1. (12 marks) Consider the properties of quasiconcavity and quasiconvexity.a. Define quasiconcavity and quasiconvexity as they relate to f(xy, ..., xp).b. Prove that convexity implies quasiconvexity...

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1. (12 marks) Consider the properties of quasiconcavity and quasiconvexity. a. Define quasiconcavity and quasiconvexity as they relate to f(xy, ..., xp). b. 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). b. 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. FLIOVE Midi cVery LALILIUIL Pid Ul Of J DCHUTOLUICLIL ISLUDIL HAA) sus An) Io d 1 AY. 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 budget 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-order conditions indeed provide for a unique absolute maximum? Totally differentiate, and apply the Implicit Function Theorem to, the first-order 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. b. 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).
Answered 1 days AfterDec 03, 2022

Answer To: 1. (12 marks) Consider the properties of quasiconcavity and quasiconvexity.a. Define quasiconcavity...

Banasree answered on Dec 05 2022
32 Votes
1.a,
A function f is}for any pair of distinct points u and v in the (convex-set) domain of f, and for 0<θ<1,
f(v)>=f(u) → f[θu+(1-θ)v){ }
to adapt this definition to strict quasiconcavity and quasiconvexity, the two weak inequalities on the right should be change into strict inequalities
Now, if c
onsider the given condition then, f(x), where x is vector variables x =(x1….xn)
Which follows, three theorems
1. If f(x) is quasiconcave(strictly quasiconcave), then -f(x) is quasiconvex (strictly quasiconvex).
2. (concavity versus quasiconcavity) Any concave (convex) function is quasiconcave (quasiconvex), but the converse is not true. Similarly, any strictly concave
(strictly convex) function is strictly quasiconcave (strictly quasiconvex),but the converse is not true.
3. (linear function) If f(x) is a linear function, then it is quasi concave as well as quasiconvex.
1.b) Prove
Theorem I follow from the fact that multiplying an inequality by -1 reverses the sense of inequality.
Let f(x) be quasiconcave, with f(v) >= f(u).
Then, f[θu + (1 -θ)v] >= f(u).
So, function -f(x) is concerned,
(after multiplying the two inequalities through by -1) -f(u) >= -f(v)
and
-f[θu +(1 -θ)v]<= -f(u).
Interpreting -f(u) the height of point N, and -f(v) as the height of M ,
the function -f(x) satisfies the condition for quasi convexity . This prove Theorem I
Theorem 2
f(x) be concave,
then,
f[θu+(1-θ)v]>=θf(u)+(1-θ)f(v)
Let assume,
f(v)>=f(u), then any weighted average of f(u) and f(v) cannot possibly be less than f(u),
θf(u)+(1-θ)f(v)>=f(u)
equating these two, we know that, by transitivity,
f[θu+(1-θ)v]>=f(u)
for f(v)>=f(u)
this satisfies the definition of quasiconcavity.
Once theorem 2 proved, 3 follows immediately.
We know that a linear function is both concave and convex. Not strictly. Therefore, theorem 2 a linear function must also be both quasiconcave and quasiconvex. Not strictly.
1.c)
A differentiable function f(x1…..xn) is { }
u = (u1…….un)
v =(v1….vn)
f(v)>=f(u) [ ] >=0
fj = df/dxi
for strict quasiconcavity and quasiconvexity, the weak inequality on the right should be change to the inequality>0
for function f(x,y)
applying bordered Hessian
|H| = |B1| + |B2|
Where,
|B1| = | | and |B2| = | |
To characterize the configuration of that function
f(x,y) = log x + log y
Here two conditions, one is necessary and the other is suffiecient.
Both relate to quasiconcavity on domain domain consisting only of the nonnegative orthant.
Where,
|B1|<= 0
|B2|>=0
Also
|B1|<0 and |B2|>0
These automatically satisfy the theorem 1
1.d)
Assue,
F(v)>=f(u)
Or
v>=u
(v,u >=0
So partial derivatives
f1 = x1 and f2(x2)
f1(u)(v1-u1)+f2(u)(v2-u2) = u2(v1-u1)+u1(v2-u2)>=0
rearrangement,
u2(v1-v2)>=u1(u2-v2)
where,
1. u1=u2=0
2. u1=0, u2>0
then,
u2v1>=0, which satisfied u2 and v1 both non negative
3. u1>0 and u2=0
then 0>=-u1v2
so,
v2(vl -u) >= u1(u2 -v2)
three possibilities,
1. u2=v2 the v1>=u1
2. u2>v2 the v1>u1
then,
u2(v1 -u1) >- u2/v2 *u1| (u2...
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