Q UEEN ’ S U NIVERSITY , E CONOMICS D EPARTMENT
ECON 212 - Microeconomic Theory I - Winter 2016 - Section 1
a. Draw the utility function for utility level U = 75 and U = 150 . Formulate the optimal choice problem for the consumer.
60
50
40
30
20
10
0 5 10 15 20 25 30 35 40 45 50 55 60
1

The red utility line is for U = 75 and the blue utility line is for U = 150.
The optimal choice problem for the consumer is to maximize utility with respect to x and y subject to the budget constraint.
max x , y
U ( x , y ) s .
t .
P x x + P y y = I ncome max x , y
3 x + 5 y s .
t .
5 x + 10 y = $320 b. Determine the optimum consumption basket.
Here, x and y are perfect substitutes.
(1)
(2)
Comparing the slope of the utility function, which is the negative of the M RS x , y
, to the slope of the budget line −
P x
P y
, both in absolute value, we get:
M RS x , y
=
3
5
>
1
2
=
P x
P y
(3)
Therefore, since they are not equal, we have a corner solution. The budget line (black line in graph) will be flatter than the utility line (green line in graph). The consumer will thus consume only 64 units of good x and 0 unit of good y.
48
40
32
24
16
8
0 8 16 24 32 40 48 56 64
2

a. Calculate the optimal consumption bundle ( c
1
, c
2
) . Now assume that r = 1 , I
1
= 50 and I
2
= 100 . Determined the level of ( c
1
, c
2
) .
L
= 3 c
1
1
3 c
2
3
2
+
λ
·
I
1
+
I
2
1 + r
− c
1
− c
2
1 + r
¸
(4)
First order conditions
With respect to c
1
: c
1
− 2
3
With respect to c
2
: 2 c
With respect to λ : I
1
1
3
1
+ c
2
2
3 c
2
− 1
3
− λ = 0
λ
I
2
−
1 + r
1 + r
− c
1
−
= 0
1 c
2
+ r
Dividing equation (5) by equation (6):
= 0 c
1
− 2
3
2 c
1
1
3 c
2
3
2 c
2
− 1
3 c
2
=
λ
λ
1 + r
2 c
1
= 1 + r c
2
= 2 c
1
(1 + r )
Substituting equation (10) into equation (7) and rearranging:
(8)
(9)
(10)
(5)
(6)
(7)
I
2
I
1
+
I
1
1 + r
I
2
I
1
+
1 + r
I
2
+
1 + r
=
=
= c c
3
1
1 c
+
+
1
2
2 c c
1
1
1 c
1
=
I
1
+
I
2
1 + r
3 c
1
=
50 +
3
100
1 + 1
(1
+
=
+ r r
3
)
100
Substituting equation (14) into equation (10), we get: c
2
= 2
I
1
+
I
2
1 + r
3 c
2
= 2
50 +
3
100
1 + 1
(1 + r )
(1 + 1) =
400
3
(14)
(15)
(16)
(17)
(11)
(12)
(13)
3

The level of ( c
1
, c
2
) is (
100
3
,
400
3
).
b. Calculate the substitution effect of an increase in r to 4.
The substitution effect is the change in consumption from the initial basket (at old prices, r = 1) to the decomposition basket (at new prices, r = 4), holding utility constant at the level achieved by the initial basket.
Let’s first calculate the final basket.
Using equation (14) and equation (16), we can calculate the final basket. We know from part a) that the initial basket is (
100
3
,
400
3
).
Optimal consumption of c
1
: c
1
= c
1
=
I
1
+
I
2
1 + r
3
50 +
100
1 + 4
3
=
70
3
Optimal consumption of c
2
: c
2
= 2
I
1
+
I
2
1 + r
3 c
2
= 2
50 +
3
100
1 + 4
(1 + r )
(1 + 4) =
700
3
The final basket is (
70
3
,
700
3
).
Now, let’s calculate the level of utility achieved by the initial basket (
100
3
,
400
3
).
U ( x , y ) = 3 c
1
3
1 c
2
3
2
U ( x , y ) = 3
µ 100 ¶
1
3
3
µ 400 ¶
2
3
3
= 251.98421
Using the relationship we found in part a) that c
2
= 2 c
1
(1 + r ), we then know that: c
2
= 2(1 + 4) c
1
= 10 c
1
Using this into the utility function and setting the utility level at 251.98421, we get:
U ( x , y ) = 3 c
1
3
1
(10 c
1
)
2
3 = 251.98421
(21)
(22)
(23)
(24)
(25)
(26)
(18)
(19)
(20)
4

Solving for c
1
: c
1
= 18.096
(27)
Substituting this value of c
1 to get the value of c
2
: c
2
= 2(1 + 4) c
1
= 10 c
1
= 10 × 18.096
= 180.96
(28)
The substitution effect is the difference between the initial basket and this intermediate basket
(18.096, 180.96). Therefore, consumption of tion of c
2 will increase from
400
3 c
1 will be reduced from
100
3 to 18.096 and consumpto 180.96. Thus, the substitution effet is ( − 15.237, + 47.63).
350
300
250
200
150
100
50
0 25 50 75 100 125 150
• Light blue is the utility function from part a) where r = 1.
• Black line is the budget constraint from part a) where r = 1.
• Red dot is the initial consumption basket.
• Red line is the budget constraint from part b) where r = 4.
• Dark blue is the utility function from part b) where r = 4.
• Green dot is the final consumption basket.
• Blue dot is the decomposition basket.
=⇒ The substitution effect is from the red dot to the blue dot.
5