Supply

Traditional topic to cover after neoclassical consumer theory is neoclassical producer theory: theory of profit-maximizing production choices by a firm.

In this presentation, we will not consider

• imperfect competition (Industrial Organization Theory)
• internal behavior and organization of firms (Organizational Economics)

Plan for this chapter:

• General Analysis with production sets
• Production and cost minimization with a single output
• Topics in Producer Theory
  • Aggregation of Supply
  • Possible: Efficiency of Production and FTWE

The production set

As in the previous chapter, we consider an economy with $L$ commodities.

The task of the firm is to transform inputs into outputs and its production plan (or vector) is $y=(y_1,y_2,\cdots ,y_L)\in {\mathbb{R}}^L$.

Definition

Definition (Production Set). The starting point of any analysis of the firm is to identify the production vectors which are technologically possible. Those vectors constitute the production set, denoted $Y\subset {\mathbb{R}}^L$​.

The production set $Y$ can be described using a transformation function $F(·)$. This function is such that $F(y)\le 0$ iff $y\in Y$ and $F(y)=0$ iff $y$ is an element of the boundary of $Y$​.

Definition (Transformation Frontier). The set of boundary points of $Y$ ,$\{y\in {\mathbb{R}}^L:F(y)=0\}$ is called the transformation frontier.

When there is a single output, say $y_L$, the production set is described using a production function $f(·)$​—where inputs are now positive numbers— defined by $q-f(y_1,\cdots ,y_{L-1})\le 0$.

Consider the production plan $\bar{y}$​ such that $F(\bar{y})=0$​​. Then, the slope of the transformation frontier with respect to commodities $\ell$​ and $k$​ is

$$MR{T}_{\ell k}(\bar{y})=\frac{\partial F(\bar{y})/\partial y_{\ell }}{\partial F(\bar{y})/\partial y_k}.$$

Definition (marginal rate of transformation). The absolute value is known as the marginal rate of transformation of good $\ell$​ for good $k$​ at $\bar{y}$​​.

When consider only a single-output, then the production set: $Y=\{(-z_1,\cdots ,-z_{L-1}):q-f(z_1,\cdots ,z_{L-1})\le 0\}$​ and $z=(z_1,\cdots ,z_{L-1})\ge 0$.

Definition ( marginal rate of technical substitution). Using the production function $f(·)$, the $MRT{S}_{lk}$ is called the marginal rate of technical substitution defined by

$$MRTS(\overline{z})=\frac{\partial f(\overline{z})/\partial z_{\ell }}{\partial f(\overline{z})/\partial z_k}$$

Properties of Production Sets

There are different properties that one can attribute to production sets. Some of those properties will hold for all production sets while others will only apply to certain production sets.

1. Y is nonempty.
2. Y is closed. This means that Y contains its boundary.
3. No free Lunch. This means that to produce any strictly positive level of output, one must use some input.
4. Free disposal. The firm can always throw away inputs if it wants. Therefore, for any $y\in Y$ , any point $y^{\prime }\le y$ is also in $Y$ .

Two other properties of production sets, although common, are not as general (or important) as the previous ones.

1. Irreversibility. This property says that the production process cannot be undone. Formally, if $y\in Y$ and $y\ne 0$ then $-y\notin Y$ .
2. Possibility of Inaction. This property says that $0\in Y$.

The following properties refer to the entire production set Y. However, many production sets will exhibit none of these.

1. Nonincreasing returns to scale. Y exhibits NIRS if any feasible production plan $y\in Y$ can be scaled down : $\alpha y\in Y$ for all scalars $\alpha \in [0,1]$.
2. Nondecreasing returns to scale. Y exhibits NDRS if any feasible production plan $y\in Y$ can be scaled up : $\alpha y\in Y$ for all scalars $\alpha \in 1$. Note that if a firm has fixed costs, it may exhibit NDRS but cannot exhibit NIRS.
3. Constant returns to scale. Y exhibits CTRS if it exhibits both NIRS and NDRS at all production levels. Formally, for all $\alpha >0$, if $y\in Y$ , then $\alpha y\in Y$.

