HW4 solution

Exercise 14.4 Business Application: Brand Names and Franchise Fees: Suppose you are cur- rently operating a hamburger 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 labor ¢ 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 recurring 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 Exercise Graph 14.4 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. fl (Q) Exercise Graph 14.4 : MacWendy’s Franchise Deal (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 low- est point. Thus, you will produce x* as indicated in panel (a) of Exercise Graph 14.4.

(c) As you are happily producing burgers in this long run equilibrium, a rep- resentative from the national MacWendy's chain comes to your restaurant and asks you to convert your restaurant to a MacWendy'’s. It turns out, this would require no effort on your part — you would simply have to al- low the MacWendy's company to install a MacWendy'’s sign, change some of the furniture and provide your employees with new uniforms — all of which the MacWendy's parent company is happy to pay for. MacWendy's would, however, charge you a weekly franchise fee of G for the privilege of being the only MacWendy's restaurant in town. When you wonder why you would agree to this, the MacWendy's representative pulls out his marketing research that convincingly documents that consumers are willing to pay $y more per hamburger when it carries the MacWendy'’s brand name. If you accept this deal, will the market price for hamburgers in your city change? Answer: The market price for hamburgers in your city would remain un- changed at p* since the industry is competitive and thus a single firm’s actions cannot change prices. However, the price you can charge for a MacWendy’s hamburger will be p* + y. (d) On your average cost curve graph, illustrate how many hamburgers you would produce if you accepted the MacWendy's deal. Answer: You would produce where p* + y intersects your marginal cost curve. In the long run, we would use the long run marginal cost curve which is illustrated in panel (a) of Exercise Graph 14.4. Output at your restaurant would then increase from x* to x'. (If capital is fixed in the short run, your initial increase in output would be less since the short run MC curve is steeper — and you would fully adjust to x’ only once you can adjust your level of capital.) (e) Next, for a given y, illustrate the largest that G could be in order for you to accept the MacWendy's deal. Answer: This is done in panel (c) of Exercise Graph 14.4 where the long run average and marginal cost curves are drawn once again. We know that you will be able to sell your MacWendy’s hamburgers at the price p*+y, and we know you will sell x’. That makes your revenue equal to the box (p* + y)x'. We also know you will incur average costs AC' — or total long run costs AC’ times x’, the smaller rectangular box. The difference between these two boxes (a+ b+ c) is the long run profit that you can make (per time period) from being a MacWendy'’s not counting the franchise fee G. Thus, a + b + c is the most you would be willing to pay per time period for the franchise fee. (f) If you accept the deal, will you end up hiring more or fewer workers? Will you hire more or less capital? Answer: Since neither the wage nor the rental rate has changed, and since the production technology is homothetic, we know your labor to capital ratio will not change as you produce more output (because all cost mini- mizing input bundles lie on the same ray from the origin in the isoquant

Solving this for x, we get x = 4320. Thus, each restaurant sells 4,320 ham- burgers per week. (b) At what price do hamburgers sell in your city? Answer: Plugging 4,320 into the AC function in equation (14.4.i), 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 priceis p =5. (c) Suppose that the weekly demand for hamburgers in your city is x(p) = 100,040 — 1000p. 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. (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 pro- duce from your long run marginal cost curve. Given the cost function C(w, r,x) = 0.028486(w’° r%°x'2%) that becomes C(x) = 0.493392x'?°> when evaluated at the input prices w = 15 and r = 20, this is 0C(x) MC(x) = =0.61674x%%. (14.4.iii) (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 = 0.61674x%%°. 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 =0.493392(8605' %) + 4320 + 5000 = 50,211. (14.4.iv) 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 re- main 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 ac- cept 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°-° r%°x!-2%), we get the conditional labor demand function 0.5,1.25 e(w,r,x) = 20D 0.014243('—5—). (14.4.v) w w05 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 20058605125 1505 = 0.014243( ) ~ 1,363 (14.4.vi) hours of labor. With the average employee working 36 hours per week, this implies 37.86 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. (i) How does this compare to the number of employees hired by the competing non-MacWendy's hamburger restaurants? In light of your answer to (f), will overall employment in the hamburger industry increase or decrease in your city as a result of you becoming a MacWendy's restaurant? Answer: Your competitors face the same input prices w = 15 and r = 20 but produce only 4,320 hamburgers per week. Plugging these values into equation (14.4.v), we get 20&54320125 1 50.5 {= 0.014243( ) =576 (14.4.vii) hours of labor per week. At an average work week of 36 hours, this im- plies 16 workers per week. Given that the number of restaurants will decrease by 1 as a result of you becoming a MacWendy’s, the city will lose 16 hamburger restaurant jobs — but it will gain about 22 jobs in your MacWendy’s (since you employed 16 workers before becoming a MacWendy’s and about 38 afterwards). Thus, there is a net increase of 6 hamburger restaurant jobs in the city as a result of you becoming a MacWendy’s restaurant.

