Producer's Equilibrium

When producing a good or service, how do suppliers determine the quantity of factors to hire? Below, we work through an example where a representative producer answers this question.

Let’s begin by making some assumptions. First, we shall assume that our producer chooses varying amounts of two factors, capital (K) and labor (L). Each factor was a price that does not vary with output. That is, the price of each unit of labor (w) and the price of each unit of capital (r) are assumed constant. We’ll further assume that w = $10 and r = $50. We can use this information to determine the producer’s total cost. We call the total cost equation an isocost line (it’s similar to a budget constraint).

The producer’s isocost line is:


10L + 50K = TC (1)



The producer’s production function is assumed to take the following form:


q = (KL)0.5 (2)



Our producer’s first step is to decide how much output to produce. Suppose that quantity is 1000 units of output. In order to produce those 1000 units of output, our producer must get a combination of L and K that makes (2) equal to 1000. Implicitly, this means that we must find a particular isoquant.

Set (2) equal to 1000 units of output, and solve for K. Doing so, we get the following equation for a specific isoquant (one of many possible isoquants):


K = 1,000,000/L (2a)



For any given value of L, (2a) gives us a corresponding value for K. Graphing these values, with K on the vertical axis and L on the horizontal axis, we obtain the blue line on the graph below. Each point on this curve is represented as a combination of K and L that yields an output level of 1000 units. Therefore, as we move along this isoquant output is constant (much like the fact that utility is constant as we move along an indifference curve).