(a) Compute the price and marginal revenue for each number of units. Table 20.1 summarizes the marginal revenues and costs for this problem.

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1 22.1 Marginal Value, Marginal Cost and Deadweight Loss (a) With decreasing marginal value for buyers (as expected with efficient allocation of goods / downward sloping demand curve) and increasing marginal costs of production, social surplus is maximized at the quantity such that Marginal Value = Marginal Cost. Here, this occurs somewhere between unit 4 and unit 5, so the optimum is corresponds to quantity = 4. At Q* = 4, the value to buyers of goods sold is = 22 and the production cost of sellers is = 6, so social surplus = 22 6 = 16. (a) Compute the price and marginal revenue for each number of units. Table 20.1 summarizes the marginal revenues and costs for this problem. Step 1: Identify the price corresponding to each quantity by reading off the price by column from left to right in the row labeled Buyer Value Step 2: Compute total revenue = Quantity multiplied by price for each value of quantity. Step 3: Compute total costs by adding the production costs from left to right in the row labeled Seller Cost. Step 4: Compute Profit = Total Revenue Total Costs for each quantity (column). Table 22.1 Marginal and Total Costs and Revenues Quantity Price Total Revenue Total Costs Profit Marginal Revenue Marginal Cost DWL (if not produced) (c) Referring to the table in (b), the optimal quantity for a monopolist is 3, with a resulting profit of 15 3 = 12. Note that if the monopolist could produce fractional quantities, there would be an even more profitable quantity between 3 and 4 since MR(Q=3) = 3 > MC(Q = 3) = 2. Alternately, it is possible to compute the profit levels for each possible quantity in order to identify the profit maximizing choice: 1. Price = 7 implies quantity = 1 with total cost 0. Profit = 1 x 7 0 = Price = 8 implies quantity = 2 with total cost 1. Profit = 2 x 6 1 = 11.

2 3. Price = 5 implies quantity = 3 with total cost 3. Profit = 3 x 5 3 = Price = 4 implies quantity = 4 with total cost 6. Profit = 4 x 4 6 = Price = 3 implies quantity = 5 with total cost 10. Profit = 5 x 3 10 = Price = 2 implies quantity = 6 with total cost 15. Profit = 6 x 2 15 = Price = 1 implies quantity = 7 with total cost 21. Profit = 7 x 1 21 = (d) The deadweight loss is the cost of lost social surplus. With production limited to integers, deadweight loss corresponds to the social value of unit 4, which the monopolist will not produce. That is, DWL = Buyer Value (unit 4) = Seller Cost (Unit 4) = 4 3 = Geometry of the Cost and Demand Functions in Monopoly Pricing (a) The monopolist maximizes Q P(Q) C(Q) = 16Q Q 2 8Q. This has first-order condition 8 2Q = 0, Therefore the optimal quantity is Q* = 4, and optimal price is P(Q*) = 16 4 = 12. (b) Now the monopolist s objective function is Π(Q) = 16Q Q 2 2 Q 2 = 16Q - 3 Q 2. This has first-order condition 16 = 6Q and yields solution Q* = 16 / 6 = 8 / 3. Although both cost functions C(Q) = 8Q and C(Q) = 2Q 2 yield C(Q) = 32 at Q = 4, (the monopoly quantity from (a)) they are not identical in terms of marginal cost. In fact, MC (a) = 8, MC (b) = 4Q, so that MC is larger in (b) if Q > 2. That is, the cost function in (b) represents an increase in marginal cost from (a) for quantities near the optimum in (a). As a result, the monopoly solution in (b) involves a lower quantity than in (a). (c) Now the monopolist s objective function is Π(Q) = 16Q Q 3 / 4 8 Q = 8Q Q 3 / 4. This has first-order condition 8 = 3Q 2 / 4 and yields solution Q* = SQRT(32 / 3), which is approximately Although both demand functions P(Q) = 16 Q and P(Q) = 16 Q 2 / 4 yield P(Q) = 12 and Q = 4 (the monopoly quantity from (a)), they are not identical in terms of marginal revenue. MR (a) = 16 2Q, while MR (c) = 16-3Q 2 / 4. Thus MR (c) > MR (a) if 2Q > 3Q 2 / 4, which holds for Q < 8/3 marginal revenue is higher for (c) than for (a) if Q < 8/3 and higher for (c) than for (a) if Q > 8/3. That is, the demand function in (c) represents a reduction in marginal revenue from (a) for quantities near the optimum in (a). As a result, the monopoly solution in (c) involves a lower quantity than in (a).

