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w10sol - Economics 11 Microeconomic Theory 1 Professor...

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Economics 11: Microeconomic Theory 1 Professor Christian Hellwig TA Session Exercises Week 10 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) Find the optimal production (and consumption) level in a close economy. Find the price ratio c) Suppose that the country trades with the rest of the world at a price ratio Px/Py=1. Find the country’s optimal consumption and production choices. What is the utility level associated with this optimum? Answer: a) 2X 2 +Y 2 =15000, 4X dX + 2Y dY = 0, so RPT=-dY/dX=(2X)/Y. MRS=Y/(3X) b) Optimality condition: MRS=RPT Æ 6X 2 =Y 2 , replacing in the PPF we have: 8X 2 =15000 Æ
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