(b) Now suppose that an increase in the capital gains tax raises the rental rate on capital k (which is fixed for each gas station in the short run). Does anything change in the short run? Answer: Since capital is fixed in the short run and the expense on capital is not a short run cost, it does not impact short run marginal cost curves and thus does not impact the market supply curve S¥. Thus neither the short run supply curve nor the demand curve change — implying p* re- mains unchanged, as do x* and X*. (c) What happens to x*, p* and X* in the long run? Explain how this emerges from your graph. Answer: Because of the increased rental rate, the AC curve will shift up and firms would make negative long run profits if they all continued to behave as before. Some firms will exit in the long run, which causes the supply curve SM to shift to the left. That leftward shift raises output price until the price reaches the new point of the new average cost curve. Thus, we know price will increase — which implies that output in the indus- try as a whole will fall. Whether x* — each gasoline station’s output — increases or decreases depends on whether the lowest point of the new AC curve lies to the right or left of x*. Without knowing more about the underlying technology of the firm, we can’t be sure. (d) Is it possible for you to tell whether you will hire more or fewer workers as a result of the capital gains tax-induced increase in the rental rate? To the extent that it is not possible, what information could help clarify this? Answer: We can be sure that, for any given output level, each gasoline sta- tion will shift away from capital and toward labor. (Put differently, within the isoquant graph, the cost minimizing input bundles will lie on a shal- lower ray (assuming k is on the vertical axis) from the origin — with a higher labor to capital ratio regardless of how much the gasoline station sells.) We cannot, however, be sure whether the lowest point of the AC curve shifts to the right or left — and thus we cannot be sure whether output in each gasoline station will increase or decrease in the new equi- librium. Even if it decreases, the fact that we are switching to a higher labor to capital ratio may imply that the gasoline stations will hire more workers as they hire less capital — and the possibility that the lowest point of the AC curve could lie to the right of x* only reinforces this possibility. It is more likely that the increase in the rental rate will cause an increase in the equilibrium number of workers in each gasoline station if capital and labor are relatively substitutable because, as we showed in chapter 13, it is then more likely that the cross-price labor demand curve (with respect to the rental rate) is upward sloping even without an equilibrium increase in output price. (e) Is it possible for you to be able to tell whether the number of gasoline sta- tions in the city increases or decreases as a result of the increase in the rental rate? What factors might your answer depend on?