3 22.3 Monopoly and Comparison of Cost Functions (a) Substituting for p 1 (Q) and p 2 (Q), p 2 (Q) > p 1 (Q) if (10 Q) > 10 Q. Solving this inequality, (10 Q) > 10 Q Q + Q 2 > 10 Q 91 19Q + Q 2 > 0 The left-hand side is strictly convex (d 2 LHS / dq 2 > 0), so dlhs / dq = 0 identifies the global minimum: dlhs / dq = 2Q 19 = 0 at Q = 19 / 2. At Q = 19/2, LHS = * 19/2 + (19 / 2) 2 = Since this is the global minimum, LHS > 0 for all Q, indicating p 2 (Q) > p 1 (Q) for Q < 10. Thus, p 2 (Q) > p 1 (Q) whenever p 1 (Q) > 0. (For Q > 10, both p 1 (Q) and p 2 (Q) are equal to 0.) (b) First note that dp 1 (Q) / dq = 1, dp 2 (Q) / dq = 2(10-Q), so both functions are decreasing for Q < 10 (the range of values where these price functions are positive). The condition for general equilibrium in an industry with constant marginal cost is simply MC = p(q). Define Q 1 * and Q 2 * to be the general equilibrium quantities for demand functions 1 and 2 respectively given MC = c. Since p 2 (Q) > p 1 (Q), it must be that p 2 (Q 1 *) > p 1 (Q 1 *) = c. Since dp 2 (Q) / dq < 0 and p 2 (Q 1 *) > c, the equilibrium condition p 2 (Q 2 *) = c requires Q 2 * > Q 1 *. (c) Revenue 1 (Q) = Q p 1 (Q) = 10Q Q 2 MR 1 (Q) = 10 2Q Revenue 2 (Q) = Q p 2 (Q) = Q(10 Q) 2 + Q = 101Q 20Q 2 + Q 3 MR 2 (Q) = Q + 3Q 2 Both MR 1 (Q) and MR 2 (Q) are declining in Q for Q < 10, so the first-order condition for profit maximization (i.e. MR(Q) = MC = c) is sufficient for profit maximization for a monopolist.

4 $ Figure 10.3 Comparing Marginal Revenue Functions MR1(Q) MR2(Q) Define Q 1M * and Q 2M * to be the general equilibrium prices for demand functions 1 and 2 respectively given MC = c. A monopolist would only choose Q 1M * > Q 2M * if MR 1 (Q) > MR 2 (Q) for some values of Q. Comparing MR 1 (Q) and MR 2 (Q), MR 1 (Q) MR 2 (Q) = (10 2Q) (101 40Q + 3Q 2 ) = 38 Q 3Q 2 91 This difference is a declining function of Q for Q < 38 / 6. Solving the quadratic equation 38 Q - 3Q 2 91 = 0 yields the relevant solution Q = (38 SQRT(352)) / At Q = 3.21, MR 1 (Q) = MR 2 (Q) = Since MR 1 (Q), MR 2 (Q) and the difference MR 1 (Q) - MR 2 (Q) are all declining in Q, 3.58 > MR 1 (Q) > MR 2 (Q) for Q > If c < 3.58, then MR 1 (Q 1M ), MR 2 (Q 2M ) = c with Q* 1M, Q* 2M > 3.21 and Q* 1M > Q* 2M. However, if c > 3.58, then MR 1 (Q 1M ), MR 2 (Q 2M ) = c with Q* 1M, Q* 2M < 3.21 and Q* 1M < Q* 2M. Choosing c* = 3.58, Q* 1M < Q* 2M if c > c* and Q* 1M > Q* 2M if c < c*. Q

