3. 1. DRAM factory. You own and operate a facility located in Taiwan that manufactures 64-megabit dynamic random-access memory chips (DRAMs) for personal computers (PCs). One year ago you acquired the land for this facility for $2 million, and used $3 million of your own money to ? nance the plant and equipment needed for DRAM manufacturing. Your facility has a maximum capacity of 10 million chips per year. Your cost of funds is 10% per year for either borrowing and investing. You could sell the land, plant and equipment today for $8 million; you estimate that the land, plant, and equipment will gain 6% in value over the coming year.
(Use a one-year planning horizon for this problem. ) In addition to the cost of land, plant, and equipment, you incur various operating expenses associated with DRAM production, such as energy, labor, raw materials, and packaging. Experience shows that these costs are $4 per chip, regardless of the number of chips produced during the year. In addition, producing DRAMs will cause you to incur ? xed costs of $500k per year for items such as security, legal, and utilities. (a) What is your cost function, C(q), where q is the number of chips produced during the year?
Answer: The $5 million you originally spent for the land, plant, and equipment is a sunk expenditure and thus not an economic cost. However, there is a “user cost of capital” associated with the land, plant and equipment, based on its current market value of $8 million and your cost of funds and the rate of depreciation or appreciation of the asset over the planning horizon. Your (opportunity) cost of investing $8 million for one year is $800k, but these assets will appreciate by $480k over the year, giving a (net) user cost of capital of $320k. (The depreciation rate is 6%. ) This is a ?
xed cost of making DRAM’s, to which we must add the other ? xed costs of $500k to get a combined ? xed cost of $820k for the year. The variable costs are a constant $4 per chip, so the cost function is C(Q) = 820k + 4 Q, in the range of 0 < Q < 1m. (One could also report that C(0) = 0, by de? nition, and that C(Q) is in? nite for Q > 1m, since your maximum capacity is one million chips per year. Of course, in practice there would likely be a way to push production beyond “rated capacity,” at some cost penalty, but that is beyond the scope of this problem. ) Assume now that you can sell as many chips as you make at the going market price per
chip of p. (b) What is the minimum price, p, at which you would ? nd it pro? table to produce DRAMs during the coming year? Answer: The average cost function is AC (Q) = 820k/Q + 4, again up to one million chips per year. This declines with Q, so the minimum AC is achieved at full capacity utilization. At one million chips per year, the ? xed costs come to $0. 82 per chip, so average costs are $4. 82 per chip. This is your minimum average cost, and thus the minimum price at which is makes sense to stay open for the year. 3. 2. mp34u. Music Ventures sells a very popular mp3 player, the mp34u.
The ? rm currently sells one million units for a price of $100 each. Marginal cost is estimated to be constant at $40, whereas average cost (at the output level of one million units) is $90. The ? rm estimates that its demand elasticity (at the current price level) is approximately -2. Should the ? rm raise price, lower price, or leave price unchanged? Explain. Answer: Optimal pricing implies m = m= p 1 |? | . MC p In this problem, we have = 100 40 = 0. 6, 100 which is greater than 1/| 2| = . 5. This tells us that the price/cost margin is too high, so a lower price would be optimal.
Note that the margin depends on MC , not AC . 3. 3. KindOfBlue jeans. Two years ago, KindOfBlue jeans were priced at $72 and 121,000 units were sold. Last year, the price was lowered to $68 and sales increased to 132,000. (a) Estimate the value of the demand elasticity. Answer: Using the percent variation method (and measuring q in 000), we have ? = 132 121 121 68 72 72 ? 1. 64 Using the logarithm method, we have ?= ln 132 ln 68 ln 121 ? ln 72 1. 52 (b) Based on your estimate of the demand elasticity, how many units would you expect to be sold if price were lowered by an additional $1?
Answer: Let q be the new value of q. Continuing with the ? rst method, I would estimate q 132 = 132 ? q = 132000 ? 1 1. 636 ? 1. 636 ? 2 67 67 68 68 68? 68 ? 135, 177 Using the logarithm method, ln(q) ln(132) = 1. 522 ? ln(67) ln(68) ? ? q = exp ln(132) 1. 522 ? ln(67) ln(68) ? 135, 011 (c) In order to increase pro? ts, should price be lowered below $68? If your answer begins — as it should! — with “it depends,” indicate as clearly as possible what additional information you would need and how you would base your answer on such additional information.
