Production and Costs in the Long Run

1. Production and Costs in the Long Run

2. The long run • The long run is the time frame longer or just as long as it takes to alter the plant. • Thus the long run is that time period in which all inputs are variable.

3. Isocost lines An isocost line includes all possible combinations of labor and capital that can be purchased for a given total cost. In equation form the total cost is TC = PLL + PKK, where TC = Total cost, PL= the wage rate, L = the amount of labor taken, PK = the rental price of capital, and K = the amount of capital taken. This equation can be re-expressed as K = TC/ PK - (PL/ PK) L.

4. example As an example say labor is $6 per unit and capital is $10 per unit. Then if we look at a total cost of $100 we see various combinations of inputs: L = 10 and K = 4 or L = 0 and K = 10 or L = 16.67 and K = 0, amoung others. On the next screen we can view the isocost line in a graph.

5. graph of isocost line K This is the isocost line at $100. If we wanted to see higher costs we would shift the line out in a parallel shift and a lower cost we have a shift in. L

6. cost and output K On this slide I want to concentrate on one level of output, as summarized by the isoquant. Input combination L1, K1 could be used and have cost summarized by 4th highest isocost shown. L2, K2 would be cheaper, and L*, K* K1 K2 K* L L1 L2 L* is the lowest cost to produce the given level of output. Here the cheapest cost of the output occurs at a tangency point.

7. cost and output K On this slide I want to concentrate on one level of cost, as summarized by the isocost line. Input combination L1, K1 could be used and have this cost but more output would be obtained if L*, K* were used. K1 K* L L1 L* Here, the most output for a given cost occurs at a tangency point.

8. cost and output On the last two screens we have seen the tangency of an isoquant and isocost line shows either 1) the cheapest way to produce a certain level of output, or 2) the most output that can be obtained for a given amount of cost. These two things are different sides of the same coin and profit maximizing firms would be expected to reach the tangency positions. The exception to reaching the tangency would be the short run when the amount of some input can not be changed to reach the tangency. In the long run all inputs can be changed in amount and thus the tangency point could be reached.

9. short run K Here the cheapest way to produce the output level as depicted in the isoquant would be to hire L*, K*. But the firm as committed to having K1 units of capital. Thus the cost of this output is indicated by the fourth highest isocost line. K1 K2 K* L L1 L2 L* We could follow K1 out and see costs of other levels of output(by putting in more isoquants).

10. Long Run • Another way to view the long run is to think about different short run situations and put them together. Think of a short run with one capital unit. Think of one with two capital units, and so on. • We would have a similar table of numbers and graphs as we did in the short run example when only one unit of capital was available.

11. Tangency In the long run when a firm is able to change all inputs we see the firm will go to a point where the slope of an isocost line is tangent to a isoquant. This means the slopes are equal. Thus Slope of isocost = - (PL/ PK) = -MRTS = slope of isoquant. Remember we said MRTS can be shown to be the ratio of marginal products of labor to capital.

12. Expansion path K Once we have a unit cost of capital and labor we can draw many isocosts, each one that is farther out has a higher cost. We can see the tangency of each isocost with an output level. L In the long run the firm will be at one of the points of tangency. When connect all those points we have the expansion path. In the long run the firm will be on the expansion path.

13. Long Run continued • When we switch from one unit of capital to two units, we have the long run because all inputs are then variable. • But with the two units we would have short run curves for that level of capital. • Now we have two sets of cost curves, one for one unit of capital and one for two units of capital.

14. Long run continued • Thus the graph of the long run is really just a bunch of curves, one for each plant size. • I will draw two ATC curves, each with a different amount of capital used.

15. Long Run Graphs ATC ATC2 ATC1 Q Q1

16. Interpretation • If output is going to be less than Q1 in the long run then only one unit of capital would be wanted because those units would be produced cheapest with one unit of capital. • Greater than Q1 would be produced cheapest with two units of capital.

17. Interpretation • The long run curve is parts of the short run curves. For each range of output the long run curve is the segment of the short run curve that is the lowest, representing the cheapest way to produce that range of output in the long run. The final long run curve is smooth. Let’s see.

18. Smooth long run curve ATC Q

19. Reason for long run shape • The long run cost curve is said to be u - shaped, just as in the short run, but for a different reason. In the short run we had diminishing returns. In the long run we have economies of scale.

20. Reason continued • The basic idea of economies of scale is that at least for a while when the plant size is increased the average cost curve is pushed down, implying average costs are lowest in a bigger plant. It may be that further increases in plant size push the average cost curve back up. This would technically be called diseconomies of scale.