5 22.4 Elasticities and the Price-Cost Margin (a) Rewriting p(q) = (q / α) -1/ε = (α / q) 1/ε. The monopolist solves Max (q) Π(q) = qp(q) c(q) = α 1/ε q (ε -1) / ε cq d Π / dq = α 1/ε [(ε -1) / ε] q -1 / ε c. If ε < 1, then both terms in d Π / dq are negative for q > 0, so in fact d Π / dq < 0, indicating that q = 0 is the optimum. But the odd part about this choice is that the with this demand function, p(q) as q 0. In fact, for q positive but very small, production costs c(q) are infinitesimal, but total revenue q p(q) = α 1/ε q (ε -1) / ε actually becomes infinite since q (ε -1) / ε = 1 / q (1 - ε) / ε as q 0. Thus, the monopolist s optimum for ε < 1 is to choose a very tiny quantity and charge an essentially infinite price. For this reason, the optimal price is not well defined. But the real problem is that the demand function is unrealistic it assumes that demand goes to zero too slowly as price becomes infinite. (b) From above, d Π / dq = α 1/ε [(ε -1) / ε] q -1 / ε c. Setting this equal to 0, we find (FOC) α 1/ε [(ε -1) / ε] q -1 / ε = c, OR c q 1 / ε = α 1/ε [(ε -1) / ε], OR q* = α [(ε -1) / cε] ε Now substitute in the formula p(q) = (q / α) -1/ε = (α / q) 1/ε to find the optimal p. p* = (α / q*) 1/ε = (ε -1) / c ε. The price-cost margin is given by (p m c) / p m = c / (ε -1). Alternately, it is also possible to work with q(p) and solve for (FOC) for the optimal price, as follows. Max (p) Π(p) = αp 1-ε cα p -ε dπ / dp = α [(1 ε) p -ε + c ε p -(ε+1) ] = α p -ε-1 [(1 ε) p + c ε ] = 0 So FOC for this problem has solution p m = c ε / (ε -1), corresponding to quantity q m = αp m -ε = α [c ε / (ε -1)] ε, consistent with our earlier results.

6 22.5 Exchange Equilibrium with Monopoly Production (a) As in the previous problem, once L 2 is fixed, person 2 has endowment (0, 8 L 2 ) and demands bundle (4qL 2, 4 L 2 ), while person 1 has endowment (2, 0) and demands bundle (1, 1/q). The equilibrium price q*(l 2 ) must clear the market for both goods. The market clearing condition for good y is given by 4 L 2 + 1/q = 8 L 2 OR q = 1 / (4 L 2 ). The equilibrium price as a function of L 2 is given by q*(l 2 ) = 1/ (4 L 2 ). Thus, person 2 can anticipate consuming the bundle (1, 4 L 2 ) after accounting for the effect of the choice of L 2 on q*. (b) Person 2 maximizes v(x 2, y 2, L 2 ) = (4L 2 * 1) 1/2 L 2 2 = 2 L 2 1/2 L 2 2. The first-order condition for the optimal value of L 1 is 1/2 1 / L 2 = 2 L 2 OR 3/2 L 2 = 1/2, which has solution L 2 = 1 / 4 1/ and associated price q*(l 2 ) = 4-2/ (c) Monopoly Outcome In the monopoly outcome, person 2 chooses L 2 = 1 / 4 1/3 and each consumer chooses the same equilibrium bundle (1, 4 2/3 ). Person 2 s utility is then v(x 2M, y 2M, L 2M ) = (4 2/3 * 1) 1/2 (1/4 1/3 ) Person 1 s utility is u(x 1M, y 1M ) = (4 2/3 * 1) 1/ Competitive Outcome In the competitive outcome, person 2 chooses L 2 = 1 and each consumer chooses the same equilibrium bundle (1, 4). Person 2 s utility is then v(x 2C, y 2C, L 2C ) = (4 * 1) 1/2 (1) 2 = 1. Person 1 s utility is u(x 1C, y 1C ) = (4 * 1) 1/2 = 2. Revealed Preference Logic As monopolist, person 2 can choose from any production level and the equilibrium (i.e. marketclearing) price associated with that production level. One such option is the competitive equilibrium with L 2 = 1, q*(l 2 ) = 4 L 2 = 4. By revealed preference, person 2 must do at least as well in the monopoly outcome as in the competitive equilibrium. Further, we should expect person 2 to gain higher utility in the monopoly outcome because the competitive outcome neglects the effect of production on marginal revenues to person 2 strongly suggesting that the

