rev2(2) - General Equilibrium 1. A small island country...

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General Equilibrium 1. A small island country produces and consumes goods X and Y. The country has available 15,000 labor hours per day. The production functions for producing X and Y are X=(0.5L X ) 0.5 and Y=(L Y ) 0.5 , where L X and L Y are the number of hours of labor devoted to X and Y, respectively. All consumers have the same preferences of U=X 0.25 Y 0.75 . a) Find the formulas for the island’s production possibility frontier (PPF), rate of product transformation (RPT) and its marginal rate of substitution (MRS). b) Suppose that the country trades with the rest of the world at a price ratio P x /P y =1. Find the country’s optimal consumption and production choices. What is the utility level associated with this optimum? c) Now suppose that the country discovers a technological innovation changes the production function for X to X=(L X ) 0.5 while the production function for Y remains unchanged. World prices are also unchanged. Find the country’s new optimal consumption and production choices. What is the utility level associated with this new optimum? d) Discuss why the country adjusted its production and consumption decisions in part c relative to part b. Discuss how the innovation affected the country’s opportunity set and well-being (utility). Answer: a) 2X 2 +Y 2 =15000, 4X dX + 2Y dY = 0, so RPT=-dY/dX=(2X)/Y. MRS=Y/(3X) b) Consumption: X=37.5, Y=112.5. Production: X=50, Y=100. U=85.48 c) Production: X=86.6, Y=86.6. Consumption: X=43.3, Y=129.9. U=98.71 d) The innovation makes it relatively cheaper for the country to produce X than before, so the PPF fans out to the right. The price vector is not tangent to the new PPF at the old production combination, and the country adjust by producing more X and less Y than before to reach a tangency where the new RPT equals world prices. The budget set shifts to the left – the country has more opportunities than before because the innovation allows it to produce goods more cheaply (i.e., more goods with the same resources as before). With greater opportunities, the country can move to a higher indifference curve and improve well-being. 2. Otto lives alone in the mountains. He consumes a diet of berries (B) and fish (F). His production functions for B and F are B=(0.5L B ) 0.5 and F=(L F ) 0.5 , where L B and L F are the number of hours of labor devoted to picking berries and fishing, respectively. Otto works exactly 162 hours per month and his preferences are given by U=B 0.5 F 0.5 . a) Find the formulas for Otto's production possibility frontier (PPF), rate of product transformation (RPT), and marginal rate of substitution (MRS). b) Suppose that Otto trades with a nearby city and the price ratio is P B /P F =0.5. Find the Otto's optimal consumption and production choices. What is the utility level associated with this optimum? c) Now suppose that a flood closes the road between Otto's house and the city, so he can no longer trade with the local community. Find Otto's new optimal consumption and production choices. What is the utility level associated with this new optimum?
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This note was uploaded on 05/06/2010 for the course ECON econ 101 taught by Professor Buddin during the Spring '10 term at UCLA.

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rev2(2) - General Equilibrium 1. A small island country...

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