Theory of Production

General Setup

A firm uses labor and capital as inputs to produce an output. If \(L\) is the amount of labor input and \(K\) is the amount of capital input, then the amount of output \(Y\) is given by:

\[Y = f(K,L)\]

We call \(f(K,L)\) the production function because it maps the choice of inputs to the amount of output.

Firms can hire one unit of labor at a wage rate \(w\) and it can rent one unit of capital at a rental rate \(r\). For simplicity, we assume that the price of the firm’s output is \(1\). (We can simply assume that labor and capital prices are priced in terms of the output unit.) The firm is a price-taker in both the input and output markets.

The firm chooses \(L\) and \(K\) to maximize profits. The firm’s optimization problem is:

\[\max_{K,L} ~ f(K,L) - wL - rK\]

First order conditions

This is an unconstrained maximization problem. The optimal choice of \(K\) and \(L\) occurs when the partial derivatives with respect to \(K\) and \(L\) are equal to zero. The first order conditions are:

\[\begin{align} f_K(K,L) &= r \\ f_L(K,L) &= w \end{align}\]

The first equation says that at the firm’s optimal choice, the marginal product of capital, \(f_K(K,L)\), is equal to the rental rate of capital, \(r\). And the marginal product of labor, \(f_L(K,L)\) is equal to the wage rate.

Economic Insight

In the equilibrium of a market with price-taking, profit-maximizing firms, the wage rate will equal the marginal product of labor, and the capital rental rate will equal the marginal product of capital.

Alternative Setup: Cost Functions

An alternative, but equivalent, way to formulate the problem is to first derive a cost function. We define the cost function \(c(q)\) as the minimum total cost to produce \(q\) units of output:

\[c(q) = \min_{K,L} ~ wL + rK ~ ~ \text{ s.t. } f(K,L)=q\]

Once we have the cost function, the firm’s optimization problem becomes:

\[\max_{q} ~ q - c(q)\]

(Remember, we’re assuming that the output price \(p=1\).)

This way of formulating the problem is equivalent

Cobb Douglas Production Function

In this class we will work only with a

Returns to scale in Cobb Douglas production functions

  • If \(a+b=1\), then the production function exhibits constant returns to scale. That is, if you increase both \(L\) and \(K\) by a factor of \(m\), output also increases by a factor of \(m\).

Proof

\[\begin{align} f(mL, mK) &= A (mL)^a (mK)^b \\ &= A m^a L^a m^b K^b \\ &= m^{a+b} A L^a K^b \\ &= m^{a+b} f(L,K) \\ &= m f(L,K) \end{align}\]
  • If \(a+b<1\), then the production function exhibits decreasing returns to scale. That is, if you increase both \(L\) and \(K\) by a factor of \(m\), output increases by a factor less than \(m\).

Proof

\[\begin{align} f(mL, mK) &= m^{a+b} f(L,K) < m f(L,K) \end{align}\]
  • If \(a+b>1\), then the production function exhibits increasing returns to scale. That is, if you increase both \(L\) and \(K\) by a factor of \(m\), output increases by a factor more than \(m\).

Proof

\[\begin{align} f(mL, mK) &= m^{a+b} f(L,K) > m f(L,K) \end{align}\]