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Assignment 1

1. Kelly Anne has 100 hours to divide between work and leisure. She receives a wage of $100 per hour. With her income, she buys a good, c, whose price is equal to $1.

(a) If Kelly Anne takes r hours of leisure, how much income does she receive?

(b) Draw the budget set with r on the horizontal axis and c on the vertical axis. What are

the values of the intercepts and the slope of the budget line?

(c) Now suppose that the government passes a law requiring overtime pay for each hour above 40. In particular, workers must be paid 50% more. What are the values of the intercepts now? Find the bundle at the kink point.

(d) Finally, suppose that the government taxes income at a at rate of 20%. (There is still overtime pay.) What are the values of the intercepts now? Find the bundle at the kink point.

2. For each of the following functions, nd the slope and the equation for a typical level set.

(Express the level set in the form y = g(x).)

(a) U(x; y) = ax + by, where a and b are positive real numbers. What is the slope if a = b?

(b) U(x; y) = ln(1 + x) + y

(c) U(x; y) = x

ayb, where a and b are positive real numbers. What is the slope if a = b?

3. Andre has the utility function u(x1; x2) =px1+ x

(a) Derive expressions for Andre’s indifference curves that pass through (x

2) = (25; 10) and (x1; x

) = (25; 15).

(b) What are the marginal rates of substitution at these two bundles?

24. Goods X and Y have prices p

X= p= 1. Consumers have strictly convex preferences.

(a) Sketch a budget set, indifference curves, and optimal bundle for a single consumer. (Assume that there is an interior solution.) Label this bundle A.

(b) Suppose that the government decides to over a subsidy on good Y at the rate of s per unit, making its new price 1 s. Sketch the new budget line and the new optimal consumption bundle. Label this point B.

(c) Suppose that the total subsidy paid to the consumer in (b) is $S. Now suppose that, instead of subsidizing consumption of good Y , the government had given the consumer a lump-sum subsidy of $S. Sketch the corresponding budget line and explain why it passes through the point B.

(d) Label the consumer’s optimal bundle C. Is the consumer better-o_ with the lump-sum subsidy or the per-unit subsidy?

(e) Suppose that the government again offers a subsidy on good Y at the rate of s per unit.

At the same time, it imposes a lump-sum tax of T on consumers. Consider a consumer (“the average consumer”) for whom the tax is equal to the total value of the subsidy at

the new equilibrium: T = sy

: Does this scheme make her better-o_ than she was with no tax or subsidy? Explain.

5. Michael has the utility function uM (x1; x2) = aln(x1) +(1 a)ln(x

21; x). Taylor has u) =xb1x1b

Michael has $mM to spend while Taylor has $m 1T.2T(x1; x2

(a) Find each consumer’s optimal bundle when the prices are p

(b) Find the aggregate demand curve for good 1.

21 and p2