9 November 2018 Homework 3 (draft 2) Due 15 November 2018
1
1. Level sets, superlevel sets and consumer choice
Alex has utility function
1 1 2 2( ) ( ) ( )
A
A AU x av x a v x where ( )Av is a differentiable strictly concave
function.
Bev has utility function
1 1 2 2 1 1 2 2( ) ( ) ( ) ( ( )) ( ( ))
B
B B A AU x av x a v x ah v x a h v x where ( )h is a
differentiable strictly concave function.
(a) Show that if
1 2x x then
1
2
( ) ( )A B
a
MRS x MRS x
a
.
(b) Show that if
1 2x x then
1
2
( ) ( )A B
a
MRS x MRS x
a
and that if
1 2x x then
1
2
( ) ( )A B
a
MRS x MRS x
a
.
(c) Show that if the two consumers have the same endowment and the prices satisfy 1 1
2 2
p a
p a
, then the
choices of the two consumers
Ax and Bx satisfy 2 2
1 1
1
A B
A B
x x
x x
.
(d) Is it also true that when 1 1
2 2
p a
p a
, then the choices of the two consumers Ax and Bx
satisfy 2 2
1 1
1
A B
A B
x x
x x
.
(e) Explain why it must be the case that, as depicted, the vertically lined superlevel set for Bev,
{ | ( ) ( , )}B B BS x U x U is a subset of the vertically and horizontally lined superlevel set
for Alex, { | ( ) ( , )}A A AS x U x U .
line
9 November 2018 Homework 3 (draft 2) Due 15 November 2018
2
2. Pareto efficient Allocations
Alex has utility function
1 1 2 2( ) ( ) ( )
A
A AU x v x v x where ( )Av is a differentiable strictly increasing,
strictly concave function. Bev has utility function
1 1 2 2( ) ( ) ( )
B
B BU x v x v x where ( )Bv is a
differentiable strictly increasing, strictly concave function. Let 0A and 0B be the individual
endowments and let
1 2( , ) 0 be the aggregate endowment.
For parts (a) and (b) 1 2 .
(a) Explain why the PE allocations are on the diagonal of the square Edgeworth Box.
(b) Explain why a price vector that supports a PE allocation must have a ratio 1 1
2 2
p a
p a
.
Henceforth 1 2
(c) Explain why any PE allocation in which 0Ax and 0Bx must be between the two dotted 45
lines and not in the shaded regions depicted below.
(d) Suppose that the utility function for Alex is
1/2 1/2
1 1 2 2( )
AU x a x a x and for Bev is
1/2 1/2
1 1 2 2( ) ( ) ( )
BU x a h x a h x where ( )h is a differentiable strictly concave function.
Show that that along the diagonal the marginal rates of substitution are different. Must the PE
allocations must all lie below the diagonal hence closer to the dotted 45 line of Bev than that for Alex?
Edgeworth Box with dotted lines
9 November 2018 Homework 3 (draft 2) Due 15 November 2018
3
3. Walrasian Equilibrium Allocations
Alex and Bev have the utility functions of question 2(a)-(c). The aggregate endowment is 1 2( , ) .
(a) If 1 2
1 2
p p
explain clearly why the sum of Alex and Bev’s demand for commodity 2 exceeds the sum
of demands for commodity 1.
(b) Hence show that if 1 2 , then the WE price ratio
1 1
2 2
p
p
. Equivalently, 1 2
1 2
p p
.
(c) Suppose that there are three commodities and H consumers where
3
1
( ) ( ), 1,…,h hj h j
j
U x a v x h H
and ( )hv is a strictly increasing, strictly concave differentiable
function. Is it necessarily the case that 1 2 3 implies that
31 2
1 2 3
pp p
a a a
? Explain carefully.
4. A two asset portfolio
An investor has a wealth of 120. There are two states. The dividends of asset A are uncertain. In state 1
the total dividend is 5000 and in state 2 it is 2000. The two states are equally likely. A 1% shareholding
in asset A costs 120 so if the investor only purchases share in asset A his uncertain outcome is the lottery
1 1
2 2
(50,20; , ). Asset B has a dividend of 2000 in state 1 and 8000 in state 2. A 1% shareholding costs
120 so if the investor purchases only shares in asset B his uncertain outcome is the lottery 1 1
2 2
(20,80; , ) .
The two outcomes are depicted below.
(a) If instead the investor spends a fraction z of his wealth on asset A and 1 z on asset B show that his
uncertain outcome is
1 1 1 11 2 2 2 2 2( , ; , ) (20 30 ,80 60 ; , )x x z z . (1.1)
line
9 November 2018 Homework 3 (draft 2) Due 15 November 2018
4
(b) Choose 1p and 2p so that 1 1 2 2 120p x p x .
Thus the outcomes from diversifying are the points on the line joining the non-diversified outcomes
(50,20)AN and (20,80)BN .
This line is thus an implicit budget constraint.
To solve for the optimal portfolio ( ,1 )z z we can first solve for the optimal outcome. This is the best
point satisfying the implicit budget constraint.
(c) Suppose ( ) lnv x x so that expected utility 1 11 22 2( ) [ ( )] ln lnU x v x x x .
Solve for the optimal outcome satisfying the implicit budget constraint.
(d) Use (1.1) to solve for z .
(e) Suppose ( ) ln(8 )v x x . Solve for the new optimal outcome x and hence for the share of wealth
spent on asset A.
(f) Suppose ( ) ln(40 )v x x . Solve for the new optimal outcome x and hence for the share of wealth
spent on asset A.
(g) Is it the case that for any differentiable strictly concave function ( )v x , the optimal outcome will be
above the 45 line?
1 point bonus
(h) Suppose ( ) ln(50 )v x x . What is the optimal portfolio?