Answer: It depends on what the value of marginal cost is. To be more precise, if marginal cost is lower than marginal revenue, then we should further increase output, that is, decrease price. Marginal revenue is given by ? ? 1 MR = p 1 + ? Depending on which value of elasticity we use, we get MR ? 26. 5 or MR ? 23. 3 (when p = $68). So, if marginal cost is lower than 23. 3 we should further decrease price. If marginal cost is greater than 26. 5, we should instead increase price. Otherwise, it is not clear. 3. 4. EZjoint. After spending 10 years and $1. 5 billion, you have ?
nally gotten Food and Drug Administration (FDA) approval to sell your new patented wonder drug, which reduces the aches and pains associated with aging joints. You will market this drug under the brand name of EZjoint. Market research indicates that the demand elasticity for EZjoint is -1. 25 (at all points on the demand curve). You estimate the marginal cost of manufacturing and selling one more dose of EZjoint is $1. (a) What is the pro? t-maximizing price per dose of EZjoint? Answer: The optimal pricing rule (one version of it) states that p= MC 1+ 1 ? where ? is the price elasticity of demand. We thus have p= 1
1 1 1. 25 =5 Note that R&D expenditures are a sunk cost and thus do not enter into the pricing decision. (b) Would you expect the elasticity of demand you face for EZjoint to rise or fall when your patent expires? Answer: The level of demand for EZJoint must fall now that there are many very close substitutes in the form of generic versions. Hopefully, your brand will still allow you to command a premium price, but surely at any given price you will sell less as a result of the presence of the generic competition. 3 The elasticity of demand for EZjoint will very likely rise now that closer substitutes are available.
Customers will presumably be more price sensitive, which in turn induces you to set a lower price. However, to the extent that the price elasticity is lower at higher price levels of the demand curve; and to the extent that competitions from generics forces the branded drug ? rm to specialize on the high-valuation segment; then it is possible that the elasticity faced by the former monopolist now be less elastic. Suppose that, after patent expiry, a generic version of EZjoint was introduced in the market (under the chemical name clorophospartane). Reacting to entry, EZjoint decided to increase price.
(c) Can this behavior be consistent with rational pro? t maximizing? Answer: One possible explanation is that the market is divided into two segments, one more price sensitive than the other. While a monopolist, EZjoint was setting a price that balanced the goal of achieving both market segments. Once entry by a generic manufacturer has taken place, EZjoint decided to “give up” on the price-sensitive segment and target exclusively the segment with more inelastic demand — thus the price increase. 3. 5. Las-O-Vision. Las-O-Vision is the sole producer of holographic TVs, 3DTVs.
The weekly demand for 3DTVs is D(p) = 10200 100 p. The cost of producing q 3DTVs per day is q 2 /2 (note this implies that MC = q). (a) What is Las-O-Vision’s total revenue schedule? Answer: Total Revenue is given by p(q) q, that is, the revenue that Las-O-Vision receives when it sells q units. To get p(q), we invert the demand function q = 10. 2k 100 p by solving for p in terms of q, or p(q) = 102 q/100. Substituting this into our total revenue equation, we obtain TR (q) = (102 q/100) q = 102 q q 2 /100. (b) What is Las-O-Vision’s marginal revenue schedule?
Answer: Marginal revenue is the derivative of Total Revenue with respect to q, so MR (q) = 102 q/50; or, since our demand equation is linear in q, we can obtain it by recalling that the marginal revenue curve is twice as steep as the inverse demand curve and starts at the same point on the vertical axis. (c) What is the pro? t-maximizing number of 3DTVs for Las-O-Vision to produce each day? What price does Las-O-Vision charge per 3DTV? What is its daily pro? t? Answer: The pro? t maximizing quantity, q ? , is that quantity at which marginal cost and marginal revenue are equal. Setting MR (q) = MC , we have 102 q ? /50 = q ?
, or q ? = 100. The pro? t maximizing price is that which generates q ? = 100 in sales or, substituting into the inverse demand function calculated in (a), p(100) = 102 (100/100) = 101. When selling 100 units, Las-O-Vision generates Total revenues equal to TR (100) = 102? 100 1002/100 = $10. 1k. Its total cost is 1002 /2 = 5k. Therefore its total pro? t when it sells 100 units is 10. 1k 5k = $5. 1k. 4 Table 3. 2 Monsanto’s Roundup: price, sales, revenue and pro? t Year Price Quantity Revenue Pro? t 1995 45 13 599 400 1996 44 16 718 473 1997 40 20 823 517 1998 35 28 1003 578 1999 33 33 1092 592 2000 28 40 1105 509 2001
25 45 1139 464 Table 3. 3 Monsanto’s Roundup: demand elasticity Year Elasticity 1996 -7. 13 1997 -2. 59 1998 -2. 51 1999 -2. 11 2000 -1. 07 2001 -1. 32 3. 6. Monsanto. With reference to the discussion in Box 3. 1: (a) How do you know that cutting the price of Roundup was a good idea for Monsanto? Answer: The key issue here is elasticity: Does a lower price produce greater revenue and pro? t? The case suggests it does. Did pro? t go up when price fell? You can guesstimate revenue (take price and quantity from the ? gures) and cost (the international price is an upper bound). Almost any estimate will show pro? t going up.
