1. (a) Derive the Marshalian demand functions and the indirect utility function for the following utility function:
u(x_{1,}x_{2},x_{3}) x1^{1/6} x2^{1/6} x3^{1/6} x_{1} ≥ 0x_{2} ≥ 0x_{3} ≥ 0
(b) Using the indirect utility function which you obtained in part (a), derive an expenditure function and from it, derive Hicksian demand function for good 1.
(c) Jack and his grandfather are sitting at the dinner table, discussing their lives. Both share the same utility function as given in part (a). Jack is boasting about his $5000 a month salary, with per-unit prices of x_{1,}x_{2} and x_{3} being $4, $4 and $16 respectively. Jack’s grandfather claims that the old days were much better because although his salary was $500 a month, the per-unit prices of x_{1}, x_{2} and x_{3} were all only $1. Do you agree with his grandfather?
2. (a) Derive the Marshalian demand functions for the following utility function:
u(x_{1,}x_{2},x_{3}) = x_{1} + δ ln(x_{2}), x_{1} ≥ 0, x_{2} ≥ 0
Does one need to consider the issue of “corner solutions” here?
(b) Derive the Hicksian demand functions and the expenditure function for the following utility function:
u(x_{1,}x_{2},x_{3}) =min{√x_{1}, 2√x_{2}, 4√x_{3}},x_{1} ≥ 0, x_{2} ≥ 0, x_{3} ≥ 0
Using the expenditure function and the Hicksian demand functions that you obtained, derive the indirect utility function and the Marshalian demand function for good 1.
[Utility theory is a powerful tool, and has been used to study a wide range of issues. The next two problems will expose you to just a few of the many applications of this theory; you may see more applications in other economics courses.]
3. Consider a person’s decision problem in trying to decide how many children to have. Although she cares about children and will like to have as many as possible, she knows that children are “costly” in the sense that there are costs to their upbringing as well as the time that she would have to take off from work in order to have children. Her utility function over her own consumption (x), her own leisure (l) and the number of children (n) is given by the following utility function:
u(x, l,n) = x^{1/6}l ^{1/6} n ^{1/6}
For tractability (and to be able to use calculus), we will assume that the number of children, n, is a continuous variable (i.e. it could take any nonnegative value, including decimal values like 2.15 etc.).
This individual is endowed with a total of T units of time in her life, which she can divide between working, leisure and having children. For having each child, she will have to take time t off from work, during which she will not earn anything. In addition this, there is a per child cost of Ε for upbringing expenses.
Her wage rate is w; she uses her total income to purchase good x for her own consumption, as well as to provide for the upbringing expenses of her children. Suppose that good x is priced at p per unit.
(a) prepare consumer’s optimization problem with suitable resource constraint, and derive her Marshalian demand for children n.
(b) Assume the government introduces child benefits such as for every child she has, the government provides her an amounts. How would this affect her decision on how many children to have i.e. is dn/ds greater or less than 0?
4. Assume an individual lives for two periods, t = 1, 2. He consumes only one good,X The price of this good is p in period t = 1 and is p(1 + i) in period t = 2. Thus i is the inflation rate.
His income at period t = 1 is I, but he has no income in period t= 2 and should depend on his savings from the first period. Savings earn an interest rate of r from the bank.
The individual’s life-time utility is given by:
u(x_{1}, x_{2}) = x_{1} + δ ln(x_{2})
where x_{t} is his consumption of the single good in period t.
(a) What is the individual’s optimal savings decision?
(b) What is the effect of inflation on the individual’s savings decision i.e. does it increase or decrease in i? What is the intuition for your result?
(c) Assume the government introduces a pension plan in which the individual’s income in period t = 1 is taxed at the rate τ, but he would be given an amount B in period t = 2. What is the individual’s savings decision now? Can it occur now that he may not save at all?