Class 15 (answers) Consumer’s intertemporal choice

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Microeconomics II 
Undergraduate degree in Economics 
 
Class nr. 15 
 
Subject: 2. Intertemporal choice 
 2.1. Consumer’s intertemporal choice (practice class) 
 
Exercise 10.1. from Bergstrom and Varian’s book (2006) “Workouts in Intermediate 
Microeconomics”, p. 130 
 
Peregrine Pickle consumes ( )21 ,cc and earns ( )21 ,mm in periods 1 and 2 respectively. 
Suppose the interest rate is r. 
 
a) Write down Peregrine’s intertemporal budget constraint in present value terms. 
 
b) If Peregrine does not consume anything in period 1, what is the maximum value 
he can consume in period 2, that is, what is the future value of his initial 
endowment? 
 
c) If Peregrine does not consume anything in period 2, what is the maximum value 
he can consume in period 1, that is, what is the present value of his initial 
endowment? What is the slope of his budget constraint? 
 
 
Answers: 
a) ( ) ( )r
mm
r
cc ++=++ 11
2
1
2
1 
b) ( ) 21 1 mrm ++ 
c) ( )r
mm ++ 1
2
1 ; ( )r+− 1 
 
Exercise 10.3. from Bergstrom and Varian’s book (2006) “Workouts in Intermediate 
Microeconomics”, pp. 131-132 
 
Nickleby has an income of $2,000 this year, and he expects an income of $1,100 next 
year. He can borrow and lend money at an interest rate of 10%. Consumption goods 
cost $1 per unit this year and there is no inflation. 
 
a) What is the present and future value of Nickleby’s endowment? Sketch, in a 
graph, his endowment, budget constraint and budget set. 
 
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b) Suppose that Nickleby has the utility function ( ) 2121 , CCCCU = . Write an 
expression for Nickleby’s marginal rate of substitution between consumption 
this year and consumption next year. 
 
c) What is the slope of Nickleby’s budget constraint? Write an equation for his 
optimal condition (this equation states that the slope of Nickleby’s indifference 
curve is equal to the slope of his budget constraint when the interest rate is 
10%). Also write down Nickleby’s budget constraint. 
 
d) Solve the two previous equations in order to determine Nickleby’s optimal 
intertemporal consumption bundle (combination of present and future 
consumptions). 
 
e) Will he borrow or save in the first period? How much? 
 
f) Sketch in a graph the combinations of present and future consumption that 
Nickleby will not chose if the interest rate increases to 20%. 
 
g) Solve for Nickleby’s optimal choice when the interest rate is 20%. 
 
h) In this case, will he borrow or save? How much? 
 
 
Answers: 
a) Present Value: ( )21
1,1002,000 3,000
1 1 0.1
mm
r
+ = + =+ + 
Future Value: ( ) ( )1 21 2,000 1 0.1 1,100 3,300m r m+ + = × + + = 
 
 
 
 
 
 
 
 
 
 
 
 
 
M
1c 
2c 
300 600 900 1,200 1,500 1,800 2,100 2,400 2,700 3,000 
500 
1,000 
1,500 
2,000 
2,500 
3,000 
3,500 
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Undergraduate degree in Economics 
 
 
b) 
1
1 2
2
1 2
,
1
2
C
C C
C
U
MU C CMRS UMU C
C
∂
∂= − = − = −∂
∂
 
c) 
( ) ( ) 1,11,011 −=+−=+− r 
( )
( ) ( )
1 2
1 2
,
2 1
2
1
p 1
p
1,1002,000
1 0.1 1 0.1
C C
CMRS r
C
CC
⎧ = − ⇔ − = − +⎪⎪⎨⎪ + = +⎪ + +⎩
 
 
d) 
( )
( ) ( )
2
2 1
1 2 1 2
2
2 1 1 11
1
1 0.1 1.1 1.1 1,650
1,100 3,000 1,5003,0002,000 1.11 0.1 1 0.1
C
C CC C C C
CC C C CCC
⎧− = − + = ×⎧⎪ = × =⎧ ⎧⎪ ⎪⇔ ⇔ ⇔⎨ ⎨ ⎨ ⎨+ = =+ = ⎩ ⎩⎪ ⎪+ = + ⎩⎪ + +⎩
 
e) 
1
1 1
1
1,500
500
2,000
C
m C
m
=⎧ ⇔ − =⎨ =⎩
 
Nickleby saves $500 in the first period. 
 
