ECOS3021 Business Cycles and Asset Markets University of Sydney
2021 Semester 1
Tutorial #3
1. Consider a simple household consumption and savings problem:
max logC1 +βlogC2 C1 ,C2 ,S
s.t C1+S=Y1 C2 = Y2 + SR
a) Derive the first order condition(s) for the household’s problem, and solve for an expression that describes the household’s savings behaviour.
ANSWER:
• Substitute the period 1 budget constraint into C1 and the period 2 budget constraint into C2: max log(Y1 − S) + β log(Y2 − SR)
S
• Taking the first order condition with respect to S yields:
−1 +βR 1 =0
• Rearranging, we can solve for S:
Y1 − S Y2 + RS
S= β Y1− 1 Y2 (1) 1 + β R(1 + β)
b) Rearrange the expression for savings to derive the savings supply curve. This curve expresses the interest rate R as a function of savings S and incomes Y1,Y2. Draw the savings supply curve on a graph, with S on the x-axis and R on the y-axis. Now assume that the savings demand curve is constant/inelastic with respect to the interest rate. Add this demand curve to the figure. Label the equilibrium interest rate in the market for savings.
ANSWER:
• The savings supply curve is:
R= Y2 (2) βY1 −(1+β)S
• The figure describing the supply and demand for savings is below:
c) On two separate figures, show the effect of (I) a decrease in period 1 income Y1, (II) a decrease in period 2 income Y2. Provide economic intuition for equilibrium outcomes following the the changes in incomes.
ANSWER:
1
R
SD
R∗
Y2 βY1−(1+β)S
S
R
Y2
βY1′ −(1+β)S
Y2 βY1−(1+β)S
S
• A decrease in income in period 1 means that the household has less resources available to consume in period 1 than was previously the case. The household wants to transfer resources from period 2 to period 1 in order to help smooth consumption in period 1. In order to do this, the household saves less in period 1 at any given level of the interest rate. Hence, the savings supply curve shifts left. In equilibrium, the decrease in the supply of savings relative to the constant demand for savings leads to an increase in the equilibrium interest rate.
R
′
SD
R∗
• A decrease in income in period 2 means that the household has less resources available to consume in period 2 than was previously the case. The household wants to transfer resources from period 1 to period 2 in order to help smooth consumption in period 2. In order to do this, the household saves more in period 1 at any given level of the interest rate. Hence, the savings supply curve shifts right. In equilibrium, the increase in the supply of savings relative to the constant demand for savings leads to a decrease in the equilibrium interest rate.
2
R
SD
Y2 βY1−(1+β)S
′
Y2 βY1−(1+β)S
S
R∗
′
R
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2. Consider the following three-period household consumption and savings problem. In this problem, a household lives for three periods. In the final period of life the household retires, earns no income, and must consume any available retirement savings, S3. Retirement savings must be put aside in the first period, but nothing can be saved from period 2. In period 1, the household can also put aside savings (or borrow), for period 2, S2. In all, the household makes two savings decisions in period 1 – S2,S3 – and one consumption decision in each of the three periods – C1, C2, C3. The household problem can be described as follows:
max logC1 +βlogC2 +β2 logC3 C1 ,C2 ,C3 ,S2 ,S3
s.t C1+S2+S3=Y1 C2 = Y2 + S2R2
C3 = S3R3
where R2 is the interest rate earned on savings in period 2, and R3 is the interest rate earning on
retirement savings available in period 3.
a) Substitute each of the budget constraints into the utility function in order to rewrite the household
problem as an unconstrained optimization problem in the two choice variables S2 and S3. ANSWER:
• Substitute the period 1 budget constraint into C1; substitute the period 2 budget constraint into C2; and substitute the period 3 budget constraint into C3. Then the household problem can be written as:
max log(Y1 − S2 − S3) + β log(Y2 − S2R2) + β2 log(S3R3) S2 ,S3
b) Take the first order conditions for the two savings decisions (S1, S2). Note: Because there are two choice variables, there are two separate first order conditions in this case.
ANSWER:
• The two first order conditions are:
1 = βR2 (3)
Y1 −S2 −S3 1
Y1 −S2 −S3
Y2 +S2R2
= β2R3 (4)
S3R3
c) Use the first order conditions to solve for the two savings decisions as functions of incomes and interest rates. Note: It may be easiest to solve for S2 explicitly, but leave the expression for S3 as a function of S2.
