ECON3102 Tutorial 04 - Week 5
Question 1
Suppose households solve the following two-period consumption-saving problem with taxes:
maxc1,c2 u(c1) + βu(c2)
Subject to:
a1 = a0 + y1 − τ1 − c1
c2 = y2 − τ2 + (1 + r)a1
with u(c) = 1
− σ/c1 − σ
. Notice that a0 is the initial wealth and τ represents the lump sum taxes.
(a) Calculate the optimal values of the c1, c2, a.
(b) How does
y1/c1 depend on y2? What would happen if households become optimistic about the future?
(c) How does
y1/c1 depend on a0? Interpret.
(d) How does
y1/c1 depend on β? Interpret.
(e)Suppose y2 = τ1 = τ2 = 0. Compute ∂r/∂c1. How does the answer depend on σ? Interpret.
Question 2
Suppose households solve the following two-period consumption-saving problem with taxes:
maxc1,c2 u(c1) + βu(c2)
Subject to:
a1 = a0 + y1 − τ1 − c1
c2 = y2 − τ2 + (1 + r)a1
with u(c) = 1 − σ/c1 − σ
. Notice that a0 is the initial wealth and τ represents the lump sum taxes. Assume that τ1 = τ2 = a0 = 0, β = 1, r = 0.
(a) Solve for c1 as function of y1 and y2.
(b) Suppose the incomes of each households are given as follows:
y1 y2
Household A 2 4
Household B 6 4
Compute c1 and c2 for each of the households.
(c) Suppose that we are are trying yo decide what is a reasonable model for consumption behavior. and only has data for period 1. Is the data supportive of the Keynesian view of consumption function? Explain.
(d) Now we can access the data for period 2. Is the data supportive of the Keynesian view of consumption function? Explain.
Question 3
Suppose households solve the following two-period consumption-saving problem with taxes:
maxc1,c2 u(c1) + βu(c2)
Subject to:
a1 = a0 + y1 − τ1 − c1
c2 = y2 − τ2 + (1 + r)a1
a1 ≥ −b
with u(c) = 1 − σ/c1 − σ
. Notice that a0 is the initial wealth and τ represents the lump sum taxes.
(a) What does a1 ≥ −b mean? What does b represent?
Assume that a0 = 0.
(b) Plot the budget constraint along with the constraint a1 ≥ −b.
(c) Solve for c1, c2, a.
(d) Show that constraint a1 ≥ −b is more likely to be binding if y2 - τ2 is high, y1 - τ1 is low, and b is low. Interpret each of these conditions.
(e) Suppose that the government decides lowering τ1 by ∆ and increasing τ2 by (1 + r)∆.
How does c1 respond to this change if we start from a situation where constraint a1 ≥ −b is not binding.
(f) How does c1 respond to this change if we start from a situation where constraint a1 ≥ −b is binding.