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C H A P T E R
Competitive Market Equilibrium
Until now, we have only proceeded through the first two of three steps of the “eco- nomic way of thinking” — the crafting of a model and the process of optimizing within that model. We now proceed to the final step: to illustrate how an equilib- rium emerges from the optimizing behavior of individuals — how the “economic environment” that individuals in a competitive setting take as given emerges from their actions. In the process, we begin to get a sense of how order can emerge “spontaneously” — an idea introduced briefly in the introduction as one of the big ideas that we should not loose as we dive into technical details of economic models.
Chapter Highlights
The main points of the chapter are:
- An equilibrium arises in an economic model when no individual has an in- centive to change behavior given what everyone else is doing. In a competi- tive model, it means that no individual has an incentive to change behavior given the economic environment that has emerged “spontaneously”.
- The short run for an industry is the time over which the number of firms in the industry is fixed because firms have not had an opportunity to enter or exit the industry. The short run industry (or market) supply curve is there- fore the sum of the firm supply curves for the (short run) fixed number of firms in the industry, and the short run equilibrium is driven by the price at which the short run industry supply curve intersect the market demand curve (which is simply the sum of all individual demand curves).
- The long run for an industry is the time it takes for sufficient numbers of firms to enter or exit the industry as conditions change. The long run in- dustry (or market) supply curve therefore arises from the condition that the marginal firm in the industry must make zero profits so that no firm in the industry has an incentive to exit and no firm outside the industry has an in- centive to enter. When all firms face identical costs, this implies a horizontal
299 14A. Solutions to Within-Chapter-Exercises for Part A
long run industry supply curve at the price which falls at the lowest point of each firm’s long run AC curve. The long run equilibrium then emerges at the intersection of market demand and long run industry supply.
- To analyze what happens as conditions change in a competitive market, the most important curves to keep track of on the firm side are the (1) long run AC curve and (2) the short run firm supply curve that crosses the long run AC curve at its lowest point (but extends below it because shut down prices are lower than exit prices.) Any change that impacts short run firm sup- ply curves will impact the short run industry supply curve, and any change that impacts the long run AC curve will impact the long run industry supply curve.
- Changes that affect only a single firm in an industry do not affect the market equilibrium in either the short or the long run.
Using the LiveGraphs
For an overview of what is contained on the LiveGraphs site for each of the chapters (from Chapter 2 through 29) and how you might utilize this resource, see pages 2- of Chapter 1 of this Study Guide. To access the LiveGraphs for Chapter 14, click the Chapter 14 tab on the left side of the LiveGraphs web site. We do not at this point have additional Exploring Relationship modules for this chapter.
14A Solutions to Within-Chapter-Exercises for
Part A
Exercise 14A.1 Can you explain why there is always a natural tendency for wage to move to- ward the equilibrium wage if all individuals try to do the best they can?
Answer: If wage were to drift below the equilibrium, firms would not be able to fill all their job vacancies because not enough workers are willing to work at a below-market wage. Thus, it is in each firm’s interest to offer a slightly higher wage in order to fill its positions — and this continues to be true until all positions are filled at the equilibrium wage. If, on the other hand, the wage were to drift above the equilibrium, some workers who want to work will be unable to find a position. It would be in their interest to offer to work for slightly less so that they can get employed when there are fewer jobs than workers wishing to take them — and this continues until the wage falls at the equilibrium where the number of jobs is exactly equal to the number of workers willing to take them.
301 14A. Solutions to Within-Chapter-Exercises for Part A
Graph 14.1: Movement to Long Run Equilibrium when price is below AC
Answer: The long run price would settle at the lowest point of the (long run) AC curve of firms. Thus, panel (f) would only need to show the long run AC curve, and the intersection of DM^ and SM^ in panel (e) would occur at the price equal to the lowest level of the AC curves of firms.