Answer: Suppose the gasoline sold in the city decreases by x% and the gasoline sold by each firm decreases by y%. Then if x = y, the number of gasoline stations remains constant. If x < y, however, the number of gasoline stations increases and if x > y, the number of gasoline stations falls. Since we cannot be sure how x and y are related to one another from what is given in the exercise, we cannot be sure whether the total number of gasoline stations increases or decreases. It depends on how the lowest point of AC curves shifts and by how much. (f) Can you tell whether employment of labor in gasoline stations increases or decreases? What about employment of capital? Answer: It is not possible to tell from the information here whether la- bor employment in gasoline stations will increase or decrease. The fact that gas stations will substitute toward labor for any output level implies it is possible that labor employment will increase if overall sales of gaso- line do not fall sufficiently to offset this. It is, however, possible to predict that less capital will be employed in the gasoline stations sector. This is because we know that overall output in the industry declines (because of the higher output price) and that firms will substitute away from capital for any output level. B: Suppose that your production function is given by f(¢,k) = 30£%4k%4, F = 1080 and the weekly city-wide demand for gallons of gasoline is x(p) = 100,040 1,000p. Furthermore, suppose that the wage is w = 15 and the current rental rate is 32.1568. Gasoline prices are typically in terms of tenths of cents — so express your answer accordingly. (a) Suppose the industry is in long run equilibrium in the absence of capital gains taxes. Assuming that you can hire fractions of hours of capital and produce fractions of gallons of gasoline, how much gasoline will you pro- duce and at what price do you sell your gasoline? (Use the cost function derived for Cobb-Douglas technologies given in equation (13.45) in exer- cise 13.5 (and remember to add the fixed cost).) Answer: The cost function for a Cobb-Douglas production process f (¢, k) ACKF is Cw,r,x)=(a+p) warpx 1/(a+fi) - (14.6.1) Aa®pBh Evaluating thisat @ = f = 0.4, A= 30, w =15 and r = 32.1568, and adding the fixed cost of $1,080, this gives us C(x) =0.62562x%° +1080. (14.6.ii) From this, we can derive the AC function 1 AC(x) = 0.62562x%-%° + ?. (14.6.iii)

(f) Will there be more or fewer gasoline stations in the city? How is your answer consistent with the change in the total sales of gasoline in the city? Answer: At the new price of $5 per gallon, the total amount of gasoline demanded in the city is x = 100,040 —1,000(5) = 95,040 — down from the initial 95,458. With each gas station producing 1,080 gallons per week, this implies that the total number of gas stations is 95040/1080=88 — up from the initial 81 gas stations. Thus, 7 new gas stations enter the market, but each gas station produces sufficiently less such that the total amount of gasoline sold in the city declines. (g) What happens to total employment at gasoline stations as a result of the capital gains tax? Explain intuitively how this can happen. Do you think this is a general result or one specific to the set-up of this problem? Answer: Initially we had 81 gasoline stations, each employing 144 worker hours. In the new equilibrium, we have 88 gasoline stations — each also employing 144 worker hours. Thus, the new equilibrium has 1,008 more labor hours employed in gasoline stations as a result of the capital gains tax. This occurs because each gasoline station, while employing the same number of workers as before, is now selling less gasoline and hiring less capital. Thus, the labor to capital ratio in each gasoline station has in- creased — as we would expect when firms substitute from the more ex- pensive capital toward labor for any given output level. While the indus- try as a whole is selling slightly less gasoline at the new higher price, it is producing that gasoline using this higher labor to capital ratio — result- ing in an increase in employment. (h) Which of your conclusions do you think is qualitatively independent of the production function used (so long as it is decreasing returns to scale), and which do you think is not? Answer: Since average costs increase, the conclusion that price will in- crease is independent of the production function employed in the anal- ysis. This also implies that the conclusion that the industry will reduce output is independent of the production function. Because each firm will substitute away from the more expensive capital and toward labor for any given output level, we know that it must be the case that the labor to cap- ital ratio increases for each firm. However, the lowest point of the aver- age cost curve could in principle shift to the left — causing each firm to produce more rather than less, which further implies that the number of gasoline stations might decrease (rather than increase). If production in each firm were to increase, it is in principle possible for capital to also in- crease (rather than decrease as we calculated) even though the labor to capital ratio increases. Similarly, the amount of labor used by each gaso- line station might increase or decrease — depending not only on whether overall output by each station increases but also on the relative substi- tutability of capital and labor in production. (Even if output decreases at each gas station, labor might also decrease if capital and labor are rela- tively complementary in production).

