Answer letter (a) to (e) only. Show solution or brief explanation

Consider an economy with the following aggregate production function:
Y =a[xKρ +(1−x)Mρ]
(1/ρ) +bP
where K is capital, M is mental labor and P is physical labor. Assume that (i) a,b > 0,
(ii) x ∈ (0,1) and (iii) ρ ∈ (−∞,1).
• (a) Take the cross-derivatives ∂2
Y/∂K∂M and ∂2
Y/∂K∂P . Determine their sign and
interpret.
• (b) Define yt = Yt/Pt and analogously kt = Kt/Pt and mt = Mt/Pt. Rewrite the
production function in lower-case letters. Then assume factors are paid their
marginal products and derive the wages for physical (wp
) and mental labor (wm).
(c) What happens to the relative wage of physical labor to mental labor as the ratio of
capital to physical labor in the economy increases?
In the model, couples live for two periods (t and t+1). They derive utility from children in
the first period and from consumption in the second period:
ut = γln(nt) + (1 − γ)ln(ct+1)
where nt is the number of children in period t and ct+1 is consumption in period t + 1. Each
parent has a time endowment of one unit and parents divide their time between work and
child rearing. Men can supply both physical and mental labor, and they earn wp + wm if
they work full time. Women cannot supply physical work, they can only supply mental
work (this is a simple way of modelling men’s advantage in physical work). Women earn
wm if they work full time. Let z be the fraction of their time endowment that one parent
must spend to raise one child.
• (d) What is the wife’s opportunity cost of raising one child? What is the husband’s
opportunity cost?
• (e) How will couples divide the child rearing amongst themselves if znt < 1? How
will they divide it if znt ≥ 1?
Since couples do not derive utility from consumption in the first period, they split their
first period income between child rearing and savings for the second period (st). In the
second period, couples don’t work and simply consume their savings: ct+1 = (1 + rt+1)st,
where rt+1 is the interest rate earned from period t to t+1.
• (f) If znt < 1, then the budget constraint of the couple in the first period is: wt
mznt +
st ≤ wt
p + 2wt
m
Explain the budget constraint in your own words.
• (g) What is the budget constraint in the first period if znt ≥ 1?
• (h) Assume znt < 1. What is the optimal number of children in this case? Set up the
couple’s utility maximization problem and solve it.
• (i) How does the optimal number of children depend on the relative wage of
physical labor to mental labor?
• (j) Summarize in your own words how the model explains the fertility reduction
during the industrial revolution.