problem set 9 solution

problem set 9 solution - Microeconomics ECON 100A Problem...

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Microeconomics ECON 100A Problem Set 9 Due Dec 1, 2010 (Solutions Posted Dec 3, 2010) 1. (This question is from a practice final) Ralph consumes only two goods, potatoes (good 1) and yams (good 2). He is not as particular with his portion sizes and has a utility function given by 2 1 2 1 ) , ( q q q q U = , where q 1 is the number of potatoes and q 2 is the number of yams he consumes in a day. He wishes to maximize utility. The price of a potato is p 1 and the price of a yam is p 2 . Ralph owns a farm that produces only potatoes and yams and selling these crops are Ralph’s only source of income. Therefore his income, which depends on the prices of potatoes and yams and on Ralph’s endowment of potatoes and yams, is given by Y = p 1 q 1 0 +p 2 q 2 0 , where q 1 0 is Ralph’s endowment of potatoes (i.e. what he harvests) and q 2 0 is his endowment of yams (i.e. what he harvests). a. (5) Solve for Ralph’s Marshallian demand function for potatoes and his Marshallian demand function for yams. Your solution should be a function only of prices and endowments. However, if you prefer, you may still use the notation “Y” in place of p 1 q 1 0 +p 2 q 2 0 to keep the equations clean. If you do this, though, don’t forget that Y is still a function of prices, that will matter! Also solve for the indirect utility function. Usual Cobb Douglas in the usual way. q 2 /q 1 = p 1 /p 2 Y = p 1 q 1 + p 2 q 2 q 1 = Y/2p 1 = (p 1 q 10 + p 2 q 20 )/2p 1 q 2 = Y/2p 2 = (p 1 q 10 + p 2 q 20 )/2p 2 V= ( Y/2p 1 )( Y/2p 2 ) b. (5) Solve for Ralph’s Hicksian demand function for potatoes and his Hicksian demand function for yams. usual cobb douglas hicksians q 2 /q 1 = p 1 /p 2 q 1 q 2 = U q 1 = (p 2 /p 1 ) 0.5 * U 0.5 q 2 = (p 1 /p 2 ) 0.5 * U 0.5 c. (5) Verify the Slutsky equation holds for a change in the price of potatoes. That is, calculate the left hand side of the equation and the right hand side of the equation directly using the Marshallians and Hicksians above, and then show the left hand side equals the right hand side. Identify the terms that represent 1) the ordinary income effect, 2) the
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endowment income effect, 3) the substitution effect, and 4) the total effect of the change in the price in potatoes. LHS. dq 1 /dp 1 = -p 2 q 20 /2p 1 2 RHS SE. dq 1 h/dp 1 = -0.5p 1 -1/2 * (p 2 *U) 1/2 = -Y/4p 1 2 = - (p 1 q 10 + p 2 q 20 )/4p 1 2 RHS OIE -dq 1 /dY * q 1 = 1/(2p 1 ) * Y/2p 1 = -Y/4p 1 2 = - (p 1 q 10 + p 2 q 20 )/4p 1 2 RHS EIE dq 1 /dY * q 10 = q 10 /(2p 1 ) RHS = - (p 1 q 10 + p 2 q 20 )/4p 1 2 - (p 1 q 10 + p 2 q 20 )/4p 1 2 + q 10 /2p 1 = - (p 1 q 10 + p 2
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problem set 9 solution - Microeconomics ECON 100A Problem...

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