Exercise 15.2 Business Application: Disneyland Pricing Revisited: In end-of-chapter exercise 10.10, we investigated different ways that you can price the use of amusement park rides in a place like Disneyland. We now return to this example. Assume throughout that consumers are never at a corner solution. v = . - a e .- A: Suppose again that you own an amusement park and assume that you have the only such amusement park in the area — i.e. suppose that you face no com- petition. You have calculated your cost curves for operating the park, and it turns out that your marginal cost curve is upward sloping throughout. You have also estimated the downward sloping (uncompensated) demand curve for your amusement park rides, and you have concluded that consumer tastes appear to be identical for all consumers and quasilinear in amusement park rides. (@) Illustrate the price you would charge per ride if your aim was to maximize the overall surplus that your park provides to society. Answer: This is illustrated in panel (a) of Exercise Graph 15.2 where de- mand intersects the MC curve at output x* and price p*. The price p* maximizes the total surplus — because, for all quantities below x*, the marginal benefit (measured by the MW TP curve) exceeds the marginal cost of production. (The demand curve is equal to the MW TP curve be- cause of the quasilinearity of rides in consumer tastes.) MC b ) (=Mw7e) ‘ (*MwTP) vides x¥* rioles x » Exercise Graph 15.2: Amusement Park Rides (b) Now imagine that you were not concerned about social surplus and only about your own profit. Illustrate in your graph a price that is slightly higher than the one you indicated in part (a). Would your profit at that higher price be greater or less than it was in part (a)? Answer: This is illustrated in panel (a) using the price p. At the original price p*, profit is equal to (d + e). At the new price p, profit is equal to

(h) True or False: The ability to charge an entrance fee in addition to per-ride prices restores efficiency that would be lost if you could only charge a per- ride price. Answer: This is true — total surplus will now be back to what it was in panel (a) of the Graph when price p* was charged — except that the con- sumer surplus appears as profit instead of consumer surplus. (i) In the presence of fixed costs, might it be possible that you would shut down your park if you could not charge an entrance fee but you keep it open if you do? Answer: Yes. Suppose that recurring fixed costs were greater than the area g in panel (b) of the graph. Then youwould earn profitif you could charge an entrance fee and possibly not if you cannot charge an entrance fee. (We don’t know yet until we talk about monopolies later on in the text ex- actly how high fixed costs would have to be in order for you not to be able to operate without an entrance fee — because, as we illustrated earlier in the problem, you can in fact increase profit by charging a price higher than p* if you cannot charge an entrance fee.) B: Suppose, as in exercise 10.10, tastes for your consumers can be modeled by the utility function u(xy, xp) = 10x‘1"5 + X2, where x) represents amusement park rides and x represents dollars of other consumption. Suppose further that your marginal cost function is given by MC(x) = x/(250,000). (a) Suppose that you have 10,000 consumers on any given day. Calculate the (aggregate) demand function for amusement park rides. Answer: Each consumer’s demand function is calculated by solving the problem gcna}cx IOx?'5 + X2 subjectto I = px;+x (15.2.1) 1,42 which gives the demand function x, (p) = 25/ p?. Multiplying by 10,000, we get the aggregate demand function 250,000 .. X(p) = 2 (15.2.11) (b) What price would you charge if your goal was to maximize total surplus? How many rides would be consumed? Answer: Inverting the aggregate demand function gives us the aggregate demand curve p = 500/(x%). Total surplus is maximized where this in- tersects the marginal cost curve. This implies that we have to solve the equation 500 x x05 "~ 250,000 which gives us x = 250,000. Plugging this back into the demand curve p = 500/ (x%>), we get a per-ride price of p = 1 at which each consumer would demand 25 rides. (15.2.1ii)

