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Tutorial 8: The Costs of Production (cont.)

 

In the last Tutorial we developed and practiced using the isocost/isoquant model to see how a firm chooses the cost-minimizing combination of inputs that would get it the desired level of output. Now we look at the differences in the cost of increasing output in the long run (when no inputs are fixed) and in the short run (when at least one input is fixed).

 

Cost Minimization in the Short Run and in the Long Run Top of page.

The Long-Run Expansion Path

Suppose the firm's production function gives it constant returns to scale (need a review?). Figure 8 illustrates the impact on the firm's costs of increasing its output by one and one-half times from 100 to 150 units/day. Because the firm faces constant returns to scale, it will have to hire 1.5 times its original number of inputs, so L and K increase from (5, 5) to (7.5, 7.5). As a result, the cost of producing the new output to increases by 1.5 times, from $20 to $30.

The two cost-minimizing combinations of labor and capital, and any number of other such combinations we could generate, can be plotted together as a line from the origin showing all the cost minimizing input combinations for various levels of output this firm could produce in the long run. It is referred to as the firm's long-run expansion path (LREP). It shows the cheapest cost of raising output in the long run.

 

Figure 8


Graph of the firm's long-run output expansion path.

 

Short-Run Expansion Path Top of page.

Suppose the firm could not expand the use of both inputs, i.e., suppose the firm is looking to expand output in the short run? Let's freeze K at 5 hours per day. Figure 9 shows the result. To increase its output from 100 to 150 units per day, given its production function, the firm will have to increase the amount of labor from 5 to 11.25 hours per day (K = 5 hours/day). This raises total cost from $20 to $32.50. Thus, when the firm expands output in the short run it faces higher costs. This is due to the inflexibility of the production process in the short run when at least one input is fixed. The short-run expansion path (SREP) illustrates the minimum cost of increasing output in the short run. Because the isocost line C1 lies above the point of tangency with isoquant Q1 , we can see that even this minimum short run cost is more expensive than the minimum cost possible in the long run.

 

Figure 9


Graph of the firm's short-run output expansion path.

 

Long-Run Average Cost Top of page.

Economies and Diseconomies of Scale

When a firm's production function gives it constant returns to scale, as does the one pictured in Figure 9, a doubling of both inputs will double output. Well it will also double cost. So average cost, C(Q)/Q, will remain unchanged. Let's see that mathemagically:

AC1(Q) = C(Q)/Q
AC2(Q) = 2 · C(Q)/2 · Q
\ AC2(Q) = AC1(Q)

 

But what about the other types of returns to scale? When a firm's production function gives it increasing returns to scale a doubling of both inputs will more than double output. Of course it will also double cost. So average cost will fall.

AC1(Q) = C(Q)/Q
AC2(Q) = 2 · C(Q)/3 · Q
\ AC2(Q) = (2/3) · AC1(Q)

 

Thus a firm enjoying increasing returns to scale is likely to experience economies of scale when it increases its output in the long run. With economies of scale, a firm can double output at less than double the cost.

Similarly, when a firm's production function gives it decreasing returns to scale a doubling of both inputs will less than double output. Of course it will also double cost. So average cost will rise.

AC1(Q) = C(Q)/Q
AC2(Q) = 2 · C(Q)/1.5 · Q
\ AC2(Q) = (1 1/3) · AC1(Q)

 

Thus a firm has decreasing returns to scale is likely to experience diseconomies of scale when it increases its output in the long run. With diseconomies of scale, a firm can double output but at more than double the cost.

Another way to look at how much more expensive it is to increase output in the short than in the long run, is to look at a graphical model comparing short run and long run average costs. Figures 7.8 and 7.9 (p. 225) in P&R provides two specific illustrations. In the short run, the firm is moving along its short run average cost curve (like the one you generated for Tutorial 8 Question 8 in the CostMin.xls workbook). In the long run it moves along a long-run average cost curve. The long-run average cost curve is an envelope function (don't worry about it) that is just tangent to a set of short run average cost curves, each of which represents a short run increase in the scale of production. The shape of the long-run average cost curve depends on whether the production function generates increasing, constant, or decreasing returns to scale.

 

In part d we used the isocost/isoquant model to discover the differences in the cost of increasing output in the short run (when at least one input is fixed) and in the long run (when no inputs are fixed). We found that increasing production in the short run was much more expensive that increasing output in the long run. This was the result of the inflexibility of the production process in the short run (when at least one input is fixed).

Now it's time to look at the revenue side of production. Then we will be in the position to determine how a profit-maximizing firm chooses how much output to produce.