EconS Endogenous Market Size

Transcription

1 EconS Endogenous Market Size Eric Dunaway Washington State University eric.dunaway@wsu.edu Industrial Organization Eric Dunaway (WSU) EconS 425 Industrial Organization 1 / 35

2 Introduction Let s contrinue our discussion of sunk costs from last time. We ll nd that they are critical for determining how many rms enter a market. Also, when is it better to have fewer rms produce more goods? Eric Dunaway (WSU) EconS 425 Industrial Organization 2 / 35

3 Market Size Up until this point, we ve taken the market size as given. 1 rm for monopoly, a lot of rms for perfect competition, etc. In reality, the number of rms in a market is endogenous. Firms will enter a market as long as their expected lifetime pro ts exceed the cost of entering a market. Eric Dunaway (WSU) EconS 425 Industrial Organization 3 / 35

4 Market Size Suppose a rm was able to earn a pro t of π i every period it produced, and would last forever. In order to enter the market, it would be forced to pay some sunk cost of K. We can express the present value of their lifetime pro ts as PV = (π 1 + δπ 2 + δ 2 π ) K = δ i 1 π i K i=1 If pro t is constant in every time period, we can actually simplify this quite a bit. Eric Dunaway (WSU) EconS 425 Industrial Organization 4 / 35

5 Market Size Let π i = π for all i. Our present value pro ts becomes = i=1 δ i 1 π K = π i=1 δ i 1 K Now, we just need a way to evaluate that in nite sum. There s a neat math trick for this, though. Eric Dunaway (WSU) EconS 425 Industrial Organization 5 / 35

6 Market Size Lets de ne the variable a as a = thus, if we multiply both sides by δ, Now, I subtract the two, n δ i 1 = 1 + δ + δ δ n 1 i=1 aδ = δ + δ 2 + δ δ n a aδ = 1 + δ + δ δ n 1 (δ + δ 2 + δ δ n ) = 1 δ n Eric Dunaway (WSU) EconS 425 Industrial Organization 6 / 35

7 Market Size a aδ = 1 δ n All that s left is to solve this expression for a, a = a(1 δ) = 1 δ n n δ i 1 = 1 δn i=1 1 δ In our example, n =. This would normally be a problem, but if we assume that people actually discount the future (i.e., δ < 1), we have lim n! δn = 0 if jδj < 1 which gives us δ i 1 = 1 i=1 1 δ Eric Dunaway (WSU) EconS 425 Industrial Organization 7 / 35

8 Market Size Substituting this result into our present value function, we have PV = π i=1 δ i 1 K = π 1 δ K and as long as the present value is positive, the rm will want to enter, i.e., π K > 0 1 δ Eric Dunaway (WSU) EconS 425 Industrial Organization 8 / 35

9 Market Size Example Consider a market with n identical rms that faces the following inverse demand function, p = a bq where Q is the aggregate quantity produced by all rms, i.e., Q = q 1 + q q n. The rms all face a common marginal cost of c. If the rms compete by setting quantities, their pro t level is π = (a c)2 (n + 1) 2 b Solve for the equilibrium number of rms. Eric Dunaway (WSU) EconS 425 Industrial Organization 9 / 35

10 Market Size Example We can insert this pro t function into our present value expression that we derived earlier, π 1 δ (a c) 2 1 (n + 1) 2 b 1 δ K > 0 K > 0 and solving this expression for n, we have s (a c) n < 2 (1 δ)bk 1 Our equilibrium number of rms is when the above statement holds with equality (or the integer just below that number) The (n + 1)th rm to enter the market would experience negative present value pro ts. Eric Dunaway (WSU) EconS 425 Industrial Organization 10 / 35

11 Market Size Example s n (a c) = 2 (1 δ)bk 1 Let s look at some comparative statics: n n a > 0 and c < 0. When the market demand increases, more rms enter the market. When cost per unit of output increases, less rms enter. n n b < 0 and δ > 0. As the demand curve becomes steeper, less rms enter the market. As rms become more patient, more rms enter the market. n K < 0. As the sunk cost to enter the market increases, less rms enter the market. Eric Dunaway (WSU) EconS 425 Industrial Organization 11 / 35

