Economics 58 Fall 2009 Solutions to PS # 10

Economics 58 Fall 2009 Solutions to PS # 10 1. a. b. Q1 = 55 -P1 R1 = (55 - Q1)Q1 = 55Q1 - Q12 MR1 = 55 -2Q1 = 5 Q1 = 25, P1 = 30 ⎛ 70 − Q 2 ⎞ 2 Q2 = 70 -2P2 R 2 = ⎜ ⎟ ⋅ Q 2 = (70 Q 2 − Q 2 )/2 MR = 35 - Q2 = 5 ⎝ 2 ⎠ Q2 = 30, P2 = 20 π = (30 - 5)25 + (20 - 5)30 = 1075 Producer wants to maximize price differential in order to maximize profits but maximum price differential is $5. So P1 = P2 + 5. π = (P1 - 5)(55 - P1) + (P2 - 5)(70 - 2P2) Set up Lagrangian £ = π + λ (5 − P1 + P2 ) ∂£ = 60 − 2 P1 − λ = 0 ∂ P1 ∂£ = 80 − 4 P2 + λ = 0 ∂P2 ∂£ = 5 − P1 + P2 = 0 ∂λ Hence 60 -2P1 = 4P2 - 80 and P1 = P2 + 5. 130 = 6P2 c. P1 = P2 So π = 140P - 3P2 - 625 P= d. P2 = 21.66 P1 = 26.66 π = 1058.33 140 = 23.33 6 Q1 = 31.67 ∂π = 140 - 6P = 0 ∂P Q2 = 23.33 π = 1008.33 If the firm adopts a linear tariff of the form T( Qi ) = α i + mQi , it can maximize profit by setting m = 5, α1 = .5(55 - 5)(50) = 1250 α2 = .5(35 - 5)(60) = 900 and π = 2150. Notice that in this problem neither market can be uniquely identified as the "least willing" buyer so a solution similar to Example 14.5 is not possible. If the entry fee were constrained to be equal in the two markets, the firm could set m = 0, and charge a fee of 1225 (the most buyers in market 2 would pay). This would yield profits of 2450 - 125(5) = 1825 which is inferior to profits yielded with T(Qi). 2. Taxation of a monopoly good. The inverse elasticity rule is P = MC (1 + 1 e) . When the monopoly is subject to MC 1 an ad valorem tax of t, this becomes P = . ⋅ (1 − t ) 1 + 1 e a. With linear demand, e falls (becomes more elastic) as price rises. Hence, Paftertax = P 1 1 MC MC ⋅ < ⋅ = pretax (1 − t ) 1 + 1 (1 − t ) 1 + 1 (1 − t ) eaftertax e pretax b. With constant elasticity demand, the inequality in part a becomes an equality so Pafter tax = Ppretax /(1 − t ) . c. If the monopoly operates on a negatively sloped portion of its marginal cost curve we have (in the constant elasticity case) Paftertax = d. MCaftertax (1 − t ) ⋅ 1 1 1+ e > MC pretax (1 − t ) ⋅ 1 1 1+ e = Ppretax (1 − t ) . The key part of this question is the requirement of equal tax revenues. That is tPaQa = τ Qs where the subscripts refer to the monopoly’s choices under the two tax regimes. Suppose that the tax rates were chosen so as to raise the same revenue for a given output level, say Q. Then τ = tPa hence τ > tMRa . But in general under an ad valorem tax MRa = (1 − t ) MR = MR − tMR whereas under a specific tax, MRs = MR − τ . Hence, for a given Q, the specific tax that raises the same revenue reduces MR by more than does the ad valorem tax. With an upward sloping MC, less would be produced under the specific tax, thereby dictating an even higher tax rate. In all, a lower output would be produced, at a higher price than under the ad valorem tax. Under perfect competition, the two equalrevenue taxes would have equivalent effects. 3. a. For a price setter, profits are given by pD ( p ) − C[ D ( p )] . First order condition is π ′( p ) = pD ′ + D − C ′D ′ = 0 . 4. b. Dividing by D’ yields p + D D ′ = C ′ c. Differentiating again yields π ′′ = pD ′′ + 2 D ′ − C ′D ′′ − D ′2 C ′′ . Assuming C ′′ > 0 (MC increasing) the final term is definitely negative. With concave demand the other three terms will be definitely negative since p > C ′ . If D ′′ > 0 (convex demand – the way we usually draw it), however, the first two terms are opposite in sign. So, for convex demand, the situation is ambiguous. d. For linear demand D ′′ = 0 so the second order conditions hold trivially. e. D ( p ) = p e ⇒ D ′( p ) = ep e−1 < 0, D ′′( p ) = e( e − 1) p e−2 > 0 . The first order condition for a maximum requires e < -1. So the question is whether this will satisfy the second order conditions also. Since the final term in the second derivative is clearly negative we need only worry about the first three. Making the various substitutions (including one for C’ ) yields: pD ′′ + 2 D ′ − C ′D ′′ = e( e − 1) p e−1 + 2ep e−1 − ( e + 1)( e − 1) p e−1 . Combining terms yields ( e + 1) p e−1 which is negative for e < -1. Notice how one must consider the interaction of demand conditions with marginal cost to get this solution to work out. a. It seems that such a subsidy could prompt the monopoly to produce more, negating the effect of monopolization. b. Profits are π ( p ) = pD ( p − t ) − C[ D ( p − t )] . The first order condition for a maximum is: π ′( p ) = pD ′ + D − C ′D ′ = 0 . c. For efficient consumption should set p − t = C ′ . Substitution yields t = − D D′ . d. Dividing the result from part c by p – t yields t ( p − t ) = − 1 e or t = − ( p − t ) e = − MC e . p (1 + 1 e) = MR = MC .