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Unformatted text preview: EXAMINATIONS 2009 TRIMESTER 1 ECON 201 MICROECONOMICS Time allowed: THREE HOURS Instructions: This is a closed book exam Answer ANY FIVE of the eight questions below Each problem is worth 20 marks You can use nonprogrammable calculators Please show your working ECON 201 Continued. . . Question 1: Consumer choice [20 marks] Rachel spends her income, Y , on Rock Concerts ( R ) and Sunglasses ( S ). Their prices are p R and p S . Rachels preferences are given by the CobbDouglas utility function: U ( R,S ) = R . 1 S . 1 . (a) Write out the Lagrangian for Rachels utilitymaximization problem. leaf leaf The Lagrangian for Rachels utility maximization problem is leaf L = R . 1 S . 1 + ( Y p R R p S S ) . (b) Use the Lagrangian to derive Rachels optimal bundle ( R * ,S * ) which maximizes her utility. leaf leaf The firstorder conditions are leaf L R = 0 . 1 R . 9 S . 1 p R = 0 L S = 0 . 1 R . 1 S . 9 p S = 0 leaf and leaf L R = Y p R R p S S = 0. leaf Rearrange and divide the first two conditions to get leaf MU R MU S = S R = p R p S . leaf Thus, p R R = p S S . After substituting in the third firstorder condition, we obtain leaf S = . 5 Y p S , R = . 5 Y p R . (c) For a given utility level, U , find Rachels compensated demands for Rock Concerts and Sunglasses. You can use any method you want. leaf leaf To calculate the compensated demands, we use the following pair of conditions: leaf MU R MU S = S R = p R p S leaf and leaf R . 1 S . 1 = U . leaf From the first condition, we get R = p S S/p R . Substitute in the constraint to get leaf parenleftbigg p S p R parenrightbigg . 1 S . 2 = U . leaf Therefore, leaf S = U 5 parenleftbigg p R p S parenrightbigg . 5 . leaf Similarly, we get leaf R = U 5 parenleftbigg p S p R parenrightbigg . 5 . ECON 201 2 Continued. . . (d) Use the compensated demands from part (c) to derive Rachels expenditure function E ( p R ,p S ,U ). leaf leaf The expenditure function is leaf E ( p R ,p S ,U ) = p R R + p S S = p R U 5 parenleftbigg p S p R parenrightbigg . 5 + p S U 5 parenleftbigg p R p S parenrightbigg . 5 = 2 p R p S ( U ) 5 . Question 2: Short run versus long run costs [20 marks] A firm produces output according to the following production function: q = f ( L,K ) = L 1 / 3 K 2 / 3 . The cost of labour w is $ 27 per hour and the rental cost of capital r is $ 2 per hour. (a) With the given prices, compute the costminimizing capitaltolabour ratio ( K/L ). leaf leaf The costminimizing input bundle satisfies leaf MP L MP K = (1 / 3) L 2 / 3 K 2 / 3 (2 / 3) L 1 / 3 K 1 / 3 = w r = 27 2 , leaf This condition simplifies to leaf 1 2 K L = 27 2 ....
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This note was uploaded on 05/24/2011 for the course ECON 201 taught by Professor Paulclacott during the Fall '10 term at Victoria Wellington.
 Fall '10
 PaulClacott
 Microeconomics

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