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MODEL ANSWERS QUESTION 1 [20 marks] A consumer has the utility function 2 1 2 1 y x 6 U = where x and y are the two goods . (a) Find the marginal rate of substitution between the two goods when x = 25 and y=200 . [5 marks] Solution: See Besanko and Braeutigam p.83 Along an indifference curve with given U , the slope is given by x y y x 3 x y 3 MU MU x y 2 1 2 1 y x U U - = - = - = = When x=25 and y=200 , the MRS xy = -(200 ÷ 25) = -8. (b) Suppose the consumer has a budget of \$2,700 to spend, and that the price of good x is \$90 and the price of good y is \$15. What is the optimal consumption bundle of x and y ? [5 marks] Solution: See Besanko and Braeutigam pp.132-133. The Lagrangian is: ( 29 y 15 x 90 2700 y x 6 2 1 2 1 - - + = λ Λ First-order conditions: 2 1 2 1 2 1 2 1 x 30 y 0 90 x y 3 x = = - = 2 1 2 1 2 1 2 1 y 5 x 0 15 y x 3 y = = - =

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Whence x 6 y and 6 y x y 5 x 30 x 30 y y 5 x 2 1 2 1 2 1 2 1 = = = = 15 x and 90 y 2700 y 30 0 y 15 y 15 2700 = = = = - - = λ Λ (c) What is the demand curve equation for x if the price of y is fixed at \$1 and the budget remains \$2,700? [5 marks] Solution: Optimality requires xX x y x y x P y P x y P P MU MU = = = Substituting into the budget constraint gives the demand curve: x X P 1350 x 0 x P 2 2700 = = -
(d) Holding the price of y at P y =1 , use a linear approximation to the demand curve to calculate roughly the change in consumer surplus when the price of x falls from P x = 10 to P x = 6. Solution: Diagrammatically, we need the sum of areas A and B in the diagram: A = \$4 x 135 = \$540 B = 0.5 x 4 x 90 = \$180 A + B = \$720 which is roughly the consumer surplus. QUESTION 2 [20 marks] (a) Using clearly-labelled diagrams, show how the labour supply curve is derived from the labour/leisure tradeoff faced by a representative worker. [5 marks] SOLUTION See Besanko and Braeutigam pp.172-175. The student is required to draw some variant of Figure 5.24 or Figure 5.26. A backward-bending segment is not necessary but credit goes to students who show it and relate it to the income and substitution effects. x P x 10 6 True demand curve, x=1350 ÷ P x Linear approximation 135 225 A B

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(b) Suppose a tax on wage income is imposed, and that as a result labour supply falls. Does this mean that the substitution effect is more powerful than the income effect, or the other way round? Explain your answer carefully. [5 marks] From the worker’s point of view the tax is a reduction in the wage rate. This relates directly to slide 67 in my Lecture 6, which showed the case where the tax increased labour supply (see also Besanko and Braeutigam p.175 Figure 5.26): To get the opposite result asked for here– a fall in labour supply – the substitution effect must dominate over the income effect (J is the “decomposition bundle”): Careful and thorough labelling of the diagram is to be rewarded along with quality
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