Economics Assignment

1. Consider the household head model from class. Assume that individual A is the altruistic household head, with a utility function defined over A's own consumption, denoted by x_A, and individual B's consumption, denoted by x_B:

U(x_A, x_B) = lnx_A + α ln x_B

where α is the altruism parameter.

a) What does the "altruism parameter" measure?

Let A's initial annual income be $70,000, and B's initial annual income be $30,000. The unit price of private consumption is $1.00

b) What is total household income?

c) In a diagram with B's consumption on the vertical axis, sketch the household budget constraint. What is the slope of the budget line? How do you know this?

d) Identify the point on the budget constraint where each individual has consumption equal to their own income; label this bundle C.

Suppose that α = 2/3. With this utility function, it can be shown that the household head will always choose to allocate income such that individual B's consumption is equal to 0.4 of total household income.

e) Given the initial incomes of the two individuals, label the consumption bundle chosen by the household head as D on your diagram in (c).

f) Does individual B prefer their consumption at point D, or consumption equal to their own income? Briefly explain how you determined this.
Suppose now that the household must decide to take one of the following two options:

i) Individual B increases market work so that B's annual income increases to $40,000, while individual A's income is unchanged;

ii) individual B decreases market work so B's annual income decreases to $20,000; this will free up time for individual A to increase market work and earn a new annual income equal to $100,000.

f) In a new diagram, sketch the household budget lines for these two options, and identify the consumption bundles chosen by the household head under each option - label them E (for option (i)) and F (for option (ii)).

g) Which option would B prefer? Explain your reasoning.

2. Exercises and Questions for Discussion

Sarah and Josh are contemplating marriage, and in the event that they marry their utility possibility frontier is given by x + y = 12, where x and y, respectively, denote the utility levels that Sarah and Josh attain in their married state. Their respective outside options generate a utility of X = 0 for Sarah and Y = 4 for Josh.

(a) Graph the region of feasible utility combinations in marriage, which presumably is a cooperative endeavor.

(b) Define the notion of Pareto efficiency. Why might we expect the outcome here to be Pareto efficient?

(c) If Sarah could unilaterally decide the allocation of utilities, what allocation would she choose? And what allocation would Josh choose?

(d) Describe (with the help of a diagram) the nature of the Nash bargaining solution (NBS), where both Sarah and Josh influence the outcome.

(e) Compute the utilities of Sarah and Josh in the NBS by graphing the Nash product. Based on your solution, can you predict whether Sarah and Josh will opt to marry?

(f) Redo part e when X = 4 and when X = 8 (assuming that Y is fixed at 4).

(g) Using the solutions you have computed, graph Sarah's utility in the NBS as a function of her threat utility. On the same figure, also graph Josh's utility in the MIS. What do these graphs say to you?