Table 3. 2 is based on a marginal cost of 15, which is probably on the high side (and thus understates the increase in pro? t). (b) How might you estimate the elasticity of demand and the pro? tmaximizing price for 1995. Do you think Monsanto set the right price? Answer: The ? rst issue to address is that the price drop is not a controlled experiment: other things might have been changing at the same time. Did the market for herbicides overall grow? We don’t know. Once you’ve made this point, you can think about estimating the value of elasticity. Some year-on-year estimates can be found in Table 3. 3. If you take 2.
5 as a ballpark estimate, the ratio of price to marginal cost should be 5 1. 67, so with a cost of 15 the price should be about 25, that is, lower than they chose. There is an important quali? cation to this estimate of optimal price: One possibility is that there is some form of price discrimination going on here. Roundup Ready seeds were a complementary product to Roundup. If seed buyers were willing to pay more than other users of Roundup, Monsanto could target both markets by charging less for Roundup and more for the seeds. Some observers suggest this is precisely what Monsanto did. More on this in Chapter 6.
(c) If cutting price was a good idea, why didn’t Monsanto do it earlier? Answer: While we have data looking at the real e? ect of the price cut, Monsanto was working o? estimates, which may have suggested that what they were doing was optimal given the information available. Even if Monsanto had an estimate of demand elasticity similar to the ones we computed, there are several considerations to take into account. First, it may have been the case that Monsanto’s executives had never thought about changing prices (this possibility is not very appealing to an economist, who always assumes that ? rms maximize pro? ts).
Second, we know that success breeds laziness: while their product was successful and pro? ts high Monsanto’s executives had little incentive to work harder. Finally, one plausible explanation for waiting until the mid-1990s is risk: experimenting with di? erent prices might be costly. Early on during the Roundup patent life, there is a lot at risk. When the patent is about the expire, the costs of making a mistake are lower. In other words, “If it ain’t broke don’t ? x it; and if it’s about to break, try ? xing it now. ” 3. 7. Windows. Microsoft is the dominant player in the market for desktop operating systems.
Suppose each copy of Windows is sold for $50. Suppose moreover that the marginal cost of production and shipping is $5. What value of the demand elasticity is consistent with this price? Does it make sense? What elements of the OS market may be missing from the elasticity rule? Answer: In order for the price of $50 to be consistent with the simple formula for optimal pricing, the price elasticity of demand, ? , would need to be such that 50 5 1 = 50 ? which implies ? ? 1. 11. This means that, if the price of Windows were to go up by 10%, more than 10% would stop buying Windows.
But 10% of $50 is $5, which is a very small fraction of the overall price tag a consumer pays for a computer. It seems unreasonable that so many users would move away from Windows (to a Mac? Linux? ) because of such a small price increase. To put it di? erently: if Microsoft were to apply the optimal price formula with the true value of elasticity, it would probably set a higher price for Windows. Why then such a “small” price for the Windows operating system? One possible answer is that, by selling more units of Windows, Microsoft is able to sell more of other products (O ce?
), so that there is some cross-subsidization going on. Another possibility is that, although the short run elasticity is small, in the long run setting a higher price risks “tipping” the industry towards a di? erent operating system. More on this in Chapter 18. 6 3. 8. Pro? t maximization. not reasonable? Explain why the assumption of pro? t maximization is or is Answer: The answer to this question may be found in Section 3. 3. The main reason why we might think that the assumption of pro? t maximization is not reasonable is that ? rm managers are frequently not the ?
rm owners; and the goals of managers frequently di? er from those of the owners. However, it can be argued that the discipline imposed by shareholders, the labor market, the product market and the capital market are su cient to enforce pro? t maximization. In particular, the threat of a takeover has been found to have a signi? cant e? ect on value maximization. 3. 9. Car parts. Two parts in an automobile taillight are the plastic exterior cover and the light bulb. Which of these parts is a car company more likely to manufacture in-house? Why? Answer: Light bulbs are generally a homogeneous good.
