A monopolist (Picasso) can sell a good either in period 1 or in period 2 . The good is durable and consumers are indifferent between buying in the two periods. Hence the price they are willing to pay depends only on the amount that they expect to buy over the two periods. Call Q1 the amount sold in period 1 and Q2 the amount sold in period 2. Buyers correctly anticipate Q2 when buying in period 1. Since consumers are indifferent between buying in the two periods and correctly anticipate Q2, the price in either period (the indirect demand curve) is: p=100(Q1+Q2). It costs the monopolist 0 to sell in period 1 and c per unit in period 2 , where 100>c0. Young Picasso doesn't discount the future. His profit over the two periods is thus: =[100(Q1+Q2)](Q1+Q2)cQ2 Prices and quantities can't go negative. 1. Say that Young Picasso can commit at the beginning to values of Q1 and Q2. What values will he choose? (Beware of a corner solution. Would he ever set Q2>0 ?) What is his lifetime profit? 2. Say that Young Picasso has already sold Q1. Old Picasso is now free to choose how much Q2 to sell. Young Picasso's profit is at this point a bygone. Given Q1 Old Picasso can make an additional profit: 2=[100(Q1+Q2)]Q2cQ2 in period 2. What value of Q2 maximizes 2 as a function of Q1 ? 1 3. Go back to period 1 and imagine that Young Picasso cannot commit to a value of Q2 now, but anticipates what Old Picasso will do in period 2 as a function of the Q1. from your answer to b. What Q1 should Young Picasso sell to maximize his profit over the two periods, which is still given by (1), anticipating Old Picasso's behavior? 2 4. What is Picasso's profit as a function of c ? Show that if c=0 it's the same as that of a committed Picasso in 1. Calculate profit at c=10,20,30. At what level of c is profit at a minimum. Why does it decrease and then increase in c ? 5. Do consumers benefit from an increase in c ? Explain. 3 1 This function is Old Picasso's reaction function to Young Picasso's choice of Q1. 2Young Picasso and Old Picasso are playing a von Stackelberg game, with Young Picasso the leader and Old Picasso the follower. A von Stackelberg game differs from a Nash game in that the players choose their strategies, here Q1 and Q2, sequentially rather than simultaneously. The follower takes the leader's choice as given while the leader correctly anticipates the reaction of the follower. Think about tic-tac-toe. 3 The answers to 4 and 5 illustrate the dilemma of the durable goods monopolist recognized by Ronald Coase. A monopolist competes with previous versions of its products. Think about various versions of the iPhone. Think about how this problem might be extended to give Picasso an incentive to differentiate his work over different periods of his career. 1 6. Explain how this problem might provide a reason for the "death effect" on the prices of paintings by a famous artist that economists of the arts look for. A monopolist (Picasso) can sell a good either in period 1 or in period 2 . The good is durable and consumers are indifferent between buying in the two periods. Hence the price they are willing to pay depends only on the amount that they expect to buy over the two periods. Call Q1 the amount sold in period 1 and Q2 the amount sold in period 2. Buyers correctly anticipate Q2 when buying in period 1. Since consumers are indifferent between buying in the two periods and correctly anticipate Q2, the price in either period (the indirect demand curve) is: p=100(Q1+Q2). It costs the monopolist 0 to sell in period 1 and c per unit in period 2 , where 100>c0. Young Picasso doesn't discount the future. His profit over the two periods is thus: =[100(Q1+Q2)](Q1+Q2)cQ2 Prices and quantities can't go negative. 1. Say that Young Picasso can commit at the beginning to values of Q1 and Q2. What values will he choose? (Beware of a corner solution. Would he ever set Q2>0 ?) What is his lifetime profit? 2. Say that Young Picasso has already sold Q1. Old Picasso is now free to choose how much Q2 to sell. Young Picasso's profit is at this point a bygone. Given Q1 Old Picasso can make an additional profit: 2=[100(Q1+Q2)]Q2cQ2 in period 2. What value of Q2 maximizes 2 as a function of Q1 ? 1 3. Go back to period 1 and imagine that Young Picasso cannot commit to a value of Q2 now, but anticipates what Old Picasso will do in period 2 as a function of the Q1. from your answer to b. What Q1 should Young Picasso sell to maximize his profit over the two periods, which is still given by (1), anticipating Old Picasso's behavior? 2 4. What is Picasso's profit as a function of c ? Show that if c=0 it's the same as that of a committed Picasso in 1. Calculate profit at c=10,20,30. At what level of c is profit at a minimum. Why does it decrease and then increase in c ? 5. Do consumers benefit from an increase in c ? Explain. 3 1 This function is Old Picasso's reaction function to Young Picasso's choice of Q1. 2Young Picasso and Old Picasso are playing a von Stackelberg game, with Young Picasso the leader and Old Picasso the follower. A von Stackelberg game differs from a Nash game in that the players choose their strategies, here Q1 and Q2, sequentially rather than simultaneously. The follower takes the leader's choice as given while the leader correctly anticipates the reaction of the follower. Think about tic-tac-toe. 3 The answers to 4 and 5 illustrate the dilemma of the durable goods monopolist recognized by Ronald Coase. A monopolist competes with previous versions of its products. Think about various versions of the iPhone. Think about how this problem might be extended to give Picasso an incentive to differentiate his work over different periods of his career. 1 6. Explain how this problem might provide a reason for the "death effect" on the prices of paintings by a famous artist that economists of the arts look for