(c) In the absence of fixed costs, what would your profit be at that price? Answer: At that price, your total revenues would be $250,000. Your costs would be the area under the MC curve — which is $125,000. (This is easy to calculate since the MC curve is linear and thus just half the total rev- enues. More generally, you would calculate this as the integral of the MC curve.) Thus, your profit (in the absence of fixed costs) is $125,000. (d) Suppose you charged a price that was 25% higher. What would happen to your profit? Answer: If you charged a price of $1.25 per ride (rather than the $1 per ride you just calculated), the number of rides demanded would be 250,000 X(2) = =160, 000. 15.2.iv (2) 1 252 ( ) This implies that your revenue would be 160,000(1.25) = $200,000. Your marginal cost at 160,000 would be MC(160,000) = 160,000/250,000 = 0.64. This implies that your total costs would be 0.64(160,000)/2) = $51,200 — which implies your profit would be 200,000 — 51,200 = $148,800. This is an increase in profit from the $125,000 we calculated for the per-ride price of $1. (e) Derive the expenditure function for your consumers. Answer: We first have to solve the expenditure minimization problem ;nigl px1 + x2 subjectto u= IOx(l)‘5 + Xo (15.2.v) 1,42 which gives us the compensated demand functions 25 50 . x1(p)=— and x(p,u) =u—-—. (15.2.vi) p p Using these, we can then derive the expenditure function E(p,u) = p(z—g) + (u— @) =u- 2—5 (15.2.vii) p p p (f) Use this expenditure function to calculate how much consumers would be willing to pay to keep you from raising the price from what you calculated in (b) to 25% more. Can you use this to argue that raising the price by 25% is inefficient even though it raises your profit? Answer: The amount that each consumer is willing to pay to avoid this price increase is E(1.25,u)—-E(1,u) = (fl— 12—5) - (fi— —) = $5.00, (15.2.viii) where we can leave the utility amount u unspecified since it cancels out. (You could also assume some sufficiently high income, calculate the util- ity the consumer gets at the initial price, and then us it to derive the

MW TP curve) is less than the marginal social cost (as measured by the supply curve) — which implies we would incur a negative social surplus for any unit of output above x™. Thus, a social planner that wishes to maximize social surplus would choose to produce x. (d) Now suppose that, for every unit of x that is produced, an amount of pol- lution that causes social damage of 6 is emitted. If you wanted to illustrate not just the marginal cost of production (as captured in supply curves) but also the additional marginal cost of pollution (that is not felt by produc- ers), where would that “social marginal cost” curve lie in your graph? Answer: This is illustrated in panel (b) of Exercise Graph 15.8 where the social marginal cost curve SMC lies § above the supply curve — because, for every unit of output, society now incurs not only the marginal cost faced by producers (as represented in the supply curve) but also the ad- ditional marginal cost § of pollution. (e) In the absence of any non-market intervention, do firms have an incentive to think about the marginal cost of pollution? Will the market equilibrium change as a result of the fact that pollution is emitted in the production process? Answer: Firms have no incentive to take the cost of pollution into account because they do not have to pay for it. Thus, the market equilibrium will remain unchanged and will continue to occur at the intersection of sup- ply and demand — resulting in output x. (f) Would the social planner who wishes to maximize social surplus take the marginal social cost of pollution into account? Illustrate in your graph the output quantity that this social planner would choose and compare it to the quantity the market would produce. Answer: Since the social planner wants to maximize the total social sur- plus, he would want to take into account the cost of pollution. Thus, the social planner would want to produce so long as SMC is below MW TP — or until x* in panel (b) of Exercise Graph 15.8. (8) Re-draw your graph with the following two curves: The demand curve and the marginal social cost curve (that includes both the marginal costs of producers and the cost imposed on society by the pollution that is gener- ated). Also, indicate on your graph the quantity x* that the social plan- ner wishes to produce as well as the quantity x™ that the market would produce. Can you identify in your graph an area that is equal to the dead- weight loss that is produced by relying solely on the competitive market? Answer: This is drawn in panel (c) of Exercise Graph 15.8. If the so- cial planner’s production level x* is produced, the total surplus would be equal to area a. If the market produces x¥ instead, we would still get the area a of social surplus but we would incur a negative social surplus for the output units between x* and x™. That negative area is equal to b in the graph — giving us an overall surplus under the market of (a — b). Thus, the deadweight loss from the overproduction in the market is equal to area b.