External suppliers enjoy economies of scale and specialization; and supply the entire industry. In contrast, the plastic exterior cover must be custom-designed and manufactured for each make and model. Because it requires more Relationship Speci? c Investment (RSI), it is more likely to be made in-house. 3. 10. Jet engines. There are three main suppliers of commercial jet engines, Pratt & Whitney, General Electric, and Rolls-Royce. All three maintain extensive support sta? at major (and many minor) airports throughout the world. Why doesn’t one ? rm service each airport? Why do all three feel they need to
provide service and support operations worldwide themselves? Why don’t they subcontract this work? Why don’t they leave it entirely to the airlines? Answer: Jet engines are marvelously idiosyncratic. The knowledge, tools and parts needed to service one family (brand) of engines do not transfer fully across brands. One ? rm does not typically service each airport because the economies of scale (across brands) are small and the economies of specialization (within brand) are large. The only thing worse for an airline than an AOG (an aircraft sitting on the ground with a broken engine) is an aircraft ? ying with a broken engine or two.
To ensure their reputation and revenues and to avoid ex post hold up, airlines demand before purchasing an aircraft that engine makers pre-commit capital to ensure that parts and service are available at major stations world-wide. Because the skills to do this are RSIs, and because the engine owner’s reputation is at stake, to sell engines and credibly commit to keeping them running, each manufacturer must provide service and support at major stations. Subcontracting would be di cult because of the RSI required (the subcontractor would fear hold-up) and because a poor subcontractor would impose a negative externality on the manufacturer.
When the jet goes down, the manufacturer’s reputation will su? er on a scale beyond any contractual penalty a subcontractor could likely be held to, so the work is not usually subcontracted. In addition, the manufacturers bene? t directly from direct feedback within the ? rm on the performance of the engines they produce. This information may ? ow more readily within the ? rm than across ? rms. Some airlines with su cient scale do perform their own routine engine maintenance at 7 their own maintenance bases. However, the airlines cannot e ciently do emergency engine repairs away from an airline’s main bases.
While there are enough GE engines going through Karachi International Airport to justify an on-site GE technical support sta? , most airlines do not have enough ? ights through Karachi to justify the investment. The economies of scale in non-routine work are site and engine speci? c, not generally airline speci? c. 3. 11. Smart car. The Smart car was created as a joint venture between Daimler-Benz AG and Swatch Group AG. Although Micro Compact Car AG (the name of the joint venture) was originally jointly owned, in November of 1998 Daimler-Benz AG took complete control by buying Swatch’s share.
11 The deal put an end to a very stressed relationship between Daimler and Swatch. What does Section 3. 4 suggest as to what the sources of strain might have been? Answer: Section 3. 4 suggests that, when two parties invest in speci? c assets and contracts are incomplete, the equilibrium solution is ine cient in every situation short of vertical integration. It is likely that some of this happened in the “stressed relationship” between Daimler and Swatch. Since none of the parties was in complete control (and ownership) of the future developments in the joint venture, the incentives for each party to invest were less than e cient. 3.
12. Franchise retailing. Empirical evidence from franchise retailing suggests that, even when stores have similar characteristics, the mother company resorts to a mix between company-owned stores and franchised ones. 12 How can this be justi? ed? Answer: Franchisers face a problem in judging the performance of their franchisees. Keeping some retail locations in-house provides the parent company with a baseline of more readily accessible and less biased information against which the performance of the franchises can be measured. This information then helps to set standards in negotiating and administering future franchise contracts.
Franchising the majority of retail locations limits the parent’s direct ? nancial outlay and exposure. Franchisers might also have an interest in direct control of locations that could have a particularly strong impact on its brand or reputation. 3. 13. Body Shop. The U. K. Body Shop franchise network consists of three types of stores: franchised, company owned and partnership stores. All stores that are distant from headquarters by more than 300 miles are franchised. More than half of the company-owned stores are within 100 miles of headquarters. 13 How can you explain these fact?
Answer: Owning a store has the advantages of vertical integration discussed in Section 3. 2. However, it also has the problem that it requires increased monitoring by the store owner. We would expect the costs from monitoring to be lower the closer the store is to headquarters. Consequently, we would expect vertical integration to be more likely when the store is located closer to headquarters. The empirical evidence seems consistent with this hypothesis. 3. 14. Intel. Explain why Intel has maintained, if not increased, its competitive advantage 8 with respect to rivals.