7 competitive outcome will involve too much production relative to the optimal level for a monopolist. (d) The monopoly outcome is not Pareto optimal. To see this, note that the marginal utility for person 2 for additional consumption of good y from the monopoly outcome is given by v / y 2 = (x 2M / y 2M ) 1/2 / 2 = (1 / 4 2/3 ) 1/2 / Since each unit of labor devoted to production yields eight units of good y, the marginal cost of an additional unit of good y is equal to (1/8) v / L 2 = L 2 / 4= 1 / 4 4/ The marginal utility for additional production of good y is greater than its cost. Therefore the outcome is not Pareto optimal.

8 23.1 Price Discrimination with Two Types of Consumers (a) A competitive market will set the same price for men and women with price equal to marginal cost = 4. Soving for the quantities for men and women that correspond to this price, we have q m = 30 p m = 26, and similarly q w = (24 p w ) / 2 = 10, for total quantity of 36. (b) Now we definitely want to work in quantity as a function of price so that we can add the two demand curves: Q m (p) = 30 p m ; Q w (p) = 12 p w / 2. So if price p < 24, both groups consume positive amounts. For this range of prices, we can simply add the two demand functions to get total demand: Q total (p) = 42 3p/2 if p < 24. However, if p > 20, only men consume widgets, so Q total (p) = Q m (p) = 30 p if p > 24. Note that these two separate formulas give the same outcome Q = 6 at p = 24 where we switch from a regime of having both types buying to having only men buying the good. Note also that if we add the two demand functions for p > 24, we will be including negative consumption for women in the total demand, which does not make sense. Now, we have a two-part kinked demand curve for men and women with quantity as function of price. So we can solve for the monopolist optimum by deriving first-order conditions, but we have to be careful about comparing the option of selling to both consumers or selling to neither group of consumers. PART 1: Both men and women consume positive amounts: Max Q(p) (p c) = (42 3p/2) (p 4) = 48 p 3p 2 / Taking the derivative with respect to p yields the first-order condition 48 3p = 0, corresponding to optimum p* = 16. This price is in the given range p < 24, so in fact it must be the best choice of price so that both men and women will purchase the good corresponding to quantity 18 (14 for men and 4 for women), revenue of 18 * 16 = 288 and profit of (16 4) * 18 = 216. PART 2: Only men consume positive amounts. Max Q(p) (p c) = (30 p) (p 4) = 34 p p This has first-order condition 34 2p = 0 or p* = 17. But that price does not meet the condition where only men are consuming (p > 24), so it cannot be the optimum.

9 That is, FOC cannot hold for sales only to men in the range of prices (24 < p < 30) where only men purchase widgets. This indicates a boundary solution p = 24 as the optimal price for the monopolist in this range of outcomes. At p = 24, the monopolist sells 6 units to men and none to women, for a profit of 6 * (24 4) = 120, but this is not as large a profit as the price identified above, p* = 16. That is, with these particular demand functions, it is optimal to sell to both consumers with p* = 16 and total quantity 18. (c) Here, the goal should be to distribute quantity X to the men and women who have highest value for it. That means that at the margin, the last man and last woman who consume the good should have the same value for it. Thus, the optimal division among men and women satisfies p m (q m ) = p w (q w ) And q m + q w = X. This yields two equations in two unknowns, so it should be straightforward to identify a unique solution. First, 30 q m = 24 2q w OR q m = 6 + 2q w. This already indicates an element of the solution. The first six units produced should be distributed to men. Intuitively since six men value units at 24 or more per unit, but no woman values a unit at more than 24, the first six units should be distributed to men. Second, q m + q w = X OR q m = X q w. Assuming that total units produced X > 6, this means that we have two equations that characterize the optimal distribution of units: q m = 6 + 2q w q m = X q w Substituting the second equation into the first, we have q m = 6 + 2(X q m ) OR 3 q m = 6 + 2X OR q m = 2 + 2X / 3, and therefore q w = X/3 2, for X > 6. Note that with X = 18, the optimal quantity in (b), these formulas indicate that it is socially optimal to divide this quantity by giving 14 units to men and 4 units to women, exactly matching the result in (b).

