assumptions on production sets.

\(Y\) is nonempty
\(Y\) is closed.
No free lunch. If \(y \in Y\) and \(y \ge 0\) then \(y = 0\).
\(0 \in Y\). (This might not hold if there are sunk costs.)
Free disposal. For \(y \in Y\), \(y' \in Y\) if \(y' \le y\).
Irreversibility. \(y \in Y\) implies that \(-y \notin Y\) for \(y \ne 0\).
Non-increasing returns to scale. \(y \in Y\) implies that \(\alpha y \in Y\) for \(\alpha \in [0,1]\)
Non-decreasing returns to scale. \(y \in Y\) implies that \(\alpha y \in Y\) for \(\alpha \ge 1\).
Constant returns to scale \(y \in Y\) implies that \(\alpha y \in Y\).
Convexity. If \(y,y' \in Y\), then \(\alpha y + (1-\alpha)y' \in Y\).

Given a production set \(Y\) and a price vector \(p \gg 0\), the firm’s problem is

\[\pi(p) = \max_{y \in Y} \; p \cdot y\]

The value function \(\pi(\cdot)\) is called the profit function and the set of maximizers \(y(p) = \{y \in Y : p\cdot y = \pi(p)\}\) is the supply correspondence.

Proposition 3.1 (5.C.1) Let \(\pi(\cdot)\) be the profit function of a production set \(Y\) and \(y(\cdot)\) be the supply correspondence. If \(Y\) is closed and satisfies free disposal then

\(\pi\) is homogeneous of degree one.
\(\pi\) is convex.
If \(Y\) is convex, then \(Y = \{y \in \mathbb{R}^{L}: p\cdot y \le \pi(p) \text{ for all } p >> 0 \}\)
\(y\) is homogeneous of degree zero
If \(Y\) is convex, then \(y(p)\) is a convex set for all \(p\). Moreover, if \(Y\) is strictly convex, then \(y(p)\) is single valued.
(Hotelling’s lemma) If \(y(\overline p)\) consists of a single point, then \(\pi(\cdot)\) is differentiable at \(\overline p\) and \(D_{p} \pi(\overline p) = y(\overline p)\)
If \(y(\cdot)\) is a function differentiable at \(\overline{p}\), then \(Dy(\overline{p}) = D^{2}\pi(\overline{p})\) is a symmetric and positive semidefinite matrix with \(Dy(\overline{p})\overline{p} = 0\).

For a single-output technology with production function \(f\), the firm’s cost-minimization problem given factor prices \(w \gg 0\) and output level \(q\) is

\[c(w,q) = \min_{z \ge 0} \; w \cdot z \quad \text{subject to} \quad f(z) \ge q\]

The value function \(c(\cdot)\) is called the cost function and the set of minimizers \(z(w,q)\) is the conditional factor demand correspondence.

Proposition 3.2 (5.C.2) Let \(c(w,q)\) be the cost function of single output technology \(Y\) with production function \(f\) and \(z(w,q)\) be the conditional factor demand correspondence. Assume also that \(Y\) is closed and satisfies free disposal.

\(c(\cdot)\) is homogeneous of degree one in \(w\) and non-decreasing in \(q\).
\(c(\cdot)\) is a concave function of \(w\).
If the set \(\{z\ge0: f(z)\ge q \}\) are convex for every \(q\), then \(Y = \{(-z,q): w\cdot z \ge c(w,q) \text{ for all } w >> 0\}\)
\(z(\cdot)\) is homogeneous of degree 0 in \(w\).
If the set \(\{z \ge 0: f(z) \ge q\}\) is convex, then \(z(w,q)\) is a convex set. Moreover, if \(\{z \ge 0: f(z) \ge q\}\) is a strictly convex set, then \(z(w,q)\) is single-valued.
(Shepard’s lemma) If \(z(\overline{w},q)\) consists of a single point, then \(c(\cdot)\) is differentiable with respect to \(w\) at \(\overline{w}\) and \(D_{w} c(\overline{w},q) = z(\overline{w},q)\).
If \(z(\cdot)\) is differentiable at \(\overline{w}\), then \(D_{w}z(\overline{w},q) = D^{2}_{w}c(\overline{w},q)\) is a symmetric and n.s.d. matrix with \(D_{w}z(\overline{w},q)\overline{w} = 0\).
If \(f(\cdot)\) is homogeneous of degree one (i.e. C.R.S.) then \(c(\cdot)\) and \(z(\cdot)\) are homogeneous of degree one in \(q\).
If \(f(\cdot)\) is concave, then \(c(\cdot)\) is a convex function of \(q\) (in particular, marginal costs are nondecreasing in \(q\)).

Exercise on fixed costs and non-convexities of production.