f) 
Present Value: ( )21
1,1002,000 2,916.(6)
1 1 0.2
mm
r
+ = + =+ + 
Future Value: ( ) ( )1 21 2,000 1 0.2 1,100 3,500m r m+ + = × + + = 
 
 
 
 
 
 
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g) 
( )
( ) ( )
2
2 1
1 2 1 2
2
2 1 1 11
1
1 0.2 1,2 1.2 1,750
1,100 2,916.(6) 1,458.(3)2,916.(6)2,000 1.21 0.2 1 0.2
C
C CC C C C
CC C C CCC
⎧− = − + = ×⎧⎪ = × =⎧ ⎧⎪ ⎪⇔ ⇔ ⇔⎨ ⎨ ⎨ ⎨+ = =+ = ⎩ ⎩⎪ ⎪+ = + ⎩⎪ + +⎩
 
h) 
1
1 1
1
1, 458.(3)
541.(6)
2,000
C
m c
m
=⎧ ⇔ − =⎨ =⎩
 
Nickleby saves $541.(6) in the first period. The increase in the interest rate led him to 
save more. 
 
Exercise 10.10. from Bergstrom and Varian’s book (2006) “Workouts in Intermediate 
Microeconomics”, pp. 138-139 
 
In an isolated mountain village, the only crop is corn. Good harvests alternate with bad 
harvests. This year the harvest will be 1,000 bushels1. Next year it will be 150 bushels. 
There is no trade with the outside world. Corn can be stored from one year to the next, 
but rats will eat 25% of what is stored in a year. The villagers have Cobb-Douglas 
utility functions, ( ) 2121 , ccccU = where 1c is consumption this year, and 2c is 
consumption next year. 
 
 
1 A measure of volume used in the United States that is roughly equivalent to 35 liters. 
M
≡ set of consumption goods that will not be 
chosen 
X
300 600 900 1,200 1,500 1,800 2,100 2,400 2,700 3,000 
500 
1,000 
1,500 
2,000 
2,500 
3,000 
3,500 
1c 
2c 
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Undergraduate degree in Economics 
 
a) Sketch in a graph the consumption possibilities (budget set) for the village. 
 
b) How much corn will the villagers consume this year? How much will the rats 
eat? How much corn will the villagers consume next year? 
 
c) Suppose that a road is built to the village so that now the village is able to trade 
with the rest of the world. Now the villagers are able to buy and sell corn at the 
world price, which is $1 per bushel. They are also able to borrow and lend 
money at an interest rate of 10%. Sketch (in a graph) the new budget constraint. 
Determine the new optimal consumption point for both periods. 
 
d) Suppose that all is as in question c) except that now there is a transportation cost 
of $0.1 per bushel for every bushel of grain hauled into or out of the village. 
Sketch in a graph the new budget constraint for the village under these new 
circumstances. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
max
1
max
2
1,000
150 1,000 0.75 900
c
c
⎧ =⎪⎨ = + × =⎪⎩
 
 
 
 
 
 
 
( )1 2,M m m≡ 
100 200 300 400 500 600 700 800 900 1,000 
150 
300 
450 
600 
750 
900 
1,050 
1c 
2c 
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Undergraduate degree in Economics 
 
b) 
( ) ( )
( )
1 2
1
,
1 2 2
1 2
0.75 0.75
0.75 1,000 0.75 150
0.75 900
c c
U
cMRS
U
c c c
c c
∂⎧⎪ ∂= − ⎪⎧ − = −⎪ ⎪ ∂⇔ ⇔⎨ ⎨× + = × +⎪ ⎪ ∂⎩ ⎪ × + =⎪⎩
 
( )
2 2 1
2 1 2
1
11 1 1
1 2
0.750.75 0.75 450
900
0.75 0.75 900 600
0.75 900 1,5
c c c
c c cc
cc c c
c c
⎧ = ×⎧= = × =⎧ ⎧⎪ ⎪⇔ ⇔ ⇔ ⇔⎨ ⎨ ⎨ ⎨=× + × = =⎩ ⎩⎪ ⎪× + = ⎩⎩
 