Indicate the explanatory power of the di? erent causes considered in the text (impediments to imitation, causal ambiguity, strategy, history). Answer: This is a complex question. In fact, as argued in this chapter, this is the question in strategy. A good source for the particular case of Intel is the HBS case “Intel Corporation: 1968–1997,” No. 9–797–137 (Rev. October 21, 1998). Challenging exercises 3. 15. Input shares. Consider a ? rm with production function (Cobb-Douglas) q = ? K? L Suppose that output price p and input prices r, w are given. Show that a pro? t maximizing ?
rm chooses inputs levels such that the ratio between labor costs and total revenue w L/p q is equal to . Answer: Firm pro? t is equal to p ? K? L rK +wL The ? rst-order condition for maximization with respect to L is given by p ? K? L 1 w=0 Noting that ? K? L 1 = q/L the ? rst-order condition may be re-written as p q/L w=0 or simply = wL pq 3. 16. Cost minimizing input mix. Consider a ? rm with production function (CobbDouglas) 1 1 q = K 2 L2 Suppose that one unit of capital costs r = 12. 5, whereas one unit of labor costs w = 8. (a) Determine the optimal input mix that leads to an output of q = 2. Answer: The ?
rm’s problem is min K,L rK +wL subject to 1 1 q = K 2 L2 9 Solving the constraint for K I get q2 L Substituting this for K in the objective function, the problem becomes K= min q2 +wL L r L (3) (4) The ? rst-order condition for this minimization problem is given by r q2 +w =0 L2 Solving for L, I get L=q r r w (5) Substituting 2, 12. 5 and 8 for q, r and w I get L = 2. 5. Substituting 2. 5 for K in (3), I ? nally get K = 1. 6 (b) Determine the ? rm’s cost function, that is, the minimum cost required to produce output q. Answer: From (5) for L in (4) and simplifying I get ? p ? C = 2 rw q Substituting 12.
5 and 8 for r and w, respectively, we get C = 20 q. 3. 17. Pricing with capacity constraints. Consider again the case of a monopoly facing linear demand and constant marginal cost. Demand is q=a b p, and marginal cost is MC = c. Suppose moreover that the monopolist has a limited capacity of K. In other words, it must be q ? K. What is the optimal price? Answer: The easiest way to solve this problem is to ? rst derive the optimal output level under the assumption that there are no capacity constraints. And then to compare the candidate output level to the capacity constraint. Solving the demand curve for p, we have p= a
b 1 q b This leads to a pro? t function ?(q) = p q cq= ? a b 1 q b ? q cq The ? rst-order condition for pro? t maximization (ignoring capacity constraints) is given by 1 a q+ b b 1 b 10 c = 0, or simply q= Now we have two possibilities. If a The optimal price is then given by bc 2 a p= b 1 b a bc 2 < K, then q = ? a bc 2 ? = a bc 2 is the optimal output level. a+bc 2b If however a 2b c < K, then the optimal output level is q = K. The reason is that marginal revenue is greater than marginal cost up to q = a 2b c , and so, if q < a 2b c then we want q as high as possible. Finally, in this case optimal price is given by p=
a b 1 a K K= b b 3. 18. Optimal bidding. You are one of two companies bidding to try to win a large construction project. Call your bid b. You estimate that your costs of actually performing the work required will be $800k. You are risk neutral. j You will win if and only if your bid is lower than that of the other bidder. You are not sure what bid your rival will submit, but you estimate that the rival’s bid is uniformly distributed between $1m and $2m. k What bid should you submit? Answer: A risk-neutral bidder will use a bidding strategy that maximizes the expected value of its bid b.
This entails picking a bid value b that balances two o? setting e? ects: changes in the value of winning due to changes in the bid (the larger your bid is the more valuable the contract is) and changes in the chances of winning due to changes in b (the larger your bid is the less likely you will win). Formally, the expected value of from placing a bid b can be expressed E[b] = (b 800k) ? Prob(b < br ) where br is the rival bidder’s bid and Prob(b < br ) is the probability that its bid, b, is less than its rival’s bid. The ? rst term in this equation is simply the payo? when a bidder wins.