10 (d) Now it s easiest to solve the problem by computing marginal revenues for each group separately. Since men have p m (q m ) = 30 q m, Revenue(men) 2 = 30 q m q m MR(men) = 30 2 q m. Since women have p w (q w ) = 24 2q w, Revenue(women) 2 = 24 q w 2q w MR(women) = 24 4 q w. With constant MC = 4, the optimal price discriminating scheme sets MR(men) = MR(women) = MC = 4, so q m = 13, q w = 5, implying p m = 17, p w = 14. This actually indicates an identical quantity to the non-discriminating outcome in part (b), but at the same time, an inefficient distribution (from the perspective of social surplus) since men get fewer units than in the outcome in (b) and in (c) when X=18. (e) It is simplest to repeat the analysis from (b) but with the new value for c. PART 1: Both men and women consume positive amounts: Max Q(p) (p c) = (42 3p/2) (p 20) = 72 p 3p 2 / Taking the derivative with respect to p yields the first-order condition 72 3p = 0, corresponding to optimum p* = 24. At p* = 24, men consume 6 units and women consume 0 for a total of 6. Total profit is given by Q(p* MC) = 6 (24 20) = 24. PART 2: Only men consume positive amounts. Max Q(p) (p c) = (30 p) (p 20) = 50 p p This has first-order condition 50 2p = 0 or p* = 25, corresponding to q = 5, all consumed by men, with profit 5 (30 25) = 25. Comparing these two possible optimal outcomes, the monopolist does best to set p * = 25 and q* = 5, with only men purchasing widgets. (f) From part (d), MR(men) = 30 2 q m. MR(women) = 24 4 q w.

11 With MC = 20, the optimal price discriminating scheme sets q m = 5 and q w = 1, so that MR = 20 for each group. The outcome is identical to that for men in (e) when price discrimination was not allowed, but is a clear improvement for women, who can now enjoy some consumer surplus since some women value widgets at more than their cost (20) of production Domestic Monopoly and Exports (a) The profit maximization problem is Max y (80 y) + z * 30 F (y + z) 2 /4 This produces two first-order conditions, one from differentiating with respect to y and one from differentiating with respect to z: 80 2y (y+z) / 2 = 0 30 (y+z) / 2 = 0. At an interior solution with both y* > 0 and z* > 0, both first-order conditions hold. The second equation identifies the value for Q = (y + z*) = 60. Then substituting this into the first equation, we can identify the specific values for y and z. That is, y* = 25 and z* = 35. (b) If y* = 0, then the first-order condition 30 (y+z) / 2 = 0 identifies the optimal value for z. Specifically z* = 60. Note that at y* = 0, z* = 60, MC = (y + z) / 2 = 30, which is equal to the marginal revenue on the world market. (c) If z* = 0, then the first-order condition 80 2y - (y+z) / 2 = 0 identifies the optimal value for y. Specifically y* = 32. Note that at y* = 32, z* = 0, MC = (y + z) / 2 = 16, while marginal revenue in the domestic market equals 80 2y* = 16, so these quantities equate MR and MC on the domestic market. (d) Profit = Total Revenue Total Cost In (a): y* = 25, z* = 35, and domestic price = 55. Total revenue = 25 * * 30 = Total cost = F + ( ) 2 /4 = F. In (b): y* = 0, z* = 60. Total revenue = 60 * 30 = Total cost = F + ( ) 2 /4 = F. In (c), y* = 32, z* = 0, and domestic price = 48. Total revenue = 32 * 64 = Total cost = F + (32) 2 /4 = F.

