If a parent is altruistic toward his p children he has

Question

Assignment 1 -

Q1. Efficient sorting in marriage markets Consider a marriage market with four men and four women. Women are indexed by i ∈ {1, 2, 3, 4} and men by j ∈ {1, 2, 3, 4}. The 4x4 matrix of marital outputs is given by:

		men
		1	2	3	4
women	1	7	8	8	5
	2	6	5	6	4
	3	2	7	7	0
	4	5	6	7	2

Note that you can get full marks for (a) and (c) if you just give me the right answer but if you give the wrong answer, an explanation will be worth partial marks and an unexplained wrong answer is a zero.

(a) What is the efficient sorting?

(b) Without proving it, explain intuitively why a set of marital incomes exist for which the sorting from (a) will be stable.

(c) Which man and which woman do you expect to receive the highest incomes in the efficient and stable sorting?

(d) The result that the efficient sorting is the equilibrium sorting in this marriage market is due to the assumption of transferability of income within marriage: the only restriction on marital incomes is that Z^m_if_j = Z^m_i + Z^f_j. Suppose instead that marital output is a non-transferable public good, which means that in a marriage between man i and woman j, Z^m_if_j = Z^m_i = Z^f_j. Will it still be the case that the efficient sorting is stable?

Q2. Sorting on traits Economic theory suggests that couples should sort positively on traits, such as personality, that help convert income into utility but negatively on traits, such as wages, that allow them to generate income. Explain.

Q3. Demand for children

Suppose you have a utility function for children given by U(n, q) = n^.95 + q^.95 where n is the number of children and q is the quality per child (we ignore demand for non-child commodities Z). Your income is I = 100, and your budget constraint is given by p_nn + p_cnq = I, where p_n is the fixed cost of having another child regardless of its quality, and p_c is the cost of a unit of quality for one child. Equality of the budget constraint comes from the non-satiation of the utility function in both arguments. If p_c = 5 and p_n = 2, what is the solution for n and q? What if p_c = 5 and p_n = 3? Do you find your mathematical answers to be intuitively credible?

Assignment 2 - Investments in children

Q1. Investments in children I:

Suppose I have one child and I value my child's adult income and my own consumption with utility function U(Z_t, I_t+1) = Z^α_t/a + I^1-α_t+1/1-α. I can make only transfers of wealth to my child y_t where the return at t + 1 is (1 + r_t)y_t and my child's adult income is I_t+1 = e_t+1 + (1 + r_t)y_t. (For simplicity, there is no luck u_t+1.) Also, I can borrow against my kid's future resources e_t+1 freely at market interest rt. Derive the parent's optimality condition relating Z_t to I_t+1 given U_t(·). What happens to I_t+1 and Z_t if my kid's natural ability e_t+1 increases by 1%?

2. Investments in children II:

Following the analysis in the Chapter 7 notes, if parents are not able to borrow against their children's future earnings, it is likely that rich families will remain (relatively) rich while poor families will remain poor from one generation to the next, even if the heredity of endowments h is fairly low (close to zero)?

3. Altruism:

If a parent is altruistic toward his p children, he has utility:

U_h = U_h(Z_h, U₁, U₂, ..., U_p);  ∂U_h/∂Z_h > 0; ∂U_h/∂U_i > 0; i = 1, ..., p

The budget constraint of the altruistic head is:

Z_h + _k=1∑^p y_k = I_h

where I_h is the head's private income and y_k are the transfers to each child k. Each of the children k = 1, ..., p therefore have private income I_k and consumption I_k + y_k = Z_k.

(a) Explain what is meant by an "effectively altruistic" head.

(b) Suppose Tom, Jane, and Mary are the three children of an altruistic head. Explain why, when Tom's income rises, the head's consumption rises even though I_h (and I_M and I_J) does not change. Would this also be the case if the altruistic head was not an effective altruist?

(c) If the head is an effective altruist toward all his children, would Tom be willing to accept a reduction of $500 in his income I_T if doing so would raise Jane's income I_J by $300 and Mary's income I_M by $250? Explain.

(d) Now suppose that, in part (c), the head is not altruistic toward Mary. Now will Tom be willing to accept a reduction of $500 in I_T if doing so would raise I_J by $300 and I_M by $250? Explain.

Attachment:- Lecture Notes.rar