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Problem set 2 answers ECO310 Fall 07

# Problem set 2 answers ECO310 Fall 07 - ECO 310 Fall 2007...

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ECO 310 - Fall 2007 Microeconomic Theory - A Mathematical Approach Problem Set 2 - Answer Key Question 1: Since you are being asked to do essentially the same calculations for two diﬀerent values of the net earnings per hour, one strategy is to ﬁnd a general formula, denoting the net hourly earnings rate by say w , and then substitute the two particular values w = \$50 and \$40 into the formula. (Of course it is perfectly ﬁne if you do two separate numerical calculations.) (a) The budget constraint is I = w H - 25000. (b) and (c) : The utility when you work H hours is U = ln( w H - 25000) + 2 ln(5000 - H ) . (You could use Lagrange, but in this case substitution from the constraint to reduce the problem to a one-variable optimization is clearly simpler.) Utility is a concave function of hours, so the FONC gives the unique max. The only possible problems are (1) an end-point solution at H = 0, but that turns out not to be relevant. (2) the FONC yields H > 5000, this is examined below. Setting dU dH w w H - 25000 - 2 1 5000 - H = 0 gives 5000 w - w H = 2 w H - 50000 , or H = 5000 10 + w 3 w . You were not asked to do this, but here is some extra information: The formula gives H > 5000 if 10 + w > 3 w , or w < 5. If w < 5, the problem is that even working for 5000 hours is not enough to service the debt. But no self-respecting doctor would even contemplate earning less than \$5 per hour. Now we can answer the numerical questions: When w = 50, H = 5000 × 60 / 150 = 2000. When w = 40, H = 5000 × 50 / 120 = 2083. So you work more when the net earnings rate goes down. The intuition is that the income eﬀect of the net earnings rate decrease (which leaves you poorer and therefore reduces your leisure choice) exceeds the substitution eﬀect (which makes you want to work less). Question 2:

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Problem set 2 answers ECO310 Fall 07 - ECO 310 Fall 2007...

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