Exercise 14A.7 How would you think the time-lag between short and long run changes in labor markets is related to the “barriers to entry” that workers face, where the barrier to entry into the PhD economist market, for instance, lies in the cost of obtaining a PhD.
Answer: The greater the barriers to entry, the longer it will take for the labor market to reach the long run equilibrium.
Exercise 14A.8 Can you explain why the previous sentence is true?
Answer: As long as the lowest points of the AC curves of firms are all at the same vertical height in the graph, all firms have the same exit (or entry) price — and thus exit and entry decisions by firms will drive long run price to that level. It does not matter whether these lowest points of AC curves occur at different output levels for different firms — i.e. whether these lowest points are horizontally different. If they are, it simply implies that different firms will produce different levels of output in long run equilibrium, but their exit/entry prices are all the same (so long as the AC curves do not differ vertically.)
Exercise 14A.9 Suppose market demand shifts inward instead of outward. Can you illustrate what would happen in graphs similar to those of Graph14.6?
Answer: Graph 14.2 (next page) illustrates this. In panel (a), DM^ and SM^ inter- sect at the original equilibrium A — with price at p ∗^ and industry output at X ∗. At
Competitive Market Equilibrium 302
that price, any firm with AC at or below p ∗^ is producing x ∗^ — as, for instance, both the firms in panel (b). (This is assuming the lowest point of all AC curves occurs at the same output quantity). When demand shifts to DM
′ as illustrated in panel (a), the initial short run equilibrium shifts to B. Since the marginal firms were making zero profit before, they are now making negative long run profit — implying that they will begin to exit, which in turn causes the short run market supply curve in panel (a) to shift to the left. This continues to happen until the marginal firm left in the industry makes zero profit. This is illustrated as firm 2 with average cost curve AC^2 in panel (b). Since the highest cost firms exit, the new equilibrium price p ′′ will fall below p ∗^ — leading to the upward sloping long run market supply curve in panel (c).
Graph 14.2: Inward shift in DM
Exercise 14A.10 True or False : The entry and exit of firms in the long run insures that the long run market supply curve is always shallower than the short run market supply curve.
Answer: This is true. One way to see this is to think about changes in demand for an industry that is initially in long run equilibrium (before the change in de- mand). Suppose demand increases. This implies that industry output will rise as each firm produces more at the higher price (that results from the new intersection of (short run) market supply and demand. But, since we started in long run equi- librium, all firms that were initially making zero profit must now be making higher profit — which gives an incentive to firms outside the industry to enter. This will shift the short run market supply curves, driving down the price and increasing in- dustry output until the marginal firm makes zero profit again. Thus, the entry of new firms causes the long run output increase to be larger than the short run in- crease. The reverse happens when demand falls. In that case, firms will produce less as price falls to the new intersection of demand and short run market supply. But since they were initially making zero profit, this implies they will not make neg-
Competitive Market Equilibrium 304
Graph 14.3: Short Run Profit in Long Run Equilibrium
creases. Put differently, if the increased fixed fee is F , the average increased fixed fee is F / x — which is F when x = 1 but converges to zero as x gets large.
Exercise 14A.15 If you add the firm’s long-run supply curve into panel (b) of the graph, where would it intersect the two average cost curves? Would the same be true for the firm’s initial short-run supply curve? ( Hint : For the second question, keep in mind that the firm will change its level of capital as its output increases.)
Answer: The firm’s long run supply curve would intersect the two long run AC curves at their lowest points (because the long run supply curve for a firm is the portion of its long run MC curve that lies above its long run AC curve.) In fact, it would initially begin at the lowest point of the blue AC curve — and would get shorter as a result of the increase in the recurring fixed cost, starting at the lowest point of the green AC ′^ curve after the increase in the recurring fixed cost. The firm’s initial short run supply curve would also cross the lowest point of the initial blue long run AC curve (since the industry is initially in long run equi- librium). But it would (almost certainly) not cross the lowest point of the green AC ′^ curve. This is because the level of capital that the firm has in the initial long run equilibrium is unlikely to be the level of capital it will end up with in the new long run equilibrium — and its short run supply curve therefore takes a (long-run) sup-optimal level of capital as fixed.
