Description
Suppose a firm produces a single output on a competitive market using two inputs, labor L and
capital K. The technology of the firm can be described by a twice differentiable production function
f(L,K). Denote wage by w and the price of capital by r. The price of the output p is normalized
to one.
a) Formulate the maximization problem of the firm and state the first and second order condi-
tions assuming an interior solution. (max. 7 points)
Now, suppose the firm’s profit function is given by (W,r) = 01(w) + 02(r). The firm demands
labor and capital on perfectly competitive input factor markets. The supply of labor is perfectly
inelastic and given by L?. The supply of capital K% (r, m) is upward-sloping in r, and increasing in
the central bank’s money supply m.
b) Derive the labor demand and capital demand. Show that the demand of labor is nonincreasing
in w and the demand of capital nonincreasing in r. (max. 7 points)
c) Set up the system of equations that determines the equilibrium values w*, r*.
Remark: If you have not derived the factor demands in part b) assume the following
L(w,r) = g(w), K(w,r) = h(r) with g'(w), h'(r) 0.
=
=
?.
a)Calculate the Arrow-Pratt measure of
absolute risk aversion. Check if this
measure is constant, increasing or
decreasing in wealth.
b)Calculate the Arrow-Pratt measure of
relative risk aversion. Check if this measure
is constant, increasing or decreasing in
wealth.
c)Give an example of a monotonic
transformation of u that globally increases
the Arrow-Pratt measure of absolute risk
aversion. Demonstrate that the Arrow-
Pratt measure of absolute risk aversion in
your example is indeed larger for the
transformed utility function than for the
original function u.
d)Consider an individual B with a von-
Neumann-Morgenstern utility function
V(x). Suppose that her Arrow-Pratt
measure of absolute risk aversion is
decreasing in wealth. Show that v”(x) > 0
for all wealth levels x.
?.
Consider a risk-averse individual A with a von-
Neumann-Morgenstern utility function given
by: u(x) = -for x >0.
x=
x .
a)Calculate the Arrow-Pratt measure of
absolute risk aversion. Check if this
measure is constant, increasing or
decreasing in wealth.
b)Calculate the Arrow-Pratt measure of
relative risk aversion. Check if this measure
is constant, increasing or decreasing in
wealth.
a
?
c)Give an example of a monotonic
transformation of u that globally increases
the Arrow-Pratt measure of absolute risk
aversion. Demonstrate that the Arrow-
Pratt measure of absolute risk aversion in
your example is indeed larger for the
transformed utility function than for the e
original function u.
d)Consider an individual B with a von-
Neumann-Morgenstern utility function
v(x). Suppose that her Arrow-Pratt
measure of absolute risk aversion is
decreasing in wealth. Show that v”(x) > 0
for all wealth levels c.
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