EconS Theory of the Firm

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1 EconS Theory of the Firm Eric Dunaway Washington State University eric.dunaway@wsu.edu October 5, 2015 Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

2 Introduction Theory of the rm, also known as producer theory, is the counterpart to our previous unit: consumer theory. Now, we will learn how to derive supply curves. At the end of this unit, we will be able to put both supply and demand together again to nd competitive market equilibria. Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

3 The Firm What is a Firm? A rm is something that produces something else. It takes many di erent kinds of inputs for a rm to produce one (or more) outputs. For the most part, we will be dealing with inputs like capital, labor, and technology with a general output. Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

4 The Firm What kind of rms are there? In the US, we typically see four types of rms: Sole proprietorships are when one person owns a rm. Essentially, the rm is an extension of the owner, and no legal separation exists. Partnerships are the same as sole proprietorships except that multiple people own the rm. Corporations are when multiple people own shares of a rm, but a legal separation exists. Limited Liability Corporations are the same as corporations, except that there are typically few owners. This list is by no means exhaustive. Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

5 The Firm Unlike consumers, rms are not interested in utility. Instead, they are interested in pro ts! We are talking about economic pro ts, not accounting pro ts. The big di erence has to do with how costs are gured in, which we will talk about next time. The rm s goal is to maximize their pro ts, which we de ne as total revenue minus total cost Pro ts = Total Revenue Total Cost π = TR TC Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

6 The Firm π = TR TC Furthermore, we can break down total revenue into the price of the output, p, times the total output level, q. TR = pq Total costs depend on several di erent things, so we will just leave that alone until next time. Substituting the total revenue equation into the pro t function, we have π = pq TC Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

7 The Firm We also assume that rms use e cient production. This means that the rms aren t dumb. They use the optimal, cost minimizing level of inputs to produce their outputs. In other words, if it took 2 cups of our and a cup of sugar to make one cake, I wouldn t put 8 cups of our and one cup of sugar and expect one cake. Essentially, Firms use only what is necessary, and do not waste inputs. Also, that would be a really gross cake. Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

8 Production Function The way that the rms transform inputs to outputs is determined by the production function. There are numerous di erent types with even more types of results. We de ne a production function by specifying some transformation of our inputs, capital (K) and labor (L) into units of output (q). q = f (K, L) Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

9 Production Function The production function is sensitive to the time horizon. When an input can be changed during the current time horizon, we call it a variable input. Otherwise it is a xed input. For simplicity, we will use two time horizons: the short run and the long run. In the short run, at least one input must be xed, while all the others remain variable. For our models, we will assume that capital is xed, while labor remains variable. Why? Capital takes time to build. If we had more than two inputs, the short run would be when at least one of them is xed. In the long run, all inputs are variable. Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

10 Short Run In the short run, our production function becomes q = f ( K, L) Since we are holding capital constant, we are interested in what happens as we vary labor. When labor starts out low, marginal increases will have a large e ect on output. We call this increasing marginal returns to scale. As labor increases, its e ect on output will diminish, with each additional unit giving less than the previous one. We call this diminishing marginal returns to scale. At some point, more labor will actually give less output than the previous unit. We call this negative marginal returns to scale. Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

11 Short Run q L Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

12 Short Run Why would more of an input give less output? At some point, the input becomes ine cient. For example, if we put way too many workers into a small o ce, they would spend all day bumping into each other, and nothing would get done! Optimally, the rm will never choose an input level with negative marginal returns. Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

13 Short Run As in consumer theory, we are interested in both the absolute levels of output as well as the rate of change. The good news is that we can use the same techniques that we learned in consumer theory! We are interested in the marginal product of labor. This tells us how much more output we will obtain by utilizing one more unit of labor. We obtain this value by applying the power rule to the production function with respect to L. MP L = Change in output Change in labor = f L (K, L) Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

14 Short Run Also, in addition to the marginal product of labor, we are interested in the average product of labor. This tells us how much output per unit of labor we are obtaining. Let s look at an example. AP L = q L = f (K, L) L Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

15 Example Consider the following production function (note: this production function is very close to reality) q = K 0.7 L 0.3 and let the level of capital, K, be xed at 2 in the short term. Our short run production function becomes q = L 0.3 = 1.62L 0.3 To nd the marginal product of labor, we simply apply the power rule MP L = f L (K, L) = L 0.7 = 0.486L 0.7 and to nd the average product of labor, we just divide the production function by L AP L = f (K, L) L = 1.62L0.3 L = 1.62L 0.7 Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

16 Short Run We can also graph the marginal and average products of labor just below our graph of the total product. Note, these are not the same curves that we just gured out in our example, but a general idea. Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

17 Short Run q MP L, AP L L MP L AP L L Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

18 Short Run MP L, AP L MP L AP L L Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

19 Short Run The intersection of the marginal and average product of labor curves is not coincidental. They will always intersect at the maximum value of the average product of labor curve. An intuition for this is to think of the marginal product of labor as "pulling" on the average product of labor. When the marginal product is above the average product it "pulls" the average product upward. When the marginal product of labor is below the average product, it "pulls" the average product downward. Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