Exercise 14A.16 Could the AC curve shift similarly in the case where the increase in cost was that of a long run fixed cost?
Answer: No, it could not. In the case of a long run fixed cost, the (long run) MC curve remains unchanged because the fixed cost does not change the additional cost of producing any of the units of output. The MC curve also has to cross the AC
305 14A. Solutions to Within-Chapter-Exercises for Part A
curve both before and after the increase in the fixed cost — which means that the lowest point of the AC curve must shift to the right when the fixed cost increases.
Exercise 14A.17 How would you illustrate the transition from the short run to the long run using graphs similar to those in panels (a) and (b) in Graph 14.8?
Answer: When long run costs increase, firms exit the industry. This shifts the short run supply curve to the left — driving price up until it reaches the lowest point of the firms’ AC curves. This is illustrated in Graph 14.4.
Graph 14.4: Transitioning to Long Run Equilibrium
Exercise 14A.18 Consider two scenarios: In both scenarios, the cost of capital increases, caus- ing the long run AC curve to shift up, with the lowest point of the AC curve shifting up by the same amount in each scenario. But in Scenario 1, the lowest point of the AC curve shifts to the right while in Scenario 2 it shifts to the left. Will the long run equilibrium price be different in the two scenarios? What about the long run equilibrium number of firms in the industry?
Answer: The only thing that matters for where the long run equilibrium price will settle is the vertical height of the lowest point of the long run AC curves of firms. Since this is the same in both scenarios, the long run equilibrium price will be the same for both cases. Thus, the long run market supply curve will intersect the market demand curve at the same point — which implies industry output will also be the same in both scenarios. But each firm will increase its production in the new equilibrium in Scenario 1 while each firm will decrease its production in the new equilibrium in Scenario 2. Thus, in order for the industry to produce the same in both scenarios, it must be the case that the equilibrium number of firms will be larger in Scenario 2 than in Scenario 1.
307 14B. Solutions to Within-Chapter-Exercises for Part B
Affected Costs Market Industry Firm LR # of Example SR LR Price Output Output Firms ↓ License Fee None AC − SR^ ↓ LR^ − SR^ ↑ LR^ − SR^ ↓ LR^ ↑ ↓ r None AC , MC − SR^ ↓ LR^ − SR^ ↑ LR^ − SR^? LR^? ↓ w AC , MC AC , MC ↓ SR^ ↓ LR^ ↑ SR^ ⇑ LR^ ↑ SR^? LR^? ↓ Demand None None ↓ SR^ − LR^ ↓ SR^ ⇓ LR^ ↓ SR^ − LR^ ↓
14B Solutions to Within-Chapter-Exercises for
Part B
Exercise 14B.1 Why is the demand function not a function of income?
Answer: This is because the utility function from which it was derived is quasi- linear — which removes income effects from demand.
Exercise 14B.2 Demonstrate that the average cost of production is U-shaped and reaches its lowest point at x = 1280 where AC =5. ( Hint : You can illustrate the U-shape by showing that the derivative of AC is zero at 1280, negative for output less than 1280 and positive for output greater than 1280.)
Answer: Taking the derivative of AC ( x ), we get
∂AC ( x ) ∂x
x 3/^
x^2
Substituting x = 1280, we get AC (1280) = 0, and, for x < 1280, the function is indeed negative while for x > 1280 it is positive. We therefore have a U-shaped AC curve that reaches its lowest point at x = 1280. At that output level, the average cost is
AC (1280) = 0.66874(1280)1/4^ +
Exercise 14B.3 Verify these individual production and consumption quantities.