ANSWER:
• First, rearrange Equation (4) for S3
β2
S3 = 1+β2(Y1 −S2) (5)
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• Next, substitute this expression for S3 into Equation (3), and solve for S2: 1 = βR2
Y1 −S2 −S3 Y2 +S2R2
⇒(Y2 + S2R2) = βR2(Y1 − S2 − S3)
( β2 ) ⇒(Y2 +S2R2)=βR2 Y1 −S2 − 1+β2(Y1 −S2)
⇒(Y2 +S2R2)= βR2 Y1 − βR2 S2 1 + β2 1 + β2
1 + β2
( 1 + β + β2 )
( βR)βR ⇒S2R2+ 2 = 2Y1−Y2
⇒S2
⇒S = R
1 + β2 βR2
R2 (
= 1 + β2 Y1 − Y2
221+β2 1+β212 • Which, tidying up, yields the expression for S2:
1 + β2
1+β+β2)−1( βR ) 2 Y −Y
β 1 + β2
S2 = 1 + β + β2 Y1 − R2(1 + β + β2)Y2 (6)
• Finaly, substitute (5) back into (6) to find S3: β2
S3 = 1+β2(Y1 −S2)
β2( β 1+β2 )
= 1 + β2 Y1 − 1 + β + β2 Y1 + R2(1 + β + β2)Y2
β2 ( 1+β2 1+β2
= 1 + β2 1 + β + β2 Y1 + R2(1 + β + β2)Y2
• Which, tidying up, yields:
(7) d) How does the household use the available assets to finance its retirement? How does your answer change
ANSWER:
• The household always has positive retirement savings, since (7) is always positive. Notice that S3 is a function of both period 1 income and period 2 income. But the household can only save for retirement in period 1, when it does not yet have access to period 2 income. This means that the household may borrow against period 2 income, in order to finance retirement savings in period 3. This can be seen from equation (6). We can see that S2 may be negative, if the second term is much larger than the first term. In this case, the household borrows using S2 in order to then save in the retirement savings account S3.
β2 ( Y) S3= 2 Y1+ 2
)
1+β+β R2 when second period income, Y2, is very large or very small?
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• If Y2 is very large, then the second term in equation (6) dominates the first term, and S2 is negative. This is the case where the household borrows against period 2 income in order to finance retirement savings.
• If Y2 is very small, then the second term in equation (6) is smaller than the first term, and S2 is positive. In this case, the household is very wealthy in period 1 and does not need to borrow against period 2 income in order to save for retirement. Instead, in period 1 the household saves resources to be consumed in both periods 2 and 3. Thus, when the household is very wealthy in period 1, they will smooth income across both of the subsequent periods by saving through assets S2 and S3 in period 1.
e) How are retirement savings affected by the second period interest rate R2? How are retirement savings affected by the third period interest rate, R3?
ANSWER:
• The effect of R2 can be expressed using the derivative of the savings functions with respect to R2: ∂S2 1 + β2
∂R =R2(1+β+β2)Y2 (8) 22
∂S3 =− β2 Y2 (9) ∂R2 1+β+β2 R2
This shows that an increase in R2 results in an increase in S2 and a decrease in S3. The increase in R2 increases the return to savings in period 2, so the household saves more for consumption for period 2 and saves less for retirement in period 3.
• The effect of R3 can be expressed using the derivative of the savings functions with respect to R3: ∂S2 =0 (10)
∂R3
∂S3 =0 (11)
∂R3
In this case, household savings decisions are unaffected by changes in the interest rate during retirement.
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3. Consider the following two-period household consumption and savings problem. In this problem we will use a power utility function for the household’s objective function.1
C 1−σ C 1−σ max 1 +β2
C1,C2,S 1−σ 1−σ s.t C1+S=Y1
C2 = Y2 + SR
where R is the gross interest rate.
a) Write down the inter-temporal budget constraint. With reference to the inter-temporal budget constraint,
describe how the interest rate is related to the price of consumption in period 2.
