Ps1 - Solve the problems PDF

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Solve the problems ...

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1

General equilibrium

1.1

Exchange

1. Suppose two individuals (Smith and Jones) have ice cream (x) and chicken soup (y). Smith’s utility function is given by US = x0.3 y 0.7 , whereas Jones’ is given by UJ = x0.5 y 0.5 . There initial endowments are: eS = (10, 6), and eJ = (6, 8). a. Draw an Edgeworth box with initial endowments and the two indifference level curves at the initial endowments. b. Calculate the contract curve and plot it in the Edgeworth box from (a.). c. What are the budget constraints of Smith and Jones, given their initial endowments? d. Calculate the demand functions for each Smith and Jones and for each good. e. Calculate the equilibrium price ratio. f. Letting y be the num´eraire, what are the equilibrium quantities consumed by Smith and Jones? g. Add the equilibrium allocation in the Edgeworth box and the two indifference level curves at the equilibrium, together with the budget constraint. 2. For the following two-person two-commodity pure exchange economy, the price of good y is normalised to 1. The table below gives the utility funcitons, endowments, and demands for goods x and y, where mi denotes the value of consumer i’s endowment, i = a, b. Person i ui a xa + ln ya b xb + 2 ln yb

ei (1, 4) (3, 2)

xi ma/px − 1 mb /px − 2

yi px /py 2px /py

a. Draw the set of Pareto efficient allocations in an Edgeworth box for this economy. b. Calculate the Walras equilibrium price for px and the Walras allocation for ((xa, ya), (xb , yb )). Check that the Walras allocation is Pareto efficient graphically and algebraically.

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1.2

Production possibility frontier

3. Suppose the production possibility frontier for guns (x) and butter (y ) is given by x2 + 2y 2 = 900. a. Graph this frontier. b. If individuals always prefer consumption bundles in which y = 2x, how much x and y will be produced? Calculate the solution and illustrate its determination in the figure from (a). c. At the point described in part (b), what will be the MRT and hence what price ratio will cause production to take place at that point? d. Show your solution on the figure from part (a).

1.3

Exchange and production

4. Suppose two individuals (Smith and Jones) each have 10 hours of labour to devote to producing either ice cream (x) or chicken soup (y). Smith’s utility function is given by US = x0.3 y 0.7 , whereas Jones’ is given by UJ = x0.5 y 0.5 . The individuals do not care whether they produce x or y, and the production function for each good is given by x = 2ℓx y = 3ℓy where ℓ is the total labour devoted to production of each good. a. Calculate the PPF. What is the MRT? What must the equilibrium price ratio, px /py , be? b. Set the wage equal to one. Given this price ratio from (a.), and the value of endowments which are now determined by the market value of labour, how much x and y will Smith and Jones demand? (Hint: Both individuals use their full work endowment; leisure time does not provide any utility in this problem.) c. Calculate the equilibrium point. Note that the equilibrium must satisfy the aggregated demand functions and must lie on the PPF. d. How will labour be allocated between x and y to satisfy demand? e. Make a graph that shows the PPF, the endowment transformation applied in equilibrium along the PPF and the Edgeworth box that arises from the equilibrium PPF transformation, the indifference level curves in equilibrium and the equilibrium price ratio line. 2

5. Consider an economy with representative consumer described by U (x, y) = xy 2 . Labour, L, is the only factor of production, has price w (wage rate), with total availabilty ¯ = 100. For each good, there is one firm producing it, with production functions of L y = Ly1/2.

x = L1/2 x ,

Choose the wage rate as the num´eraire, w = 1. a. What are the demand functions for the two goods, x = x(px , py , M ), and y = y(px , py , M ) (M denotes the income of the consumer)? b. What are the cost functions for production of the two goods? c. What are the supply functions? (Note: Firms’ supply functions are a consequence of their profit maximisation.) d. Using the condition for utility maximisation, the two supply functions, and the resource constraint of workers to find equilibrium quantities and prices. e. Calculate total income (wages plus profits) and show that in the equilibrium found in (d) this is equal to total expenditure for goods. f. Show that supply is equal to demand for each good at the equilibrium.

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