Question (1) (a) (10 Points) Betty has no savings, but she can borrow at r

(1) (a) (10 Points) Betty has no savings, but she can borrow at r = .05. What is the maximum that Betty would be willing to pay for an apartment building with the following characteristics. As always, you may leave your answer in expression form – no need to make the actual calculation.
• It earns $10,000 per year in rental income and Betty could expect the first $10,000 to come in exactly one year.
• It requires $3000 in regular annual maintenance and the previous owner just recently performed all the required annual maintenance.
• It will require a one-time upgrade of a new roof in 3 years at a cost of $50,000.
(b) (5 Points – Difficult) Suppose that the roof replacement is actually expected to occur every 30 years. (So that it will need to be replaced in 3 years, again in 33 years, again in 63 years, etc). By how much, if at all, would this reduce Betty’s willingness to pay?

(2) Tamar has $400 to spend on rocks and shells. The price of rocks is $1 per pound. The price of shells is $2 each for the first 50 shells, $1 each for the next 100 shells and $0.50 each for all shell beyond the first 150.
(a) (10 Points) Draw a picture of Tamar’s budget for rocks and shells.
(b) (10 Points) Suppose that Tamar’s preferences over combinations of rocks and shells is given by u(R, S) = R+S – where R is the quantity of rocks and S is the quantity of shells. Which affordable bundle of rocks and shell makes Tamar the happiest?
(c) (5 Points – Difficult) For the price structure given above, draw a picture of Tamar’s income expansion path.

(3) Consider Harold, a consumer/worker who has preferences over bundles of leisure time (xl) and material consumption (xc) given by the following utility function
u(xl, xc) = x
1
2
l x
1
2
c
Harold’s endowment is (
l,
c). Assume throughout that
l = 100 and that Harold can choose to work (descreasing his consumption of leisure for each hour that he works) any amount between 0 and 100 hours. When he faces a constant wage equal to w per hour for all the hours he works (i.e. no overtime), Harold’s partial demand equations1 are given by

Figure 1. Tamar’s Budget and Optimal Choice. Her budget line is the kinked bold black line. The figure also depicts Tamar preferences represented by the blue indifference curves. Since she has perfect substitutes preferences, her indifference curves are straight line in Rock × Shell bundle space.
xl(w,
c) =100w +
c2w xc(w,
c) = 100w +
c 2
Answer the following questions. Be sure to explain your steps and interpret your answers.
(a) (10 Points) Suppose that
c = 600 and w = 6. Calculate Harold’s demands for leisure and consumption and draw a picture of his choice

Figure 2. Tamar’s disconnected Income Expansion Path. Path is drawn with thicker brownish-orange line. For income level below $250, the income expanstion path rides along the Rocks axis. At income levels above $250, the i.e.p. switches over to the Shells axis. Budget lines for income levels below $250, exactly equal to $250 and above $250 are shown to illustrate the logic of the disconnected i.e.p. in a Leisure × Consumption space diagram. Be sure to include the budget line he faces for the given endowment and wage in your diagram.

(b) (10 Points) Suppose that Harold’s wage increases from 6 to 10. Add the new budget line and new choice to your diagram.

(c) (10 Points) Calculate the income and substitution effects of the wage increase from 6 to 10. Add the compensated budget line and compensated choice to your diagram.

(d) (5 Points – More Difficult) Derive the full demand equations. (Full, meaning not partial in the sense defined above. In other words, your full demand equations should describe how Harold behaves for any combination of w and
c.) Did using the partial demand equations given above affect the calculations compared to if one had used the full demand equations?

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