ECON 200
Midsemester Test 1
Question Two (20 marks)
– A Problem
Suggested time: 20 minutes
Consider the following CobbDouglas utility function for a consumer who must choose a
bundle of goods 1 and 2:
()
1
,
2
1
2
1
=
+
=
β
α
x
x
x
x
U
(
1
)
(a)
Very often in dealing with this type of utility function, we study the logform:
()
2
1
2
1
ln
ln
,
ln
x
x
x
x
β
α
+
=
=
How do we know that this form of the utility function represents the same
preference ordering as (1)?
(
2 marks
)
Because the ln() function is an increasing monotonic transformation of U, it
preserves the ranking of U, and so represents the same preference ordering.
The commodity bundle that maximizes u will also maximize U.
(b)
The marginal utility with respect to good 1 and good 2 are given by:
2
2
1
1
x
x
u
x
β
α
=
∂
∂
=
∂
∂
What happens to marginal utility as we consume more of each commodity?
(
2 marks
)
Student should note that the marginal utility of both goods is diminishing as
we consume more of both. Answers which show that thought went into will
be given one more mark than answers that just saying – they are both
diminishing.
↓
=
∂
∂
↑⇒
↓
=
∂
∂
↑⇒
2
2
2
1
1
1
x
x
u
x
x
x
β
α
(c)
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View Full DocumentECON200 Midsemester 1  Solutions 2007
indifference curves should look like. Your drawing need not be exact. (Hint:
consider what happens to the MRS
21
as we change x
1
and x
2
in particular.)
(
5 marks
)
MRS = alpha x
2
/ beta x
1``
(2 marks)
Slope = negative
(1 mark)
Shape = convex – as for high x
2
– low x
1
, MRS is steep, and for high x
1
– low
x
2
, MRS is relatively flatter.
(1 mark)
Drawing of convex, negatively sloped indifference curves. (1 mark)
(d)
Assume prices are given by (p
1
, p
2
) and income is fixed amount ‘m’. State the
consumer’s maximization problem. What is the equilibrium condition for a
consumer with this utility function? Is it a corner or interior solution? How do you
know?
(
4 marks
)
Max U (x
1
,x
2
)
s.t. p
1
x
1
+ p
2
x
2
= m
(1 mark)
Equilibrium condition – MRS
21
= p
1
/p
2
(2 marks)
Interior solution –we know this because indifference curves are convex and
budget line is regular. Another way to show this is to say that the MRS is
undefined when the consumer has zero of either good (because MU is
undefined), so that this consumer will never be at that point. (1 mark)
(e)
The ordinary demand functions for good 1 and 2 that satisfy this equilibrium
condition are given by:
2
2
1
1
p
m
x
p
m
x
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
=
β
α
What influence do ‘alpha’ and ‘beta’ have on our demand for these commodities?
Derive the own price elasticity of demand for good 1 and interpret it.(
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 Three '05
 DONALD
 Utility, indifference curves, Alpha

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