4.4 The cost function

The cost function is $C = \sum_i p_i x_i$.

Just the quantity of inputs times their prices.

Relates the minimum cost of producing a level of output y to the prices of the inputs and the state of technical knowledge.

Objective; Find the input levels that minimize the production cost for a given level of output. (Cost minimizer assumption.)

This is an optimization problem, in particular choose input levels to minimize costs. But subject to a constraint: the inputs must produce a given level of output, y.

Mathematical technique for solution of constrained optimization: Lagrange multipliers.

It turns out that, assuming the Cobb Douglas production function, then the optimal level of inputs produce a COST FUNCTION of the form

$$C = k\, y^{1/r} p_1^{\alpha_1/r} p_2^{\alpha_2/r} p_3^{\alpha_3/r},$$

where

$$k = r ( A \alpha_1^{\alpha_1} \alpha_2^{\alpha_2} \alpha_3^{\alpha_3})^{-1/r}.$$

It looks a mess, but notice that it is multiplicative, so taking logs will achieve a linear expression ready for regression.

Further, using the fact that $\alpha_3 = r - \alpha_1 - \alpha_2$ the logged version can be rewritten as

$$\ln(C^\ast) = \beta_0 + \beta_y \ln(y) + \beta_1 \ln(p_1^\ast) + \beta_2 \ln(p_2^\ast),$$

where

• $\ln(C^\ast) = \ln(C) - \ln(p_3)$
• $\ln(p_1^\ast) = \ln(p_1) - \ln(p_3)$
• $\ln(p_2^\ast) = \ln(p_2) - \ln(p_3)$
• $\beta_0 = \ln(k)$
• $\beta_y = 1/r$
• $\beta_1 = \alpha_1/r$
• $\beta_2 = \alpha_2/r$

From this lot we can get at what's of interest, $r, \alpha_1, \alpha_2, \alpha_3$.