The second term is its chances of winning (which requires that b < br ). In this problem, the focal bidder believes that its rival’s bid can be anywhere between $1m and $2m so Prob(b < br ) = 1 (b 1m)/1m for all bids between $1m and $2m. (Note that Prob(b < br ) = 1 for all bids less than $1m since the focal bidder believes that the rival never bids below $1m; and Prob(b < br ) = 0 j . We say that an agent is risk neutral if he or she is indi? erent between receiving 100 for sure and receiving 0 or 200 with probability 50% each. More generally, a risk-neutral agent only cares about the expected value of each outcome. k .
By “uniformly distributed between a and b” we mean that all values between a and b are equally likely. 11 for all bids greater than $2m since the focal bidder believes that the rival never bids above $2m). Substituting this expression into the expected value from placing a bid b we obtain ? ? b 1m E[b] = (b 800k) 1 1m From this expression it is clear that the bidder’s payo? goes up with b but that its chance of winning declines with b. Picking the optimal b entails ? nding the maximum of E[b], which we can easily obtain by taking the derivative of E[b], setting it equal to zero and solving for b. This bid will be the point at which the two e?
ects of changing b just o? set each other. Dropping the zeros we have @ @b b E[b] = 1 .8 = 1 (b 1) (b .8) = 0 b+1 b = 1. 4m An alternative approach to this problem is to construct a demand function from the information we have about the market. You can then solve the problem in the same way as you would with more straightforward problems in which you are given an explicit demand function (i. e. , set MR = MC and solve for optimal q, then solve for optimal b). To see how this approach works, note that the bid the ? rm submits is just like a price. The higher its bid, the lower its expected demand will be.
In this case, demand falls as the price goes up because the ? rm’s chance of winning is falling. Formally, expected demand, q, at any level b is equal to q = 1 Prob(b < br ) = 1 (b 1m)/1m = 2 b/1m As this equation indicates, when the ? rm’s bid is equal to 1m, “demand” comes to 1 unit. That is, the ? rm is sure to win the contract. As the bid (the price) increases, demand falls to some fraction of a unit until at 2m “demand” is zero. Since the contract is a winner-take-all item, the idea of fractional units is not really correct, but if there were, say, N consumers instead of a single consumer, and the ? rm was bidding against other ?
rms for the business of each consumer, the aggregate demand function would then be ? ? b bN qN = N 2 = 2N 1m 1m This is like a simple linear demand function. To continue with this approach, we need to invert the demand function and solve for b to get b = 2m 1m ? q The bidder’s total revenue is then 1m ? q) q b q = (2m Taking the derivative of this total revenue function, we ? nd that the marginal revenue of the ? rm is 2m 2 ? 1m ? q 12 As we would expect, the marginal revenue curve has twice the slope of the inverse demand curve. We can then set this marginal revenue equal to the marginal cost of 800,000 to get optimal q:
800k = 2m 2 ? 1m ? q = 1. 2m 2 ? 1m ? q q = 1. 2/2 = . 6 Substituting this value into our inverse demand function, we obtain the optimal bid of b = 2m .6 ? 1m = 1. 4m 3. 19. Competitive pressure. Suppose that a ? rm’s pro? ts are given by ? = ? + (e) + ? , where ? denotes the intensity of product market competition, e e? ort by the manager, and ? a random shock. The function (e) is increasing and concave, that is, 0 > 0 and 00 < 0. In order for the ? rm to survive, pro? ts must be are greater than ?. The manager’s payo? is > 0 if the ? rm survives and zero if it is liquidated, that is, if pro?
ts fall short of the minimum target. The idea is that if the ? rm is liquidated, then the manager loses his job and the rents associated with it. Suppose that ? is normally distributed with mean µ and variance 2 , and that µ > ?. Show that increased product market competition (lower ? ) induces greater e? ort by the manager, that is, @ e/ @ ? < 0. Answer: The manager’s payo? is given by ? ? P = P ? + (e) + ? > ? e where P(x > y) is the probability that x > y. Since ? is normally distributed, we have ? ? ?? P = 1 F ? ? (e) e where F (x) is the probability that ? is less than x (cumulative distribution function).
Taking the derivative with respect to e, the manager’s choice of e? ort level, we get dP = de ? f ? ? (e) ? 1 where f (x) is the density function of ?. Since µ > ? , µ > ? ? (e). Therefore f ? ? (e) is in the increasing portion of f . It follows that an increase in ? leads to a decrease in f ? ? (e) ; and this, in turn, implies a lower dP /de. Finally, a lower dP /de implies a lower value of e. In words, a decrease in the degree of competition (higher ? ) decreases the marginal bene? t from managerial e? ort (dP /de), and ultimately leads to a lower e? ort of managerial e? ort (e). 13