Answer: Substituting p = 5 into the consumer demand equation, we get
xd^ (5) =
and substituting p = 5 into the firm supply equation xs^ ( p ), we get
xs^ (5) = 437.754(5)2/3^ = 1280. (14.4)
Competitive Market Equilibrium 308
Exercise 14B.4 We have already indicated that k = 256 is in fact the optimal long run quantity of capital when ( p , w , r ) =(5,20,10). Can you then conclude that the industry is in long run equilibrium from the information in the previous paragraph? ( Hint : This can only be true if no firm has an incentive to enter or exit the industry.)
Answer: Since we know k = 256 is the long run optimal quantity of capital, the short run expenditures of $3,840 become economic costs in the long run. (If we did not know k = 256 was long run optimal, we would not be able to conclude this since the firm would further adjust capital and therefore the long run cost of cap- ital would differ from the short run expense on capital.) Thus, from a long run perspective, the firm has $3,840 more in costs than it has in the short run. Since we concluded that short run profit is $3,840, this implies long run profit is $0. The in- dustry is therefore in long run equilibrium — with no firm in the industry wanting to exit and no firm outside the industry wanting to enter.
Exercise 14B.5 Can you graph the AC SR^ into panel (c) of Graph 14.12?
Answer: This is illustrated in Graph 14.5 where the short run average cost curve must lie below the short run MC curve.
Graph 14.5: AC SR^ with Decreasing Returns to Scale Production
Exercise 14B.6 Why does the long-run profit become negative $960 if nothing changes?
Answer: Since we started in long run equilibrium, we know that initially the firms were making zero long run profit. When the license fees are increased by $960, this then implies that long run profit must fall to minus $960 if nothing changes.
Exercise 14B.7 Verify these calculations.
Competitive Market Equilibrium 310
Answer: The graph suggests that industry output will fall while firm output remains unchanged and price increases. We calculated that industry output falls from 1,600,000 to 1,156,925 and price rises from $5 to $5.88. These results are con- sistent with the graph. We also calculated that each firm’s output will fall from 1, to 1,088 which is different from what is shown in the graph. However, in developing the graph, we noted that the lowest point of the long run AC curve might shift to the left or right as the rental rate of capital increases — and we just happened to draw it as shifting in neither direction. So, while the mathematical results in this example are not consistent with how we drew the graph, they are consistent with our discussion in part A of the chapter.
Exercise 14B.11 How much capital and labor are hired in the industry before and after the increase in r?
Answer: In Chapters 12 and 13, we calculated the input demand functions for this technolgoy to be
ℓ ( p , w , r ) = 32768 p^5 r^2 w^3
and k ( p , w , r ) = 32768 p^5 w^2 r^3
When p = 5 and w = 20, these become
ℓ ( r ) =
r^2
and k ( r ) =
r^3
Evaluate at r = 10 and r = 15, this gives us ℓ = 128 and k = 256 when r = 10 going to ℓ = 58.89 and k = 75.85 when r increases to 15.
Exercise 14B.12 Verify these calculations.
Answer: Setting short run market supply equal to market demand implies
417,586 p 2/3^ =
p^2
which implies p 8/3^ = 95.789 or p = 5.533409 ≈ $5.53. Substituting this into the firm’s new short run supply function xs ′ ( p ) = 334.069 p 2/3^ , we get that each firm produces xs ′(5.53) ≈ 1,045. The industry output can be calculated by substituting the new price into either the market demand or supply curves — both of which tell us that overall industry output will rise to 1,306,395.^1 Total revenue for each firm will then simply be the price times the output level 1,045 — or approximately $5,782 if we use the un-rounded price or $5,779 if we use p = 5.53 which is slightly rounded down.
(^1) Because of rounding error, you will actually get slightly different answers depending on whether you plug the new price into the market demand or short run market supply functions — the output level 1,306,395 arises from using the price p = 5.533409 we calculated before rounding. Up to a rounding error, this is also the same as what we get if we multiply each firm’s output of 1,045 by the total number of firms in the short run equilibrium (1,250).