12 Market Size When we look at future examples of imperfect competition, we ll take the number of rms as given unless otherwise speci ed. Intuitively, we ll assume that the entry cost, K, is su ciently high such that no other rm would want to enter the market. Let s shift our focus now to when rms aren t necessarily identical. Speci cally, we ll assume that rms have di erent cost functions. Some rms can produce their output more cheaply than their competitors. Eric Dunaway (WSU) EconS 425 Industrial Organization 12 / 35

13 Remember that the di erence between price and marginal cost is the rm s pro t margin. The shape of the average cost function (along with the market price) impacts the pro tability of the rm. Consider a situation where there are two rms, 1 and 2, and they produce an identical good. Firm 1 has a slight cost advantage over rm 2. It can produce the good slightly cheaper than rm 2 can. The market price, faced by both rms, decreases as either rm increases their output level. Eric Dunaway (WSU) EconS 425 Industrial Organization 13 / 35

14 p p AC 1 q Eric Dunaway (WSU) EconS 425 Industrial Organization 14 / 35

15 p AC 2 p AC 1 q Eric Dunaway (WSU) EconS 425 Industrial Organization 15 / 35

16 p AC 2 p AC 1 q Eric Dunaway (WSU) EconS 425 Industrial Organization 16 / 35

17 p AC 2 p AC 1 π 1 q * q Eric Dunaway (WSU) EconS 425 Industrial Organization 17 / 35

18 p p π 2 AC 2 AC 1 q * q Eric Dunaway (WSU) EconS 425 Industrial Organization 18 / 35

19 If rm 1 wanted to, he could increase his output. This could either raise or lower pro ts, depending on how sensitive the price is to that change. Moreso, rm 1 would know that his decision would also a ect rm 2 s pro ts as well. Firm 1 might be able to push rm 2 out if the price were lowered enough. Eric Dunaway (WSU) EconS 425 Industrial Organization 19 / 35

20 p AC 2 p π 1 AC 1 q * q Eric Dunaway (WSU) EconS 425 Industrial Organization 20 / 35

21 p AC 2 p π 2 AC 1 q * q Eric Dunaway (WSU) EconS 425 Industrial Organization 21 / 35

22 This is an overly simpli ed example, but it gives way to the idea of economies of scale. Economies of scale exist when an increase in output leads to a decrease in the average cost that a rm faces. This corresponds with the decreasing portion of the average cost curve. On the contrary, diseconomies of scale exist when an increase in output leads to an increase in the average cost that a rm faces. This corresponds with the increasing portion of the average cost curve. Eric Dunaway (WSU) EconS 425 Industrial Organization 22 / 35

23 p AC 1 Economies of Scale Diseconomies of Scale q Eric Dunaway (WSU) EconS 425 Industrial Organization 23 / 35

24 Economies of scale are very sensitive to the average cost curve s shape. In fact, it s possible that economies of scale could exist until very large levels of output. If the output level where economies of scale ends is su ciently high enough, it could cause a natural monopoly to form. Speci cally, if it is cheaper for one rm to produce the entire market output than it is for any number of smaller rm to produce smaller parts, then we have the case of natural monopoly. Mathematically, natural monopolies exist if where Q = n i =1 q i. c 1 (q 1 ) + c 2 (q 2 ) + c 3 (q 3 ) c n (q n ) > c(q) Eric Dunaway (WSU) EconS 425 Industrial Organization 24 / 35

25 p Economies of Scale AC 1 q Eric Dunaway (WSU) EconS 425 Industrial Organization 25 / 35

26 In a natural monopoly, the rm with the lowest cost structure will eventually drive out its competition and seize the market. Natural monopolies are actually fairly common. Most public utilities are natural monopolies. It s cheaper for one rm to lay all the necessary piping (or wiring) to supply water (or electricity) to a city than it is to have multiple competitors. Governments typically grant one rm the relevant monopoly and then regulate their price. What about cable companies? Eric Dunaway (WSU) EconS 425 Industrial Organization 26 / 35

