Homework 8 Solution1 Econ 4001.03, Fall 2013 Prof. Lixin Ye Homework 8 Solution 1. True or False, and explain briefly. 1) The effects of a price change are always understated by a partial-equilibrium analysis when compared to a general-equilibrium analysis. False. If the price change affects other markets, the partial-equilibrium analysis will be different than the general-equilibrium. However, the results may be smaller or larger under either analysis. 2) Assume a government likes a particular equilibrium along the contract curve. It can achieve that equilibrium through competition and income redistribution. True. This statement is called the Second Welfare Theorem. 3) A monopoly always operates in the inelastic portion of its demand curve. False. A monopoly never operates in the inelastic portion of its demand curve. Marginal revenue is negative in this region. 4) The less elastic is the demand for a firm's product, the greater is that firm's market power. True. The less elastic the demand for the firm's product, the greater is the firm's ability to set price over marginal cost. 2. Consider the two markets for peanut butter and for jelly. The demand and supply curves in each market are given below. 150 10 5185 15 1020 515 5dP PJdJ JPsPPsJJQ PPQ PPQPQP=−−=−−= += + 1) What are the equilibrium prices for peanut butter and for jelly? In equilibrium, demand will equal supply in both markets. This implies2 150 10 5 20 5185 15 10 15 5PJ PJP JPP PPP P− −=+−− =+ This is a system of two equations with two unknown variables. Simplifying the equations gives 26 317 2PJPJPPPP= += + Finally, solving for JP in the first equation and substituting into the second implies 17 2(26 3 )17 52 65 357PPPPPPPPPPPP=+−=+−== Substituting this result back into any of the above equations implies 5JP =. 2) At these prices, how many units of each will be purchased? Plugging the prices from part a) back into the demand and supply equations implies 55PQ = and 40JQ =. 3. Suppose the workers at John Deery’s Tractors, a company that produces farm tractors, sign a new contract that increases their wages by 20%. Discuss how this new contract might affect the price consumers pay for corn at the grocery store. As the workers wages increase, John Deery’s costs will likely rise. This will have the effect of reducing their supply of tractors, pushing the price of tractors higher. Since tractors are an input into the farming process, as the price of tractors rise, farmer’s costs will rise. This will have the effect of reducing their supply, in this case their supply of corn, pushing the price of corn higher. Thus, an increase in the wages for tractor manufacturing employees will likely have the effect of increasing the price consumers pay for corn. 4. Two consumers, Sammy and Spencer, have two goods, pizza and ice cream. Sammy and Spencer have an equal allocation of the two goods: each has 5 slices of pizza and 4 gallons of ice cream. However, Sammy and Spencer have very3 different preferences for these two goods. Sammy loves ices cream but derives no utility from pizza. Spencer loves pizza but derives no utility from ice cream. 1) Is this an efficient allocation of pizza and ice cream? For an allocation to be efficient, it must be the case that we cannot reallocate the goods without making someone worse off. In this case, any reallocation that takes pizza away from Sammy and gives some to Spencer increases Spencer’s utility without lowering Sammy’s utility. Any reallocation that takes ice cream away from Spencer and gives some to Sammy increases Sammy’s utility without lowering Spencer’s utility. Since this type of reallocation is possible, the initial allocation is not efficient. 2) Draw the Edgeworth box, the initial allocation, and the indifference curves for Sammy and Spencer. Initial Allocation 10 8 Pizza Pizza Ice Cream Ice Cream Sammy Spencer 4 8 10 4 5 5 Usammy Uspencer 3) Identify the contract curve. The contract curve shows all allocations of goods in the Edgeworth box that are economically efficient. Any allocation that gives any pizza to Sammy or that gives any ice cream to Spencer is not efficient, because giving pizza to Spencer raises Spencer’s utility without lowering Sammy’s utility, and giving ice cream to Sammy raises Sammy’s utility without lowering Spencer’s utility. Therefore, the contract curve consists of a single point, where Sammy has all 8 gallons of ice cream and 0 pizzas and Spencer has all 10 pizzas and 0 gallons of ice cream. 5. Assume that a monopolist sells a product with a total cost function 2400 QTC += The market demand curve is given by the equation 500PQ= −.4 1) Find the profit-maximizing output and price for this monopolist. Is the monopolist profitable? To find the profit-maximizing price and quantity, set MCMR =. First, ( )/ / 500 22MR d P Q dQ P Q dP dQPQQMC Q   Hence MCMR = implies 2 500 24 500125QQQQ=-== Plug Q into the demand curve to find P. 375125500500=−=−=PPQP Profit equals total revenue minus total cost. 2125(375) (400 125 )46,875 400 15,62530,852PQ TCππππ= −= −+= −−= Yes, the monopolist is profitable. 2) Calculate the price elasticity of demand at the monopolist’s profit-maximizing price. Also calculate the marginal cost at the monopolist’s profit-maximizing output. Verify that the inverse elasticity pricing (IEPR) rule holds. The price elasticity of demand at the profit-maximizing price is 3−. ,,37513125QPQPdQ PεdP Qε  The marginal cost when 125=Q equals 2 2(125) 250Q = =. Therefore, the IEPR rule holds.5 ,1375 250 1375 31133QPP MCIEPRPε−=>=−−=