EconS Firm Optimization

Transcription

1 EconS Firm Optimization Eric Dunaway Washington State University eric.dunaway@wsu.edu October 9, 2015 Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

2 Introduction Over the past two lectures, we have seen how rms can allocate capital and labor to reach both a desired output level and a desired cost level. Today, we will put both of these concepts together to derive the rm s demand for capital and labor. Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

3 Long Run Costs A few things to remember from last time: We are dealing with long run costs (since neither capital nor labor are xed). The rm maximizes pro ts (its goal) by minimizing its costs. We know the desired level of output we want to achieve (the isoquant), but we don t know what the minimum cost level is to achieve that output level (the isocost). Let s use the same techniques we used in consumer theory to solve the rm s problem. Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

4 Long Run Costs K L Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

5 Long Run Costs K L Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

6 Long Run Costs K L Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

7 Long Run Costs We can solve the cost minimization problem by rst nding our tangency point. This will be where the slope of the isoquant (the marginal rate of technical substitution) is equal to the rate of change of the isocost (the ratio of the prices). w MRTS = r MP L w = MP K r MP L = w MP K r Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

8 Long Run Costs We can also rearrange these terms to the optimal decision rule we saw in consumer theory MP L w = MP K r This rule, know as the last dollar rule, says that the increase in output from one more unit of an input divided by that input s price must be the same across all inputs. Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

9 Long Run Costs MP L MP K = w r We have our tangency rule, but we need another equation to nd a solution We can t use the isocost because we don t know what the minimal level of total costs will be. We can use the production function, which is given, as our second equation. q = f (K, L) Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

10 Example Consider a rm that uses capital (K) and labor (L) to obtain an output (q) using production function q = K 0.5 L 0.5 What is the optimal amount of capital and labor to achieve an output level of q = 20 when the price of labor is w = 2 and the price of capital is r = 32? Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

11 Example q = K 0.5 L 0.5 q = 20 w = 2 r = 32 First, we need the marginal rate of technical substitution, which we obtain by applying the power rule MRTS = MP L = 0.5K 0.5 L 0.5 MP K 0.5K 0.5 L 0.5 = K L Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

12 Example q = K 0.5 L 0.5 q = 20 w = 2 r = 32 To nd our tangency point, we set it equal to the ratio of the input prices MRTS = w r K 2 = L 32 K L = K = L This implies that we will utilize sixteen units of labor for each one unit of capital in equilibrium. Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

13 Example Now, we have two equations and two unknowns 16K = L 20 {z} q = K 0.5 L 0.5 We can substitute L = 16K into the second equation to obtain and since we know that L = 16K, 20 = K 0.5 (16K ) 0.5 = 4K K = 20 4 = 5 L = 16(5) = 80 Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

14 Example w = 2 r = 32 K = 5 L = 80 This leaves us with a total minimum cost of C = wl + rk C = 2(80) + 32(5) = 320 to achieve an output level of q = 20. Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

15 Long Run Cost Function Mathematically, this process was very similar to what happens in the consumer s utility maximization problem. Exploit that when you are solving problems, but remember the key intuition di erences. Like with its consumer counterpart, we are interested in what happens when the prices of inputs change. The key di erence is that the isoquant does not change levels, but the isocost actually expands in or out to compensate. Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

16 Long Run Cost Function K A L Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

17 Long Run Cost Function K B A L Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

18 Long Run Cost Function K B A L Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

19 Long Run Cost Function We made labor more relatively expensive than capital in the second gure, but we still need to reach the same output level as before. This will simply cost more. And we will substitute some labor for additional capital. There are substitution and "income" e ects from an input price change for production theory. They are a bit beyond the scope of this class, however. We will not be covering them. We will, however, be covering what happens when the total output level rises or falls. Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

20 Long Run Cost Function When the given output level changes, we move to a di erent isoquant. This will require either a higher or lower level or total cost, as well as di erent corresponding bundles of inputs. Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

21 Long Run Cost Function K C 2 A q 2 L Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

22 Long Run Cost Function K C 2 C 1 A B q 2 q 1 L Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

23 Long Run Cost Function K C 3 C 2 C 1 C A q 3 B q 2 q 1 L Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

24 Long Run Cost Function K C 3 C 2 C 1 Expansion Path C A q 3 B q 2 q 1 L Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

25 Long Run Cost Function The expansion path shows us how the optimal levels of capital and labor will expand when we increase the level of output. We don t ever have a case where an input is inferior, so the expansion path will never go left or down. We can also plot the di erent levels of output (q) on the horizontal axis with the di erent levels of cost (C) on the vertical axis to obtain our long run cost curve. Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

26 Long Run Cost Function C 3 C C TC C 2 A C 1 B q 1 q 2 q 3 q Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

27 Long Run Cost Function Just like with short run costs, we are interested in both the average total cost and the marginal cost functions. We use the same techniques as before to derive them. Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

28 Long Run Cost Function C/q MC AC q Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

29 Long Run Cost Function A few notes on the curvature of the long run average cost function. The short run average cost function curved upward at higher levels of output due to diminishing marginal returns in the production function (it takes more and more of a variable input to increase output) Diminishing marginal returns do not a ect the long run cost function because everything is variable. The curvature of the long run average cost function is a result of the returns to scale. When the long run average cost function is decreasing in quantity, we have economies of scale. When the long run average cost function is increasing in quantity, we have diseconomies of scale. Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

30 Long Run Cost Function C/q MC AC Economies of Scale Diseconomies of Scale q Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

31 Short Run vs Long Run How can we compare the short run vs. long run average cost curves? In the long run, rms will choose a xed input based on their long run average cost curve. This will generate a corresponding short run average cost curve. Basically, the rm attempts to optimize for the long run, but if something changes after they have xed an input, they will have to adjust in the short run. We can plot several di erent short run average cost curves on top of a long run average cost curve. Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

32 Short Run vs Long Run C/q LRAC LRMC q Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

33 Short Run vs Long Run C/q SRMC SRAC LRAC LRMC q Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

34 Short Run vs Long Run C/q SRMC SRAC LRAC LRMC q Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

35 Short Run vs Long Run C/q SRMC SRAC LRAC LRMC q Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

36 Short Run vs Long Run C/q LRAC LRMC q Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

37 Short Run vs Long Run The short run average cost curve can never go below the long run average cost curve. This is because if the rm guesses correctly for their amount of xed input, the will optimize, but if they guess poorly, they will face higher costs. Combining all of the possible short run average cost curves together, the long run average cost curve forms an "envelope" underneath them. Interestingly enough, we call this an application of the envelope theorem. Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

38 Summary Firms minimize their costs to nd the optimal demand for their inputs. We can see the relationship between short run and long run costs via the envelope theorem. Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

39 Preview for Monday Competitive Markets How do supply and demand come together with production? Perlo, Chapter 8 Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40

40 Assignment Consider a rm with the following production function q = K 2 3 L 1 3 a. Let w = 1, r = 4 and q = 10. What are the optimal levels of capital and labor? b. What is the minimum total cost to achieve the production level of q = 10? Eric Dunaway (WSU) EconS Lecture 18 October 9, / 40