Economics II - Exercise Session, December 3, Suggested Solution

Transcription

1 Economics II - Exercise Session, December 3, Suggested Solution Problem 1: A firm is on a competitive market, i.e. takes price of the output as given. Production function is given b f(x 1, x ) = x 1/4, prices of inputs are w 1 = 4, w = 4 and price of output is p = 1. Find the profit maximizing level of output using: (a) Profit-maximization approach (b) Cost-minimization approach Solution: (a) Profit-maximization approach: We maximize profit (revenues minus costs) of the firm. max p w 1 x 1 w x max 1x 1/4 4x 1 4x x F OC[x 1 ] : 4(x 1 x ) 4 = 0 3/4 x 1 F OC[x ] : 4(x 1 x ) 4 = 0 3/4 Solving these two equations with two unknowns gives: x 1 = x = 1 56 (b) Cost-minimization approach: Consists of two stages: First, we find minimum cost for producing an given level of output. Second, we find optimal value of output. First stage: find minimum cost for arbitrar level of output : min w 1 x 1 + w x min 4x 1 + 4x such that x 1/4 = x = 4 x 1 min 4x x 1 x 1 FOC: x 1 = 0 x 1 = and x = 1

2 So in this example, our cost function is: c() = 4x 1 + 4x = = 8 Second stage: find optimal level of output : max p c() max 8 FOC: 1 16 = 0 = 1 16 x 1 = x = = 1 56 Profit maximization Cost minimization. If a firm is maximizing profits and if it chooses to suppl some output, then it must be minimizing the cost of producing. If this were not so, then there would be some cheaper wa of producing units of output, which would mean that the firm was not maximizing profits in the first place. This simple observation turns out to be quite useful in examining firm behavior. Problem : Take the set-up from the previous problem. Apart from that the firm has to bu certain equipment before it starts the production. This equipment cost 000. Compute: variable costs (VC), fixed costs (FC), average variable costs (AVC), average fixed costs (AFC), average costs (AC) and marginal costs (MC). Solution: Note that the cost function is given b: c() = variable costs: V C = fixed costs: F C = 000 average variable costs: AV C = V C = average fixed costs: AF C = F C = 000 average costs: AC = c() = AV C + AF C = marginal costs: MC = c () =

3 Problem 3: Monopol: A monopolist can produce at constant average and marginal costs of AC = MC = 5. The firm faces a market demand curve given b D = 53 P. (a) Calculate the profit-maximizing price-quantit combination for the monopolist. Also calculate the monopolists profits and consumer surplus. (b) What output level would be produced b this industr under perfect competition if ever firm could produce at the same average and marginal cost as the monopol? (c) Calculate the consumer surplus obtained b consumers in part (b). Show that this exceeds the sum of the monopolists profits and consumer surplus received in part (a). What is the value of the deadweight loss from monopolization? Solution: (a) Since AC = MC = 5, we know the cost function is given b C() = 5. Also, we can solve for the inverse demand function to be: P () = 53. The profitmaximizing price-quantit is given b solving the following: max P () C() max(53 ) 5 Solving we have: max + 48 F.O.C.: + 48 = 0 m = 4 Using the demand equation and solving for P, we have: P (4) = 53 4 = 9 Firms profits are then given b: π(4) m = 4(9 5) = 576 Consumer Surplus is given b: CS m = 1 (53 9)(4) = 88 (b) In perfect competition firms produce such that price is equal to marginal cost, so P = 5. Using the demand function, this means pc = 53 5 = 48. 3

4 (c) If price is set to $5 and quantit is 48, then consumer surplus is given b: CS pc = 1 (53 5)(48) = 115 The combination of firms profits and consumer surplus from part (b) is given b: π m + CS m = = 864 Therefore, the value of deadweight loss from monopolization is given b: CS pc (π m + CS m ) = = 88 Problem 4: Suppose that a monopolist faces two markets with demand curves given b: D 1 (p 1 ) = 100 p 1 D (p ) = 100 p Assume that the monopolist s marginal cost is constant at $0 a unit. If it can price discriminate, what price should it charge in each market in order to maximize profits? What if it can t price discriminate? Then what price should it charge? Solution: To solve the price-discrimination problem, we first calculate the inverse demand functions: p 1 ( 1 ) = p ( ) = Marginal revenue equals marginal cost in each market ields the two equations: = 0 50 = 0 Solving we have 1 = 40 and = 30. Substituting back into the inverse demand functions gives us the prices p 1 = 60 and p = 35. If the monopolist must charge the same price in each market, we first calculate the total demand: D(p) = D 1 (p l ) + D (p ) = 00 3p 4

5 The inverse demand curve is: p() = Marginal revenue equal marginal cost gives us: = 0 which can be solved to give = 70 and p =