Production

Perloff 6, 7; WB 12 and 13.
We examine how firms' costs are determined by two key Supply shifters — technology and input prices. This provides a foundation for the Supply curve in our model of perfect competition (it is equal to the marginal cost curve).

Of course, costs also matter in non-competitive markets, oligopolies and monopolies. We will see that a firm's output goal changes depending on in its market power. However, all firms minimize costs.

In this section the firm uses two inputs to reach a fixed output goal (quantity to supply to the market). The firm buys labor L and capital K to produce q units of the good at the lowest cost. This cost minimization problem is very similar to the consumer's utility maximization problem.

Technology

The firm's technology is described by a production function F(L,K), which gives the output that can be made with any combination of labor L and capital K. We make two assumptions on the technology…

Free disposal and useful inputs

We assume that the production function is (strictly monotone) increasing in both L and K. The production function can never be decreasing in the inputs because the firm can freely dispose of unwanted inputs. If the production function is always increasing (as we assume), then both inputs are always useful, no matter how many are currently being used.

  • What is the free disposal assumption?

Convexity

We also assume that the marginal products MPL and MPK are decreasing. This makes the technology's isoquants strictly convex. Or, equivalently, it makes the isoquant slopes — the Marginal Rate of Technical Substitution (MRTS) — fall.

  • What is the Law of Diminishing Marginal Returns?

Graphing technology

All pairs of inputs L and K that reach the same output goal q form a curve called the isoquant for q. Our assumptions yield isoquants just like the indifference curves seen in the Consumer Choice model. Again, we will consider three cases:

  • Cobb-Douglas, which satisfies both assumptions;
  • Perfect substitutes, which violates strict convexity; and
  • Fixed-proportions, which violates strict monotonicity.
  • Graph the relevant isoquant for fixed-proportions technology, given the output goal and either the L-K proportion or the production function.
  • Graph the relevant isoquant for perfect-substitutes technology, given the output goal and either the L-K trade-off or the production function.

Returns to scale

The firm's technology has increasing returns to scale if a proportional increase in both inputs causes a more-than-proportional increase in outputs. In other words, the scale elasticity of output must be greater than one:

(1)
\begin{align} \varepsilon_{SCALE}=\frac{\Delta F(tL,tK)}{\Delta t}\times\frac{t}{q} \end{align}

With increasing returns to scale, the amount of (both) resources required per unit produced falls with output.

  • Given a simple production function, determine if it has increasing, decreasing or constant returns to scale.
  • Given a complicated production function, determine if it has increasing, decreasing or constant returns to scale.

Costs

All pairs of inputs L and K that cost the same amount wL+rK form a line called the isocost line. The slope is -w/r.

  • Graph some isocost lines, given the input prices.

Cost minimization

Goal Must have
Consumer's problem pick X*, Y* to max utility spending = I
Producer's problem pick L*, K* to min spending output = q

As with utility maximization, we need two numbers, so we need two equations. The first equation is that the production from L and K is equal to the output goal. The second condition depends on which of three cases we are looking at.

Solve the firm's cost-minimization problem given an output goal q, input costs w and r, and…

  • Cobb-Douglas technology.
  • Fixed-proportions technology.
  • Perfect substitutes technology.
  • Calculate the cost.
  • Calculate the cost as a function of input prices and the output goal.

Perfect substitutes

Compare the cost of producing using only L and using only K.

Fixed-proportions

Use the proportion as the second equation.

Cobb-Douglas

Use the slope-matching approach (MRTS = -w/r) or the cost-share shortcut.

Economies of scale

There are economies of scale when a proportional increase in output causes a less-than-proportional increase in costs. If we define the scale elasticity of cost as $\xi_{SCALE}=\frac{\Delta C}{\Delta q}\times\frac{q}{C}$, then there are economies of scale whenever this elasticity is less than one. If it is greater than one, there are ‘diseconomies’ of scale.

With economies of scale, the amount of money required per unit produced falls with output. (Note the difference between this and increasing returns to scale.)

  • Given a cost function, determine if there are economies of diseconomies of scale (or neither).

Short-run cost minimization

Suppose that, in the short-run, capital K cannot be changed. Then the firm will choose labor L to minimize cost.

  • Solve the firm's short run cost-minimization problem given an output goal q, input prices w and r, one of the three types of technology, and a fixed level of K; and calculate the cost.

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