311 14B. Solutions to Within-Chapter-Exercises for Part B
Exercise 14B.13 How much does the industry production change in the short run?
Answer: Industry production falls from the original 1,600,000 calculated earlier to the 1,306,395 we calculated in the previous exercise — a short run drop of a little less than 300,000 output units.
Exercise 14B.14 Verify these calculations and compare the results to our graphical analysis of an increase in the wage rate in Graph 14.10.
Answer: First, to calculate the long run equilibrium price, we need to determine the lowest point of the long run average cost curve. The cost curve C ( w , r , x ) = 2( wr )1/2( x /20)5/4^ + 1280 (given at the beginning of part B of the text) becomes C (30,10, x ) = 0.819036 x 5/4^ + 1280 when the new wage (and original rental rate) are substituted for w and r. Thus, the average (long run) cost curve after the wage increase is
AC ( x ) = 0.819036 x 1/4^ +
x
This reaches its lowest point when
∂AC ( x ) ∂x
x 3/^
x^2
Solving for x , we get that x = 1088.36 ≈ 1088 — and the lowest average cost level reached at that output level is AC (1088.36) ≈ $5.88. Thus, the price rises from the original $5.00 to $5.53 in the short run to $5.88 in the long run. Each firm, which originally produced 1280 units, reduces its output to 1045 in the short run, but that output level increases to 1088 in the long run for those firms that remain in the industry. At the new long run equilibrium price, the market demands DM^ = 40,000,000/(5.88^2 ) ≈ 1,156,925 — down from the original 1,600,000 and from the short run equilibrium industry output of 1,306,395. This implies that the number of firms that remain in the industry falls from the original 1,250 to 1,156,925/1088≈ 1,063 firms. The graph in part A predicted that the industry would produce less in the short run and even less in the long run, and that the price will rise in the short run and even more in the long run. Both these predictions hold up in this example. The graph furthermore predicted that output by each firm will initially fall in the short run but will rise back to its original quantity in the long run for firms that stay in the industry. The latter does not hold in this example, but we had pointed out in part A that the long run output could in principle go up or down — and we simply graphed it as unchanged from the original quantity solely for convenience. Thus, the prediction in this example that firm output for those that remain in the industry will initially go down and then recover somewhat but not fully is not inconsistent with the discussion surrounding the graph in part A.
Exercise 14B.15 How much does individual consumption by consumers who were originally in the market change in the short run?
313 14C. Solutions to End-of-Chapter Exercises
same output quantity, the only way more can be produced in the industry is for more firms to have entered — i.e. the number of firms in the industry has increased. (c) Now consider an increase in the wage rate and suppose first that this causes the long run AC curve to shift up without changing the output level at which the curve reaches its lowest point. In this case, can you predict whether the number of firms increases or decreases? Answer: In this case, the output level produced by each firm in the industry will remain the same but will be sold at a higher price. When price increases, however, less will be demanded (assuming a downward sloping market demand curve) — which implies the in- dustry produces less. With each firm in the industry producing the same as before, the only way for the industry to produce less is for some firms to have exited. Thus, the number of firms in the industry falls in this case. (d) Repeat (c) but assume that the lowest point of the AC curve shifts up and to the right. Answer: If the lowest point of the AC curve shifts up and to the right, it means that firms that remain in the industry will produce more at a higher price — but the higher price implies that less will be demanded and thus the industry produces less. The only way each firm can produce more while the industry produces less is if some firms exited — and the number of firms in the industry therefore declines. (e) Repeat (c) again but this time assume that