(h) Explain how pollution-producing production processes can result in in- efficient outcomes under perfect competition. How does your conclusion change if the government forces producers to pay § in a per-unit tax? Answer: Pollution producing production processes result in inefficient market outcomes if firms are not forced to face the marginal cost of caus- ing pollution. If the government charges the firms § per output unit, it is in effect forcing firms to take the cost of pollution into account. As a result, the supply curve would shift up by é (because the marginal cost of production has increased by §). As a result, the new supply curve would intersect demand at x* — thus restoring efficiency in the market. B: In exercise 15.2, you should have derived the aggregate demand function XP (p) = 250,000/ p? from the presence of 10,000 consumers with tastes that can be rep- resented by the utility function u(x,y) = 10x°° + y. Suppose that this accu- rately characterizes the demand side of the market in the current problem. Sup- pose further that the long run market supply curve is given by the equation X5 (p) = 250,000p. (a) Derive the competitive equilibrium price and quantity produced in the market. Answer: The competitive equilibrium occurs where demand intersects supply —i.e. where 250,000 p2 xPp) = =250,000p = X°(p). (15.8.i) Solving for p, we get the market price pM = 1. At that price, both the supply and demand functions tell us that the competitive market output will be xM = 250,000. (b) Derive the size of consumer surplus and profit (or producer surplus). Answer: The consumer surplus is 250,000 f ©0 250,000 1 250,000 p? - ) =250,000. (15.8.i) The producer surplus (or profit) is equal to total revenues minus costs. Total revenues are 1(250,000) = $250,000, and total costs are half that (given the linear supply curve). More generally, the costs are just the area under the supply curve — which is 1 f 250000p dp = 125000p?|§ = 125000 — 0 = 125,000. (15.8.iii) 0 Producer surplus is then equal to 250,000 — 125,000 = $125, 000.

Answer: This is illustrated in panel (b) of Exercise Graph 16.2 where the budget line (with slope —p;/p>) through the endowment point E causes the two individuals to trade from E to A. (e) Does the equilibrium lie in the core? Answer: Yes, it does, as illustrated in panel (b) by the fact that Alies on the section of the contract curve that falls between the (dotted) indifference curves which pass through the endowment E. (f) Why would your prediction when the two individuals have different bar- gaining skills differ from this? Answer: When individuals behave competitively (as they do in reaching A), they essentially have the same bargaining skills (in the sense that they are not bargaining but simply taking prices as given). In an economy with only two individuals, however, such “competitive” or “price taking” be- havior is not necessarily to be expected. Rather, individuals would em- ploy their bargaining skills to get to the best possible allocation in the core. If individual 1 has greater bargaining power, for instance, the bar- gaining process would lead to an allocation that lies between A and B on the contract curve, whereas if person 2 has greater bargaining skills, we would expect to end up between A and C. B: Suppose, as in exercise 16.1, that the tastes for individuals 1 and 2 can be described by the utility functions u' = x*x\'~% and u? = xf xél_p " (where a and B both lie between 0 and 1). (a) Derive the demands for x; and x, by each of the two individuals as a func- tion of prices py and p» (and as a function of their individual endow- ments). Answer: Solving the usual utility maximization problems (with income represented by the value of endowments), we get a(prel + pzel) (1-a)(pie] + p2e)) xi (p1, p2) = . and x;(p1,p2) = : (16.2.1) for individual 1 and X} (p1, p2) = Plprer + pacy and x5 (p1,p2) = (= P)(prci + paey : (16.2.11) for individual 2. (b) Let p; and p; denote equilibrium prices. Derive the ratio p; | p; . Answer: In equilibrium, the sum of the demands for good 1 has to be equal to the supply (or economy-wide endowment) of good 1. (The same has to be true for good 2, but we only have to solve one of the “demand