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# 2009-final-answers - EXAMINATIONS 2009 TRIMESTER 1 ECON 201...

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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 non-programmable calculators Please show your working ECON 201 Continued...

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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 . Rachel’s preferences are given by the Cobb-Douglas utility function: U ( R, S ) = R 0 . 1 S 0 . 1 . (a) Write out the Lagrangian for Rachel’s utility-maximization problem. leaf leaf The Lagrangian for Rachel’s utility maximization problem is leaf L = R 0 . 1 S 0 . 1 + λ ( Y p R R p S S ) . (b) Use the Lagrangian to derive Rachel’s optimal bundle ( R * , S * ) which maximizes her utility. leaf leaf The first-order conditions are leaf ∂L ∂R = 0 . 1 R - 0 . 9 S 0 . 1 λp R = 0 ∂L ∂S = 0 . 1 R 0 . 1 S - 0 . 9 λp 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 first-order condition, we obtain leaf S = 0 . 5 Y p S , R = 0 . 5 Y p R . (c) For a given utility level, U 0 , find Rachel’s 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 0 . 1 S 0 . 1 = U 0 . 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 0 . 1 S 0 . 2 = U 0 . leaf Therefore, leaf S = U 5 0 parenleftbigg p R p S parenrightbigg 0 . 5 . leaf Similarly, we get leaf R = U 5 0 parenleftbigg p S p R parenrightbigg 0 . 5 . ECON 201 2 Continued...
(d) Use the compensated demands from part (c) to derive Rachel’s expenditure function E ( p R , p S , U 0 ). leaf leaf The expenditure function is leaf E ( p R , p S , U 0 ) = p R R + p S S = p R U 5 0 parenleftbigg p S p R parenrightbigg 0 . 5 + p S U 5 0 parenleftbigg p R p S parenrightbigg 0 . 5 = 2 p R p S ( U 0 ) 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 cost-minimizing capital-to-labour ratio ( K/L ). leaf leaf The cost-minimizing 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 . leaf Thus, leaf K L = 27. (b) Suppose the firm wishes to produce 72 units of output. How much capital and how much labour does the firm employ? What is the long-run total cost of producing 72 units of output? leaf leaf From the above condition, we have K = 27 L . Substitute in the production function: leaf 9 L 2 / 3 L 1 / 3 = 72.

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2009-final-answers - EXAMINATIONS 2009 TRIMESTER 1 ECON 201...

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