ANSWER:
• First rearrange the period 2 budget constraint for savings:
S = C2 − Y2 RR
• Substitute this equation into the period 1 budget constraint and rerrange:
C1+C2 =Y1+Y2 RR
• Since this is a budget constraint, the price of consumption in period 1 is equal to 1 (i.e. the numeraire). Then the price of consumption in period 2 is equal to 1 . An increase in the interest
R
rate is associated with a decrease in the price of consumption in period 2.
b) Solve for the first order conditions of the problem. Use the first order conditions and the budget constraint(s) to find consumption demands as functions of the interest rate and income.
ANSWER:
• First, substitute the inter-temporal budget constraint into the objective function:
(Y1 + Y2 − C2 )1−σ C1−σ max R R +β2
C2 1−σ 1−σ • Next, take the first order condition with respect to C2:
1( Y C)−σ −Y+2−2 +βC−σ=0
R1RR 2
• Next, rearrange for C2. First, multiply each side of the equation by R and then raise each side of
the equation to the power of −1 : σ
1This utility function is also often to as the Constant Relative Risk Aversion utility function.
Y +Y2 −C2 =(βR)−1C 1σ2
RR
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• Next, multiply each side of the equation by R again to help simplify: RY +Y −C =β−1R1−1C
122σσ2
• Now gather the terms in C2, and solve for the period 2 consumption demand:
C = 1 (RY +Y ) 2 −1 1−1 1 2
1+βσRσ • For final tidying, multiply and divide by β:
⇒C = β (RY +Y) (12) 2 1−112
β+(βR) σ
• Now solve for consumption demand in period 1 by substituting equation (12) into the inter-
temporal budget constraint:
C + 1 β (RY + Y ) = Y + Y2 1 1−1121
Rβ+(βR) σ R (β)(Y)
⇒C1= 1− 1−1 Y1+ 2
(
β + (βR) ) σ R (βR)1−1 ( Y)
⇒C1 =
c) Show that when σ = 1, the consumption functions you derived above are identical to those under the
log-utility specification discussed in Lecture 3.
ANSWER:
• The consumption demand functions in the current problem are:
σ2
1− 1 Y1 + β+(βR) σ
R
( 1−1 )( )
(βR) σ C1 = 1− 1
Y2 R
((βR)0)( Y) C1= 0Y1+2
β + (βR) R
C2 = β (RY1 +Y2) β + (βR)0
C1=
C2=
• Where these demand functions are identical to those derived in Lecture 3.
Y1 + C = β (RY +Y )
β+(βR) σ
2 1−112
β+(βR) σ
• Set σ = 1, which yields:
• And simplifying further:
(1)(Y) Y1+ 2
β+1 R
βR( Y) Y1+ 2
β+1 R
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d) Continue to assume that σ = 1. What is the effect of an increase in the interest rate R on C1 and C2? Use math to justify your answer.
ANSWER:
• In order to show the effect of the interest rate on consumption demands, take the derivative of each function with respect to the interest rate R:
∂C (1)Y 1=− 2 ∂R β+1 R2
∂C2 = β Y1 ∂R β+1
• As the interest rate increases, consumption in period 1 declines, while consumption in period 2 increases.
• Note that since 1 is the price of consumption in period 2, an increase in the interest rate effectively R
reduces the price of C2 and increases the price of C1.
e) Suppose that income is constant across periods, so Y1 = Y2 = Y , and that R > 1. Now assume that
β = 0. Under this parameterization, is C1 or C2 more sensitive to changes in the interest rate? ANSWER:
• Setting Y1 = Y2 = Y and β = 0, and substituting into the derivatives from above:
∂C1 =−Y ∂R R2
∂C2 =0 ∂R
• Under this parameterization consumption in period 2 is totally insensitive to changes in the interest rate, while consumption in period 1 is still negatively associated with increases in the interest rate.
f) Suppose that income is constant across periods, so Y1 = Y2 = Y , and that R > 1. Now assume that β = 1. Under this parameterization, is C1 or C2 more sensitive to changes in the interest rate?
ANSWER:
• Setting Y1 = Y2 = Y and β = 1, and substituting into the derivatives from above:
∂C1 =−1Y ∂R 2 R2
∂C2 = 1Y ∂R 2
• Since R > 1, we have that 1 Y is greater than 1 Y (i.e. the absolute value of the derivative for 2 2 R2
C1). Therefore, C2 is more sensitive to changes in the interest rate than is C1.
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