Consider a two-period small open economy populated by a large number of

identical households with preferences described by the utility function

lnCT 1 + lnCN 1 + lnCT 2 + lnCN 2

where CT 1 and CT 2 denote consumption of tradables in periods 1 and 2, respectively, and CN 1 and CN 2 denote consumption of nontradables in periods

1 and 2. Households are born in period 1 with no debts or assets and are

endowed with L1 = 1 units of labor in period and L2 = 1 units of labor in

period 2. Households o?er their labor to ?rms, for which they get paid the

wage rate w1 in period 1 and w2 in period 2. The wage rate is expressed in

terms of tradable goods. Households can borrow or lend in the international ?nancial market at the world interest rate r?. Let pN 1 and pN 2 denote the

relative price of nontradable goods in terms of tradable goods in periods 1

and 2, respectively.

Firms in the traded sector produce output with the technology QT 1 = aTLT 1 in period 1 and QT 2 = aTLT 2 in period 2, where QT t denotes output in period t = 1,2 and LT t denotes employment in the traded sector in period t = 1,2.

Similarly, production in the nontraded sector in periods 1 and 2 is given by

QN 1 = aNLN 1 and QN 2 = aNLN 2 .

1. Write down the budget constraint of the household in periods 1 and 2.

2. Write down the intertemporal budget constraint of the household.

3. State the household's utility maximization problem.

International Macroeconomics, Chapter 9 299

4. Derive the optimality conditions associated with the household's max

imization problem.

5. Derive an expression for the optimal levels of consumption of trad

ables and nontradables in periods 1 and 2 (CT 1 , CN 1 , CT 2 , and CN 2 ) as functions of r?, w1, w2, pN 1 , and pN 2 .

6. Using the zero-pro?t conditions on ?rms, derive expressions for the real wage and the relative price of nontradables (wt and pN t , t = 1,2),

in terms of the parameters aT and aN.

7. Write down the market clearing condition for nontradables.

8. Write down the market clearing condition for labor.

9. Using the above results, derive the equilibrium levels of consumption ,

the trade balance, and sectoral employment (CT 1 , CT 2 , CN 1 , CN 2 , TB1, TB2, LT 1 , and LT 2 ) in terms of the structural parameters aT, aN, and r?.

10. Is there any sectoral labor reallocation over time? If so, explain the

intuition behind it.

3. (20 points) Consider a competitive market with 100 firms (producers) which produce and offer an identical good on the market. There are two types of firms, type A and type B, which differ in their cost functions. Each firm of type A has cost function CA(y) = where y 2 0 denotes an amount of individual firm's output. On the other hand, each firm of type B has cost function CB(y) = y. Suppose 50 firms are of type A, and other 50 firms are of type B. The market (industry) demand is given by X (p) = 110 - 10p for nonnegative price p 2 0. (a) Derive the individual supply function of each firm of type A. (b) Derive equilibrium price, quantity, and total surplus. (20 points) Consider a pure exchange economy with three goods, 21, 12, and 13. There a consumers who are either of type A or of type B.(c) If RC makes consumption decisions in order to maximize utility, write out the necessary first order conditions for utility maximization. (d) Find a set of prices ( P. and w) that give a competitive equilibrium in this economy. 4. Suppose the demand for a product is Q = 500-2P. A monopolist is the only producer of the good. The monopolist has marginal cost function me = 25 + 1Q. Production of the good also leads to pollution. The external marginal costs are me = 1Q. (a) What is the socially optimal level of output, Q50? (b) What level of output maximizes profits to the monopolist? Qu. (c) Calculate the Total External Costs under monopoly and compare with the socially optimal level of External Costs. (d) Compare the Total surplus under monopoly with the total surplus under a com- petitive outcome when the industry marginal cost function is mend = 25 + 10. (e) Describe what government ad valorem tax or subsidy would be needed to bring about the socially optimal level of production or consumption. 5. Mosquito control in a small town has characteristics of a public good. Each individual has demand for mosquito control described by the inverse demand/ willingness-to-pay function wip = 20 - m where m is the level of mosquito control. The cost of an additional unit of control is c = $10. (a) What is the socially optimal level of mosquito control when there are 2 individuals in the town. (b) Suppose that mosquito control is not publicly provided. If individual 1 provided no mosquito control, how much would individual 2 provide? If individual 1 provided 5 units of mosquito control, how much would individual 2 provide? What is the total amount of mosquito control provided in each of these cases? (c) Suppose there are 100 people in the town, each with the inverse demand/wtp function: wep = 20 - m. What is the socially optimal level of mosquito control? What is the cost associated with the optimal provision? (d) If the costs of the optimal level of control from (c) are divided equally among the 100 residents of the town, what is the cost per individual? Will it be possible for the town to rely on a voluntary contribution scheme to finance the mosquito control? Explain. 6. British Columbia is a large province with (outside the Lower Mainland) relatively low population density. In most areas of the province where congestion is not a concern the road network can be viewed as a public-good. Suppose that in a rural region of the province (no congestion) there are two types of users of the road network. Commercial users have a high demand for the road network, of = 1000 - 5p and private users have a low demand for the road network, q = 1000 - 20p, where q is measured in km of road available for a year and the price is measured in $/km for one year. Suppose there is one commercial user and one private user and the marginal cost of providing a km of road for one year is $90