Proposition. Suppose $f(·)$ is the production function associated with a single-output technology and $Y$ is the production set of the technology. Then $Y$ exhibits CRS if and only if the production function $f(·)$ is homogeneous of degree 1.

At last, two others common properties.

1. Additivity (free entry). This assumption means that any production that 2 firms can do separately can also be done by a single firm. Suppose $y\in Y$ and $y^{\prime }\in Y$. Then additivity requires that $y+y^{\prime }\in Y$ .

2. Convexity. It states that if $y\in Y$ , $y^{\prime }\in Y$ and $\alpha \in [0,1]$, then $\alpha y+(1-\alpha )y^{\prime }\in Y$.

Profit Maximization Program

Assume $p\gg 0$ and price-taking.

The firm’s objective is to maximize profit. The profit maximization program (PMP) is given by:

$$\underset{y}{max}p\cdot y\text{s.t.}y\in Y.$$

Since $Y=\{y|F(y)\le 0\}$, this can be equivalently stated as

$$\underset{y}{max}p\cdot y\text{s.t.}F(y)\le 0.$$

Assuming differentiability of $F(·)$, the FOC are given by $p_{\ell }=\lambda F_{\ell }(y^{\ast })$ for $i=1,\cdots ,L$ and $F(y^{\ast })=0$

Combining the FOC leads to

$$MRS{T}_{\ell k}=\frac{F_{\ell }(y^{\ast })}{F_k(y^{\ast })}=\frac{p_{\ell }}{p_k}$$

Definition (Net Supply Function). The solution to the PMP, $y(p)$, is called the firm’s net supply function (or correspondence). The value function for this PMP, $\pi (p)=max\{p\cdot y(p):y\in Y\}$ is called the profit function.

PMP with a Single Output

Since $w\ge 0$, the constraint always binds so the problem can be simply stated as $\underset{z}{max}pf(z)-w\cdot z$.

This unconstrained problem is easy to solve but one must take care of the possible corner solutions. The KT FOC are therefore: $p{f}_i(z^{\ast })-w_i\le 0$, with equality if $z^{\ast }>0$, for all $i$.

In the event that the production possibility set is convex (the production function is concave) the first order conditions are of course both necessary and sufficient.

The solution to the PMP is denoted $z(w,p)$ and called the unconditional factor demand function.

By plugging $z(w,p)$ into the production function, we get $q(w,p)=f(z(w,p))$ which is known as the supply function. The profit function can therefore be written as:

$$\pi (w,p)=pf(z(w,p))-w\cdot z(w,p)=pq(w,p)-w·z(w,p)$$

Properties of the net Supply and Profit Functions

Proposition. Suppose that $\pi (·)$ is the profit function of the production set Y and that $y(·)$ is the associated supply correspondence. Assume also that Y is closed and satisfies the free disposal property. Then

1. $y(·)$ is homogeneous of degree zero.
2. If $Y$ is convex, then $y(p)$ is a convex set for all $p$. If $Y$ is strictly convex, then $y(p)$ is single-valued.
3. $\pi (·)$ is homogeneous of degree 1 and is convex.
4. If $Y$ is convex$,Y=y\in {\mathbb{R}}^L:p\cdot y\le \pi (p)$ for all $p\gg 0$.
5. (Hotelling’s lemma) If $y(\overline{p})$ is single-valued at $\overline{p}$, then $\pi (·)$ is differentiable at $\overline{p}$ and $\mathrm{\nabla }\pi (\overline{p})=y(\overline{p})$.
6. If $y(·)$ is a function differentiable at $\overline{p}$, then $D_y(\overline{p})=D^2\pi (\overline{p})$ is a symmetric and positive semidefinite matrix with $D_y(\overline{p})\overline{p}=0$.

Cost Minimization Problem

The profit maximization can be obtained in two sequential steps:

1. Given $y$, and the choice of inputs that allows the producer to obtain $y$ at the minimum cost. This generates conditional factor demands and the cost function.
2. Given the cost function, and the profit maximizing output level.

CMP with a Single Output

Any solution of the PMP should also solve the CMP.