12 Thus, total profit = 1525 F in (a), 900 F in (b), and 1280 F in (c). Comparing (a) and (b), the firm produces the same total quantity in each case and thus has the same production costs in each case. However, the firm increases its total revenue and total profits by selling units on the domestic market until the point where MR(domestic) = 30, the world price. That is, the firm increases profits from (b) to (a) because it can sell units on the domestic market at prices greater than 30. Comparing (a) and (c), the firm sells more units on the domestic market in (c) and achieves a higher profit on the domestic market in (c). However, the firm gains in two ways by selling on the world market in (a). First, the firm prefers to sell units 26 to 32 on the world market, as it does in (a), because its marginal revenue is 30 per unit in the world market, but less than that in the domestic market. Second, the firm can achieve additional profits by producing units 33 to 60 and selling them on the world market at a price greater than marginal cost. (e) In this problem, the monopolist s profit function is simply equal to Domestic Revenue + World Revenue Cost of Production. Further, domestic revenue is a concave (quadratic) function of domestic quantity whereas world revenue is a linear function of world quantity. In economic terms, domestic revenue involves decreasing returns / decreasing marginal utility, whereas world revenue involves constant returns as a function of quantity. Ignoring the costs of production, these two revenue terms combine to produce a profit function that is identical in form to a quasilinear utility function with two goods for a consumer maximization problem. As in the consumer problem, the solution to the monopolist s problem takes the following form: 1. Produce in the domestic market so long as MR(domestic) > MR(world). 2. Once MR(domestic) = MR(world), then produce in the world market. The only distinction between the monopolist problem and the consumer problem with quasilinear utility is the stopping point. The consumer is limited by the budget constraint and the optimal solution corresponds to continual purchasing until the budget is exhausted. The monopolist is limited by the cost function and the optimal solution is determined by the point where MR = MC. Note that MR is (weakly) decreasing in quantity produced, while MC is increasing in quantity produced, so the point where MR = MC is indeed an optimum for profits. (f) As computed in (d), the firm s profit with y* = 25, z* = 35 is equal to 1525 F. Thus, the firm can operate profitably if F < Total cost is F and total quantity is 60, so ATC = (900 + F) / 60 = 15 + (F/60). (g) ATC is greater than 30 if F/60 > 15, or F > 900. Thus, both conditions hold if 900 < F < In this range of values, the firm may appear to be dumping its product on the world market because it is producing units and selling them at price below ATC. However, the real inefficiency here is that the firm is artificially reducing quantity on the domestic market,

13 where price = 55 is much greater than the price on the world market (and dumping is an artifact of that behavior) Learning by Doing (a) The first-order conditions for myopic monopoly profit maximization are given by MR = MC in each period. In Period 1, MR 1 = 48 2Q 1 and MC 1 = 24, so the myopic optimal quantity is given by Q 1m = 12. Then period 2 MC = MC 2 = 18. Setting MR 2 = MC 2 yields 48-2Q 2 = 18, so the myopic optimal quantity is given by Q 2m = 15. (b) Given Q 1 and Q 2, the monopolist has Period 1 Revenue: (48 - Q 1 ) Q 1 Period 1 Cost : 24 Q 1 Period 2 Revenue: (48 Q 2 ) Q 2 Period 1 Cost : (24 Q 1 / 2) Q 2 So the monopolist s total profit over two periods is given by π(q 1, Q 2 ) = (48 - Q 1 ) Q 1 + (48 Q 2 ) Q 2-24 Q 1 (24 Q 1 / 2) Q 2 The first-order conditions are given by π / Q 1 = 48-2Q Q 2 /2 = 0; (1) π / Q 2 = 48-2Q Q 1 /2 = 0. (2) Solving (2) for Q 2 gives Q 2 = 12 + Q 1 /4. Substituting this into (1) gives 24-2Q 1 + (12 + Q 1 /4)/2 = 0 OR Q 1 / 8 = 0 OR Q 1 = 16. Solving for Q 2 given Q 1 =16 gives Q 2 =16. Note that you could also solve (1) and (2) by inspection, observing that the symmetry of these conditions should produce an optimal pair of quantities where Q 1 = Q 2. A forward-looking monopolist increases Q 1 from the myopic value of 12 from (a) to account for the effect of Q 1 in reducing marginal costs in period 2. In addition, the monopolist also increases Q 2 from the myopic value of 15 from (a) as a result of increased production in period 1 and thus reduced marginal costs in period 2. (c) Setting 48 Q 1 = 24 yields Q 1 = 24. In turn, Q 1 = 24 reduces marginal cost from 24 to 12 in period 2.