The rats eat 100 bushels. (1000 - 600) * 0,25 
 
c) 
Start by computing the maximum amount of corn the villagers can obtain in period 1 
( )2 0c = : 
( ) ( )max max21 1 1
1501,000
1 1
mc m c
r r
= + ⇒ = ++ + 
( )21
m
r+ is the maximum amount of corn the villagers can borrow in period 1 and still be 
able to repay the loan in period 2 with an interest rate of r . 
Now compute the maximum amount of corn the villagers can obtain in period 2 
( )1 0c = : 
( )max2 1 21c m r m= × + + 
( )
( )
max max
1 1
max
max 2
2
1501,000 1,136.(36)1 0.1
12501,000 1 0.1 150
c c
cc
⎧ = + ⎧ =⎪ ⎪+ ⇔⎨ ⎨ =⎪⎪ ⎩= × + +⎩
 
 
The new budget constraint is: 
( ) ( ) ( ) ( )2 1 2 1 1 2 1 21 1 1 1c m r m c r c r c m r m= × + + − × + ⇔ × + + = × + +Microeconomics II 
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Graphical representation 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
( )
( ) ( )
1 2
21
,
1
1 2 2 1 2
1 2
2 1
2 1 2
1 1 11
1 0,1 1.11.1
1 0.1 1,000 1 0.1 150 1.1 1, 250
1.1 1,250
1.11.1 625
1,2501.1 1.1 1, 250 568.(18)
2.2
c c
U
ccMRS
U c
c c c c c
c c
c cc c c
c c cc
∂⎧⎪ ∂ ⎧⎧ = − + =⎪− = −⎪ ⎪∂⇔ ⇔ ⇔⎨ ⎨ ⎨× + + = × + +⎪ ⎪ ⎪⎩ ∂ × + =⎩⎪ × + =⎩
= ×⎧= × =⎧ ⎧⎪⇔ ⇔ ⇔⎨ ⎨ ⎨× + × = ==⎩ ⎩⎪⎩
 
d) 
Start by computing the maximum amount of corn the villagers can obtain in period 1 
( )2 0c = : 
( )
( )
2
max
1 1 2
1
1
t
c m m
r
−= + × + 
Notice that the transport cost is incurred twice (once in period 1 hauling/shipping the 
corn into the village and once again in period 2 carrying the corn out of the village into 
the market to pay for the debt incurred). 
( )
( )
2
max
1
150 1 0.1
1,000
1 0.1
c
× −= + + 
M
1,250 
150 
300 
450 
600 
750 
900 
1,050 
1,100 100 200 300 400 500 600 700 800 900 1,000 1c 
2c 
Slope: ( )2
1
1dc r
dc
= − + 
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Now compute the maximum amount of corn the villagers can obtain in period 2 
( )1 0c = : 
( ) ( )2max2 1 21 1c m t r m= × − × + + 
Once again, notice that the transport cost is incurred twice (once in period 1 hauling the 
corn out of the village and once again in period 2 carrying the corn back into the 
village). 
( )
( )
( ) ( )
( )
⎪⎩
⎪⎨⎧ =
=⇔
⎪⎩
⎪⎨
⎧
++×−×=
+
−×+=
1041
45.1110
1501.011.011000
1.01
1.011501000
max
2
max
1
2max
2
2
max
1
c
c
c
c
 
 
We now have a piecewise budget constraint with the two line segments connected with 
kink at the endowment. 
 
Line segment to the left of the endowment (lenders will choose a bundle on this line 
segment): 
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
2 2
2 1 2 1
2 2
1 2 1 2
1 1 1 1
1 1 1 1
c m t r m c t r
c t r c m t r m
= × − × + + − × − × + ⇔
⇔ × − × + + = × − × + +
 
We can also write it as ( ) ( ) ( )22 1 1 21 1c m c t r m= − × − × + + 
 
Line segment to the right of the endowment (borrowers will choose a bundle on this line 
segment): 
( )
( )
( )
( )
( )
( )
( )
( )
2 2 2 2
1 1 2 2 1 2 1 2
1 1 1 1
1 1 1 1
t t t t
c m m c c c m m
r r r r
− − − −= + × − × ⇔ + × = + ×+ + + + 
We can also write it as ( ) ( )( )2 1 1 22
1
1
r
c m c m
t
+= − × +− 
 
 
 
 
 
 
 
 
 
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M
M
1250 
150 
300 
450 
600 
750 
900 
1050 
100 200 300 400 500 600 700 800 900 1000 1100 1c 
2c 
Slope: ( ) ( )22
1
1 1dc t r
dc
= − − × + 
Slope: ( )( )
2
2
1
1
1
rdc
dc t
+= − − 
Kink in the budget constraint 
at the endowment