27 Even without a visual of the shape of the Average Cost curve, we can calculate whether we have economies of scale. Remember from before that the relationship between the Average Cost curve and Marginal Cost curves determines whether the Average Cost curve is increasing or decreasing. When MC < AC, Average Cost is decreasing. When MC > AC, Average Cost is increasing. We can create a simple ratio. De ne S as S AC MC if S > 1, we have economies of scale. If S < 1, we have diseconomies of scale. Eric Dunaway (WSU) EconS 425 Industrial Organization 27 / 35

28 S AC MC Let s look at this ratio a bit more. We can expand Average Cost and Marginal cost into their de nitions, and rearranging terms, S = c(q) q dc(q) dq S = 1 dc(q) dq q c(q) = 1 η Note that the term in the denominator is just an elasticity, the elasticity of costs with respect to output. Eric Dunaway (WSU) EconS 425 Industrial Organization 28 / 35

29 Consider the total cost function TC = q + 0.5q 2 for what value of output are there neither economies nor diseconomies of scale? Eric Dunaway (WSU) EconS 425 Industrial Organization 29 / 35

30 TC = q + 0.5q 2 Neither economies of scale nor diseconomies of scale exist when S = 1. We simply need to calculate the Average and Marginal Cost functions and plug them into our scale de nition, rearranging terms, S = AC MC = q + 0.5q2 q(10 + q) = q + 0.5q 2 = 10q + q 2 0.5q 2 = 50 q = 10 Thus, at an output level of q = 10, economies of scale disappear. As a note, this is also the minimum of the Average Cost curve. This technique may be useful later. Eric Dunaway (WSU) EconS 425 Industrial Organization 30 / 35

31 Economies of Scope Many rms produce more than one good. For example, a cattle ranch not only produces beef, but leather, glycerin, collagen, etc. We may have a situation in our cost structure where producing one product makes it actually cheaper to produce another product as well. Similar capital requirements, one product may be a byproduct of the other, etc. When it is cheaper for one rm to produce separate products than it is for individual rms to produce those products, we have economies of scope. Eric Dunaway (WSU) EconS 425 Industrial Organization 31 / 35

32 Economies of Scope Mathematically, if there are m di erent products, economies of scope exist if c(q 1, 0,..., 0) + c(0, q 2,..., 0) c(0, 0,..., q m ) > c(q 1, q 2,..., q m ) or, for a simpler case when there are m = 2 products, c(q 1, 0) + c(0, q 2 ) > c(q 1, q 2 ) Intuitively, if the single rm can produce both products at a lower cost than two smaller rms can each produce one of the products, why have two rms? Eric Dunaway (WSU) EconS 425 Industrial Organization 32 / 35

33 Summary As we will see going forward, cost structures are critical in all levels of rm decision making, going so far as they determine whether a rm will actually enter the market. Economies of Scale and Scope give us an indication of what kind of market structure we can expect. Eric Dunaway (WSU) EconS 425 Industrial Organization 33 / 35

34 Next Time Market Structure and Market Power How do government agencies measure and react to market concentration? Reading: Chapter 4. Eric Dunaway (WSU) EconS 425 Industrial Organization 34 / 35

35 Homework 2-1 Consider the following cost functions c(q 1, q 2 ) = F + cq 1 + cq 2 µq 1 q 2 c(q 1, 0) = F 1 + cq 1 c(0, q 2 ) = F 2 + cq 2 where F 1, F 2, c, and µ are all positive and F 1 + F 2 < F. 1. F 1, F 2, and F are all xed costs, c is the constant marginal cost, and q 1 and q 2 are output levels. Interpret the nal term of c(q 1, q 2 ), µq 1 q 2. What happens as µ increases? 2. For what values of µ do economies of scope exist in this production? Eric Dunaway (WSU) EconS 425 Industrial Organization 35 / 35