14 Value ($) With MC = 12 in period 2, the planner directs the firm to choose quantity Q 2 satisfying 48 Q 2 = 12 or Q 2 = 36. (d) Figure 21.3 shows the demand curve and marginal cost curves in period 1, assuming quantity Q 1 = 32. Figure 21.3(a) Learning by Doing Values p(q) = 48 - Q MC 16 Price 8 Q1 = Total Value of Goods in Period 1: The total value of goods produced in period 1 is the area under the demand curve for quantities up to 32 and is equal to the sum of (1) the triangle with endpoints (0, 48), (0, 16), (32, 16); (2) the rectangle with endpoints (0,0), (0, 16), (32, 0), (32, 16). As a function of Q 1 the triangle has height Q 1 and base Q 1 so its area is half the product or 1/2 Q 1 2. As a function of Q 1, the rectangle has area Q 1 (48 -Q 1 ). Adding these two gives the value of goods produced in period 1: Q 1 (48 -Q 1 ) + 1/2 Q 1 2 or 48 Q 1-1/2 Q 1 2. Total Value of Goods in Period 2: Iterating this reasoning, the total value of goods produced in period 2 takes the similar form 48 Q 1-1/2 Q 1 2. Total Cost of Goods in Period 1: The total cost of goods produced in period 1 is simply Q 1 * MC 1 = 24 Q 1. Total Cost of Goods in Period 2: The total cost of goods produced in period 1 is simply Q 2 * MC 2 = (24 Q 1 / 2) Q 2. Q1

15 Adding values and subtracting costs in periods 1 and 2 gives the objective function for the social planner: Social Surplus(Q 1, Q 2 ) = 48 Q 1-1/2 Q Q 2-1/2 Q Q 1 (24 Q 1 / 2) Q 2 = 24 Q 1 1/2 Q Q 2 1/2 Q Q 1 Q 2 / 2 Differentiating with respect to Q 1 and Q 2 gives the first-order conditions for the two-period planner s problem: 24 - Q 1 + Q 2 / 2 = 0, (3) 24 - Q 2 + Q 1 / 2 = 0. (4) Solving for Q 1 in (3) yields Q 1 = 24 + Q 2 / 2. Substituting this equation in (4) yields 24 - Q Q 2 / 4 = 0, OR Q 2 = 48, which in turn identifies Q 1 = 48. Comparing the results of (c) and (d), the planner also produces too little in both periods by myopically setting price = MC in each period. Note that the benevolent social planner solves the following maximization problem: (e) Given any choice of Q 1, the marginal cost in the second period will be 24 Q 1 / 2. Thus, a one unit increase (say) in Q 1 will cause a one-half unit reduction in marginal cost in period 2, corresponding to a reduction in total costs in period 2 of Q 2 / 2. Combining the direct (immediate cost) and indirect (reduced future costs) of production in the first period yields an Adjusted Marginal Cost of 24 - Q 2 / 2. With this adjustment, the planner can be seen as choosing first-period quantity to set first-period price = adjusted marginal cost: 48 Q 1 = 24 - Q 2 / 2