1. (12 marks) Consider the properties of quasiconcavity and quasiconvexity.
a. Define quasiconcavity and quasiconvexity as they relate to f(xy, ..., xp).
. Prove that convexity implies quasiconvexity as they relate to f(xy, ..., Xp).
c. Determine the quasiconcavity-quasiconvexity status of f(x,y) = Inx + Iny.
d. Letting x = (x4, ..., %,) and z = g(x), prove that f(z) is quasiconcave if g(x) is
quasiconcave and f(z) is non-decreasing.
e. Define explicit quasiconcavity and explicit quasiconvexity as they relate to
fxs, oe, 2p).
f. Determine the explicit quasiconcavity-explicit quasiconvexity status of f(x) =
{ ifx€ (—Ve, ve)
, where ¢ > 0is a constant.
x? otherwise
2. (12 marks) Consider the properties of homogeneity and homotheticity.
a. Define homogeneity of degree r as it relates to f(xy, ..., Xp).
. Let Q(L,K) be a linearly homogenous (i.e. constant-returns-to-scale) production
function where L and K are labour and capital inputs, respectively, and MPP; is the
marginal physical product of input i € {L, K}. Prove that MPP; = ¢'(k) and MPP, =
¢(k) — k¢'(k), where k = K/L and ¢(k) = Q(L,K)/L = Q(1,k), verifying that the
marginal physical products depend on the capital-labour ratio only.
c. Forx = (xy,...,%,), prove that f(x) € C* is homogenous of degree r if and only if
7-1 fix; = rf (x). For the sufficiency direction of the proof, consider using the
Fundamental Theorem of Calculus and the monotonicity of the natural logarithm.
d. Define homotheticity as it relates to a composite function h(xy, ..., xn) = f(z) where
2= g(r, Xa).
e. Define for a function h(xy, ..., x,,) the 8-intensity expansion path in the x;-x; plane.
f. Prove that every expansion path of a homothetic function h(x, ..., x) is a ray.
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3. (20 marks) Consider the general two-good consumer’s problem, where the utility function
U(x,y) € C? is strictly increasing and strictly quasiconcave and where x > 0 and y > 0. The
udget is B > 0 and the prices of goods x and y are p, > 0 and p,, > 0, respectively.
a.
Set up the utility maximization problem and derive its first-order conditions. Why is
there no need to check the second-order condition to verify that the first-orde
conditions indeed provide for a unique absolute maximum?
Totally differentiate, and apply the Implicit Function Theorem to, the first-orde
conditions to derive the comparative statics = and x where x* = x* (Par Py, B)
x x
andy* = y* (0x Py, B) are the utility-maximizing demand functions.
Use the results of Part (b) to prove that x and y must be substitutes.
ox" . . ; . ox
- and use it along with the comparative static > to
x
Derive the comparative static
prove that a Giffen good must be inferior (i.e. & <0 ifs > 0).
x
Use the first-order conditions to prove that x* (ps, Py, B) and y* (Px. 0ys B) are
homogenous of degree zero.
4. (6 marks) A general two-input constant elasticity of substitution (CES) production function
1
is given by Q(L,K) = A[6L™" + (1 — §)K =P] » where L is labour input, X is capital input,
A > 0is the technology parameter, § € (0,1) is the input weight parameter and p > —1 is
the substitutability parameter such that p # 0.
a.
.
Determine the degree of homogeneity of Q(L, K).
Verify that the isoquants of Q(L, K), denoted as K(L, Q,) where Q, > 0 represents
any fixed level of output, are negatively sloped and strictly convex.
Verify thato = w is the elasticity of substitution for Q(L, K).