the lowest point of the AC curve shifts up and to the left. Answer: In this case, the lowest point of the AC curve occurs at a lower level of output and higher price — which means that firms in the industry will produce less and sell at that higher price. A higher price in turn means that consumers will demand less. Thus, each firm produces less as does the industry. Whether this implies more or fewer firms now depends on how much less each firm produces relative to how much the quantity demanded falls with the increase in price. Suppose each firm produces x % less and the industry as a whole produces y % less. Then if x = y , the number oof firms stays exactly the same; if x < y , the number of firms falls and if x > y , the number of firms in the industry has to increase. (f) Is the analysis regarding the new equilibrium number of firms any different for a change in r? Answer: No, the analysis is no different for a change in r. Even if capital is fixed in the short run, it is variable in the long run — and treated just like the input labor. (g) Which way would the lowest point of the AC curve have to shift in order for us not to be sure whether the number of firms increases or decreases when w falls_?_ Answer: When w falls, we know the long run AC curve will shift down — which implies that the long run equilibrium price will fall. At a lower price, the quantity demanded will increase — which implies that industry output will increase. Were each firm to continue to produce the same amount as before — or were each firm to produce less, then the only way for the industry to produce more would be for the number of firms to increase. Thus, in order for us not to be sure of whether the number of firms increases, it would have to be that each firm produces more (just as the industry produces more) — and this only happens if the lowest point of the AC curve shifts to the right as w falls. B: Consider the case of a firm that operates with a Cobb-Douglas production function f ( ℓ , k ) = Aℓαkβ^ where α , β > 0 and α + β < 1_._ (a) The cost function for such a production process — assuming no fixed costs — was given in equation (13.45) of exercise 13.5. Assuming an additional recurring fixed cost F , what is the average cost function for this firm? Answer: Including the fixed cost F , the total cost function becomes
C ( w , r , x ) = ( α + β )
( wαr β^ x Aααββ
)1/( α + β )
- F (14.14) which gives us an AC function
AC ( w , r , x ) = C^ ( w , r ,^ x ) x = ( α + β )
( wαr β^ x (1− α − β ) Aααββ
)1/( α + β )
. (14.15)
Competitive Market Equilibrium 314
(b) Derive the equation for the output level x ∗^ at which the long run AC curve reaches its lowest point. Answer: The AC curve reaches its lowest point where its derivative with respect to x is zero — i.e. where
∂AC ( w , r , x ) ∂x =
(1 − α − β )
( wαr β Aααββ
)1/( α + β ) x (1−2( α + β ))/( α + β )
(^) − F x^2 = 0. (14.16)
Solving this for x , we then get the output level at the lowest point of the long run AC curve:
x ∗^ =
( Aααββ wαr β
) ( F 1 − α − β
)( α + β )
. (14.17)
(c) How does x ∗^ change with F , w and r? Answer: Given the expression for x ∗^ above, it is easy to see that x ∗^ increases with F and decreases with w and r. Put differently, the lowest point of the AC curve occurs at higher output levels as the fixed cost increases and at lower output levels when input prices in- crease. (d) True or False : For industries in which firms face Cobb-Douglas production processes with recurring fixed costs, we can predict that the number of firms in the industry increases with F but we cannot predict whether the number of firms will increase or decrease with w or r. Answer: This is true. As F increases, output price rises as does output by each firm. The higher output price, however, means that the quantity demanded — and thus the quantity supplied by the industry — decreases. The only way the industry output can decrease when firm output increases is if some firms have left the industry. When input prices increase, the equilibrium output price similarly rises (as the AC shifts up) — causing the industry to produce less. But, in the case analyzed here, each firm also produces less — so we cannot immediately tell whether the number of firms will increase or decrease.