Assume $z\ge 0$, $f(z)$ is the production function, $w\gg 0$ is the price vector of the inputs $z$.

For a given level of production $q$, the CMP writes as

$$\underset{z\ge 0}{min}w\cdot z\text{s.t.}f(z)\ge q$$

If $\lambda \ge 0$ denotes the lagrange multiplier, the optimality FOC are

$$w_{\ell }\ge \lambda f_{\ell }(z^{\ast })\text{ with equality if }{z}^{\ast }>0\text{ for all }\ell .$$

For any two inputs $\ell ,k$ and $(z_{\ell }^{\ast },z_k^{\ast })\gg 0$ (not corner solution), we have

$$MRT{S}_{\ell k}=\frac{w_{\ell }}{w_k}$$

Definition (Conditional Factor Demand Function). The solution of the problem, denoted $z(w,q)$, is known as the conditional factor demand function.

Definition (Cost Function). The value function of the optimization problem is the cost function $c(w,q)=w\cdot z(w,q).$

Properties of factor demand and cost functions

Proposition. Suppose $c(w,q)$ is the cost function associated with a single-out technology Y with production function $f(·)$ and that $z(w,q)$ is the associated conditional factor demand correspondence. Assume that Y is closed and satisfies the free disposal property. Then

1. $z(w,q)$ is homogenous of degree zero in w.
2. $c(w,q)$ is homogenous of degree one in w.
3. $c(w,q)$ is non-decreasing in q and concave in w.
4. (Shephard’s Lemma) If $z(\overline{w},q)$ is single valued at $\overline{w}$, then ${\mathrm{\nabla }}_{w}c(\overline{w},q)=z(\overline{w},q)$.
5. If $z(w,q)$ is differentiable, then the matrix of second derivatives of the cost function w.r.t. prices is a symmetric negative semi-definite matrix.

The cost function, generated by the CMP, contains in fact the same information as $Y$ (the same way as $\pi (y)$ can be used to recover $Y$ ).

6. If $f(·)$ is homogeneous of degree one, then $c(·)$ and $z(·)$ are homogeneous of degree one in $q$.
7. If $f(·)$ is concave, then $c(·)$ is a convex function of $q$.

So, we can rewrite the firm’s profit maximization problem using the cost function,

$$\underset{q}{max}pq-c(w,q)$$

Solving this problem will yield the same input usage and output production as if the PMP had been solved in its original form.

Aggregation of Supply

In consumer theory, we said that aggregation could be a serious problem. This is not the case for supply, since there are no wealth effects. To aggregate supply, it is sufficient to add up the individual supply functions.

Consider $J$ production units specified by a production set $Y_1,\cdots ,Y_J$. Assume each $Y_j$ is nonempty closed, and satisfies the free disposal property. Denote the profit function and supply correspondence by $Y_j$ by ${\pi }_j(p)$ and $y_j(p)$.

Then the aggregate supply correspondence is

$$y(p)=\sum _{j=1}^J{y}_j(p)$$

Proposition. In a purely competitive environment the maximum profit obtained by every firm maximizing profits separately is the same as the profit obtained if all J firms were to coordinate their choices in a joint profit maximization. For all $p\gg 0$, we have

$${\pi }^{\ast }(p,w)=\sum _j{\pi }_j(p)$$$$y^{\ast }(p,w)=\sum _j{y}_j(p)$$

This implies that the decentralized allocation of the production among firms is cost minimizing.

Efficiency of Production

We will say that a production plan $y\in Y$ is efficient if there is no other vector $y^{\prime }\in Y$ such that $y^{\prime }\ge y$ and $y^{\prime }\ne y$. So, we have efficiency when we cannot produce the same output with less inputs or produce a greater output with the same amount of inputs.

As in the general case (with consumption), we can state two important results.

FTWE If $y\in Y$ is profit maximizing for some $p\gg 0$, then it is efficient.

STWE If $Y$ is convex, then every efficient production $y\in Y$ is a profit-maximizing production for some nonzero price vector $p\ge 0$.

So production choices are not wasteful and any choice can be decentralized with an appropriate choice of prices.