20 Long Run Let s shift gears and talk about long run production. In this case, all inputs are able to be varied. We could think of this as the rm setting out a plan for the next ve years. This gives them plenty of time to construct their capital and plan for their labor usage. One thing to know about our production function is that there are several di erent combinations of capital and labor that can lead to the same quantity of output being produced. The combination that is selected will be based on the relative costs of the two inputs. Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

21 Long Run For example, a process where we see this is in street cleaning. In the US, capital is relatively cheap compared to labor. This is why single person operated street cleaning machines are most often seen. In Asia, capital is relatively expensive compared to labor. This is why it is more common to see many people using brooms to clean a street. The point is that a rm can select from many di erent "bundles" of inputs to reach a desired level of output. Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

22 Long Run When analyzing the rm s input decision, we want to look at isoquants. We take a desired level of output, and plot all of the combinations of capital and labor that can achieve that output level. q = f (K, L) Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

23 Long Run K q 3 q 1 q 2 L Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

24 Long Run I hope that looked familiar. Don t confuse isoquants for indi erence curves, though. In our coming analyses, we will be using very similar techniques that we used to solve problems in consumer theory. Intuitively, though, the isoquant is a constraint. It is actually closer in behavior to a budget line than an indi erence curve. Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

25 Long Run Like with indi erence curves, we are interested in the rate of change of the isoquant. We call this the marginal rate of technical substitution. This tells us the rate at which we can substitute labor for capital, but still remain at the same level of production. Fortunately, the formula is very similar. MRTS = Change in capital Change in labor = MP L MP K Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

26 Long Run Also like indi erence curves, the curvature of the isoquant tells us the degree of substitutability of the inputs. Perfect substitutes will have straight isoquants and a constant marginal rate of technical substitution. Perfect complements will have kinked isoquants and an unde ned marginal rate of technical substitution. The straighter the isoquant, the more substitutable the inputs are for one another. Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

27 Long Run K q 1 q 2 q 3 L Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

28 Long Run K q 3 q 2 q 1 L Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

29 Example Consider the following production function q = K α L β where α and β are positive numbers. We can nd the marginal rate of technical substitution by rst applying the power rule. and then apply the formula MP K = αk α 1 L β MP L = βk α L β 1 MRTS = MP L = βk α L β 1 MP K αk α 1 L β = βk αl Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

30 Returns to Scale Another thing we will talk about with regards to the production function is returns to scale. Typically, we break this into three types: Increasing returns to scale: When all of the inputs are increased by a common multiple, the output level will increase by even more than the multiple. Constant returns to scale: When all of the inputs are increased by a common multiple, the output level will increase by the same multiple. Decreasing returns to scale: When all of the inputs are increased by a common multiple, the output level will increase by less than the multiple. Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

31 Returns to Scale Why would we have di erent returns to scale? This could be due to a lot of factors. Increasing returns to scale could happen when doubling all inputs, the labor force is allowed to specialize, becoming more productive. Decreasing returns to scale could happen when doubling all inputs creates way too many managers, increasing ine ciency. There are several other possibilities, but these are two examples. Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

32 Returns to Scale We can tell what the returns to scale are by our production function. What we have been using so far is known as Cobb-Douglass production, and it has the form that we saw in the last example. q = K α L β where α and β are positive numbers. If α + β > 1, we have increasing returns to scale. If α + β = 1, we have constant returns to scale. If α + β < 1, we have decreasing returns to scale. In the rst example from today, the production function was q = K 0.7 L 0.3 where in this case α = 0.7 and β = 0.3. Adding them together equals 1, which implies that this production function has constant returns to scale. Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

33 Returns to Scale Let s look at another example. Consider the production function q = 2K + 4L Let s say we used 1 unit of capital and 1 unit of labor. Our output level would be q = 2(1) + 4(1) = 6 Now, let s double both inputs and see how much output we get. q = 2(2) + 4(2) = 12 When I doubled both the inputs, the output doubled! Is this enough to say that we have constant returns to scale? Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

34 Returns to Scale We can prove that this production has constant returns to scale by multiplying all of the inputs by some common multiple, which we call t. 2(tK ) + 4(tL) First, distribute everything through the parenthesis 2tK + 4tL and then factor out the t, = tq =q z } { t( 2K + 4L) Since multiplying all of the inputs by a common multiple yields that same multiple times our output level, we know for sure that this production function has constant returns to scale. Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

35 Technology One other thing we like to include in the production function is technology. We treat this as a positive constant that we just multiply by the rest of the production function q = AK α L β where A would represent technology. We use technology to account for changes in production over time, since according to the data we have, it s not possible that all of our increases could be attributed to improvements in capital and labor. We will talk about this more next time. Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

36 Summary Firms supply the market with goods and services based on their motivation for pro t. They attempt to gure out the optimal bundle of inputs to most e ciently produce their output. We can use similar techniques as in consumer theory to solve for the optimal decision of the rms. Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

37 Preview for Wednesday Cost functions, the other side of the rm s problem. Chapter 7 in Perlo Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38

38 Assignment Consider the following production function q = K 0.7 L 0.3 a. What is the marginal rate of technical substitution? b. I mentioned in the slides that this function has constant returns to scale. Use the technique I presented on slide 34 to prove this. Here are a few math tricks with exponents and parenthesis that might help: (ab) c = a c b c a c b c a d e d = a c+d b c e d Eric Dunaway (WSU) EconS Lecture 16 October 5, / 38