Microeconomics Revision

Question 1

What is the MRS of good j for good i

Answer 1

The ratio of their marginal utilities

Question 2

Explain what Marshallian demand functions are and how to get them

Answer 2

A solution that maximises utility subject to a budget constraint. Use Lagrangian and take FOCs

Question 3

Explain the indirect utility function

Answer 3

We summarise the relationship between u(x) and p,y

Question 4

Draw that duality graph, as well as how to get between the minimsation/maximisation problems.

Answer 4

Good luck…

You can get the Marshallian by putting the indirect into 2nd part of Hicksian. You can get the Hicksian by putting the expenditure into the 2nd part of Marshallian.

You also get:
e(p,v(p,y) = y
v(p,e(p,u) = u

Question 5

Prove homogeneity of degree zero in (p,y) for indirect utility functions

Answer 5

We must show that v(p,y)=v(tp,ty) for t>0.

We know that v(tp,ty)=max u(x) s.t. tpx≤ty
= v(p,y)=max u(x) s.t. px≤y.

We divide both sides by t

Question 6

Define Roy’s identity

Answer 6

The consumer’s Marshallian demand for good i is the ratio of partial derivatives of indirect utility wrt pi and y after a sign change

Question 7

Prove Roy’s identity

Answer 7

Use the Envelope theorem to evaluate δv(p,y)/δpi = Lagrangian wrt pi = -λx

We know that λ* = δv(p,y)/δy >0

Therefore, -(δv(p0,y0)/δpi)/(δv(p0,y0)/δy) = x*

Question 8

Prove that the expenditure function is concave in p

Answer 8

We must prove that e(p,u) is a concave function of prices. If we said that:
p1x1≤p1x and p2x2≤p2x
These relationships also hold for x*
p1x1≤p1x* and p2x2≤p2x*

Because t≥0 and (1-t)≥0, if we multiply the first by t and the second by (1-t), and add them, we get:
tp1x1+(1-t)p2x2 ≤ ptx*

Question 9

Define and derive Shephard’s lemma

Answer 9

e(p,u) is differentiable in p (p0,u0_ with p0»0 and δe(p0,u0)/δpi = xh(p0,u0)

We use the Envelope theorem but differentiate wrt pi:
δe(p,u)/δpi = δL(x,λ)/δpi = xi* =xh(p,u)

Question 10

Prove that the cost function is concave in w

Answer 10

We must prove that c(w,y) is a concave function of prices. If we said that:
w1x1≤w1x and w2x2≤w2x
These relationships also hold for x*
w1x1≤w1x* and w2x2≤w2x*

Because t≥0 and (1-t)≥0, if we multiply the first by t and the second by (1-t), and add them, we get:
tp1x1+(1-t)p2x2 ≤ ptx*
t(c1,y) +(1-t)c(w2,y) ≤ c(wt,y)

Question 11

Prove that the profit function is convex in (p,w)

Answer 11

π(p,w) = py-wx ≥ py* - wx*
π(p’,w’) = p’y’ - w’x’ ≥ p’x* - w’x*
Therefore:
tπ(p,w) + (1-t)π(p’,w’) ≥ (tp+(1-t)p’)y* - (tw + (1-t)w’)x* = π(pt,wt)

Question 12

Show the Slutsky equation

Answer 12

δxih(p,u)/δpj = δxi(p,y)/δpj + xj(p,y)δxi(p,y)/δy

Question 13

Prove Hotelling’s lemma

Answer 13

Show that δπ(p,w)/δp = y(p,w) and -δπ(p,w)/δw = xi(p,w)

Using Envelope theorem, we know that:
δπ(p,w)/δp = δL(x*,λ)/δp

L = py - wx
So δL/δp = y(p,w)

Question 14

Show the Arrow-Pratt Measure of Absolute Risk Aversion. What do the different values of Ra(w) mean?

Answer 14

Ra(w) ≡ -u’‘(w)/u’(w)

If >0, agent is risk averse
If =0, agent is risk neutral
If <0 agent is risk loving