Exercise 14.4: Brand Names and Franchise Fees
46 Business Application: Brand Names and Franchise Fees : Suppose you are currently operating a ham- burger restaurant that is part of a competitive industry in your city. A: Your restaurant is identical to others in its homothetic production technology which employs la- bor ℓ and capital k and has decreasing returns to scale. (a) In addition to paying for labor and capital each week, each restaurant also has to pay recur- ring weekly fees F in order to operate. Illustrate the average weekly long run cost curve for your restaurant. Answer: This is illustrated in panel (a) of Graph 14.6 where the long run average cost curve for your restaurant is U-shaped because of the combination of a recurring fixed cost and decreasing returns to scale in production. (b) On a separate graph, illustrate the weekly demand curve for hamburgers in your city as well as the short run industry supply curve assuming that the industry is in long run equilibrium. How many hamburgers do you sell each week? Answer: This is illustrated in panel (b) where the market supply curve SM^ is simply all short run restaurant supply curves added together. This has to intersect demand at p ∗^ which lies at the lowest point of the AC curve in panel (a). It is only at that price that long run profits for restaurants are zero and thus no incentives for entering or exiting the industry exist. At this price, you will sell hamburgers so long as p ∗^ is greater than long run average cost — and we know that MC crosses the AC curve at its lowest point. Thus, you will produce x ∗^ as indicated in panel (a) of Graph 14.6. (c) As you are happily producing burgers in this long run equilibrium, a representative from the national MacWendy’s chain comes to your restaurant and asks you to convert your restau- rant to a MacWendy’s. It turns out, this would require no effort on your part — you would
Competitive Market Equilibrium 316
origin in the isoquant graph). We know from what we did above that you will produce more — thus you will hire more labor and more capital. (g) Does your decision on how many workers and capital to hire under the MacWendy’s deal depend on the size of the franchise fee G? Answer: No — once you accept the Wendy’s deal, it is a fixed cost that has no impact on the MC curve. It therefore has no impact on how much you will produce — and thus no impact on how many workers and capital you hire. (h) Suppose that you accepted the MacWendy’s deal and, because of the increased sales of ham- burgers at your restaurant, one hamburger restaurant in the city closes down. Assuming that the total number of hamburgers consumed remains the same, can you speculate whether to- tal employment (of labor) in the hamburger industry went up or down in the city? ( Hint : Think about the fact that all restaurants operate under the same decreasing returns to scale technology.) Answer: If my increased production drove one restaurant out of business and the overall number of hamburgers sold in the city remains unchanged, it must mean that production by the other restaurant that was driven out of business went down by x ∗^ while your produc- tion went from x ∗^ to 2 x ∗. With decreasing returns to scale, however, the other restaurant needed to use less labor and capital to produce x ∗^ than you need to use to increase your output from x ∗^ to 2 x ∗. Thus, overall employment in the restaurant sector of the city in- creases.
B: Suppose all restaurants in the industry use the same technology that has a long run cost function C ( w , r , x ) = 0.028486( w 0.5^ r 0.5^ x 1.25) which, as a function of wage w and rental rate r , gives the weekly cost of producing x hamburgers.^2 (a) Suppose that each hamburger restaurant has to pay a recurring weekly fee of $4,320 to operate in the city in which you are located and that w = 15 and r = 20_. If the restaurant industry is in long run equilibrium in your city, how many hamburgers does each restaurant sell each week?_ Answer: If the industry is in long run equilibrium, each restaurant makes zero long run profit and thus operates at the lowest point of its AC curve. Given w = 15 and r = 20, the average cost function is
AC ( x ) = 0.028486(
0.5 20 0.5 (^) x 1.25) x
- 4320 x = 0.4934 x 0.25^ + 4320 x
. (14.18)
The lowest point of this AC curve occurs where the derivative with respect to x is zero — i.e. where
d AC ( x ) d x = 0. x 0.^ − 4320 x^2 = 0. (14.19)
Solving this for x , we get x = 4320. Thus, each restaurant sells 4,320 hamburgers per week. (b) At what price do hamburgers sell in your city? Answer: Plugging 4,320 into the AC function in equation (14.18), this gives us the lowest point of the AC curve on the vertical axis — which is also equal to the long run equilibrium price. That price is p = 5. (c) Suppose that the weekly demand for hamburgers in your city is x ( p ) = 100,040− 1000 p. How many hamburger restaurants are there in the city? Answer: At a price of $5 per hamburger, the total demand will be x = 100,040 − 1000(5) = 95,040 hamburgers per week. With each hamburger restaurant producing 4,320 per week, this implies that the number of such restaurants in the city is 95040/4320 = 22.
(^2) For those who find unending amusement in proving such things, you can check that this cost function arises from the Cobb-Douglas production function f ( ℓ , k ) = 30 ℓ 0.4 k 0.4.
317 14C. Solutions to End-of-Chapter Exercises
(d) Now consider the MacWendy’s offer described in A(c) of this exercise. In particular, suppose that the franchise fee required by MacWendy’s is G = 5,000 and that consumers are willing to pay 94 cents more per hamburger when it carries the MacWendy’s brand name. How many hamburgers would you end up producing if you accept MacWendy’s deal? Answer: Since you would be able to sell your MacWendy’s hamburgers for $5.94 instead of $5, we need to determine how many you would produce from your long run marginal cost curve. Given the cost function C ( w , r , x ) = 0.028486( w 0.5^ r 0.5 x 1.25) that becomes C ( x ) = 0.493392 x 1.25^ when evaluated at the input prices w = 15 and r = 20, this is
MC ( x ) = ∂C ( x ) ∂x =^ 0.61674 x
0.25 (^). (14.20)
(Note that the fixed cost is irrelevant for the marginal cost curve which is why we did not need to include it in the C ( x ) function we differentiated.) You will produce until price is equal to marginal cost — i.e. until p = 5.94 = 0.61674 x 0.25^. Solving this for x , we get that you will produce x = 8,605 hamburgers per week. (e) Will you accept the MacWendy’s deal? Answer: Yes, you will — because your long run profit is now positive. We just determined that you will sell 8,605 hamburgers per week at a price of $5.94 — which gives you total revenue of about $51,114. Your weekly cost is given by the cost function that includes the fixed cost and franchise fee:
C = 0.
( 8605 1.
)
- 4320 + 5000 ≈ 50,211. (14.21) Subtracting costs from revenues, we get a long run profit of $903 per week. (f) Assuming that the total number of hamburgers sold in your city will remain roughly the same, would the number of hamburger restaurants in the city change as a result of you accepting the deal? Answer: Since you are selling roughly twice as many hamburgers (8,605 versus 4,320) as a MacWendy’s hamburger restaurant, you will in effect drive one restaurant out of the market. The total number of hamburger restaurants therefore falls to 21 — 20 of the general kind and 1 MacWendy’s. (g) What is the most that the MacWendy’s representative could have charged you for you to have been willing to accept the deal? Answer: Since you are making $903 in weekly profit when you are paying a $5,000 weekly franchise fee, the most that the MacWendy’s representative could have charged is $5,903 per week. (h) Suppose the average employee works for 36 hours per week. Can you use Shephard’s Lemma to determine how many employees you hire if you accept the deal? Does this depend on how high a franchise fee you are paying? Answer: Applying Shephard’s Lemma to the function C ( w , r , x ) = 0.028486( w 0.5^ r 0.5 x 1.25), we get the conditional labor demand function
ℓ ( w , r , x ) = ∂C^ ( w , r ,^ x ) ∂w = 0.
( r 0.5 x 1. w 0.
)
. (14.22)
After becoming a MacWendy’s, you are producing 8,605 hamburgers per week. Plugging in x = 8605 and the input prices w = 15 and r = 20, we therefore get that you will hire
ℓ = 0.
( 20 0.5 8605 1. 15 0.
) ≈ 1,363 (14.23)
hours of labor. With the average employee working 36 hours per week, this implies 37. workers. Note that the franchise fee is irrelevant to this because it is a fixed cost — as long as you accept it, you will hire about 38 workers.
