Question

Consumption-Savings Consider a consumer with a lifetime utility function

U = u(Ct) + _u(Ct+1)

that satisfies all the standard assumptions listed in the book. The period t

and t + 1 budget constraints are

Ct + St = Yt

Ct+1 + St+1 = Yt+1 + (1 + r)St

(a) What is the optimal value of St+1? Impose this optimal value and derive

the lifetime budget constraint.

(b) Derive the Euler equation. Explain the economic intuition of the equa-

tion.

(c) Graphically depict the optimality condition. Carefully label the inter-

cepts of the budget constraint. What is the slope of the indi_erence

curve at the optimal consumption basket, (C.

t ;C.t+1)?

(d) Graphically depict the e_ects of an increase in Yt+1. Carefully label the

intercepts of the budget constraint. Is the slope of the indi_erence curve

at the optimal consumption basket, (C.t ;C.t+1), different than in part c?

(e) Now suppose Ct is taxed at rate _ so consumers pay 1 + _ for one unit

of period t consumption. Redo parts a-c under these new assumptions.

(f) Suppose the tax rate increases from _ to _ .. Graphically depict this.

Carefully label the intercepts of the budget constraint. Is the slope of

the indi_erence curve at the optimal consumption basket, (C.t ;C.t+1),

different than in part e? Intuitively describe the roles played by the

substitution and income e_ects. Using this intuition, can you de_nitively

prove the sign of @C.t@_ and @C.t+1@? It is not necessary to use math for this.

Describing it in words is fine.

Consumption-Savings Consider a consumer with a lifetime utility function

U = u(Ct) + _u(Ct+1)

that satisfies all the standard assumptions listed in the book. The period t

and t + 1 budget constraints are

Ct + St = Yt

Ct+1 + St+1 = Yt+1 + (1 + r)St

(a) What is the optimal value of St+1? Impose this optimal value and derive

the lifetime budget constraint.

(b) Derive the Euler equation. Explain the economic intuition of the equa-

tion.

(c) Graphically depict the optimality condition. Carefully label the inter-

cepts of the budget constraint. What is the slope of the indi_erence

curve at the optimal consumption basket, (C.

t ;C.t+1)?

(d) Graphically depict the e_ects of an increase in Yt+1. Carefully label the

intercepts of the budget constraint. Is the slope of the indi_erence curve

at the optimal consumption basket, (C.t ;C.t+1), different than in part c?

(e) Now suppose Ct is taxed at rate _ so consumers pay 1 + _ for one unit

of period t consumption. Redo parts a-c under these new assumptions.

(f) Suppose the tax rate increases from _ to _ .. Graphically depict this.

Carefully label the intercepts of the budget constraint. Is the slope of

the indi_erence curve at the optimal consumption basket, (C.t ;C.t+1),

different than in part e? Intuitively describe the roles played by the

substitution and income e_ects. Using this intuition, can you de_nitively

prove the sign of @C.t@_ and @C.t+1@? It is not necessary to use math for this.

Describing it in words is fine.

Answer 1

a).

Consider the given problem here the life time utility function of the individual is given by, => U = u(Ct) + b*u(Ct+1), where “Ct=consumption of period t” and “Ct+1= consumption of period t+1”. So, this is a two period intertemporal choice problem, => the savings of period t+1 should be zero.

=> St+1 = 0.

So, the budget constraint of period t+1 is given by.

=> Ct+1 + St+1 = Yt+1 + (1+r)*St, where “St+1=0”.

=> Ct+1 = Yt+1 + (1+r)*St, where “St = Yt-Ct.

=> Ct+1 = Yt+1 + (1+r)*(Yt-Ct), where “St = Yt-Ct.

=> Ct+1 = Yt+1 + (1+r)*Yt - (1+r)*Ct, => (1+r)*Ct + Ct+1 = Yt+1 + (1+r)*Yt.

=> Ct + Ct+1/(1+r) = Yt+1/(1+r) + Yt,

=> Ct + Ct+1/(1+r) = Yt + Yt+1/(1+r), be the intertemporal budget line of the individual.

b).

Here the intertemporal optimization problem is given by.

=> Maximize “U = u(Ct) + b*u(Ct+1)”, subject to “Ct + Ct+1/(1+r) = Yt + Yt+1/(1+r)”.

So, the lagrangian function is given below.

=> L = u(Ct) + b*u(Ct+1) + λ*[ Yt + Yt+1/(1+r) - Ct - Ct+1/(1+r)].

So, the FOC for maximization require “dL/dCt = dL/dCt+1 = dL/dλ = 0”

=> dL/dCt = 0.

=> du(Ct)/dCt + λ*(-1) = 0, => du(Ct)/dCt = λ……………………(1).

=> dL/dCt+1 = 0.

=> b*du(Ct+1)/dCt+1 + λ*(-1/1+r) = 0, => du(Ct+1)/dCt+1 = λ/b*(1+r)……………………(2).

=> dL/dλ = 0.

=> Yt + Yt+1/(1+r) - Ct - Ct+1/(1+r) = 0,

=> Ct + Ct+1/(1+r) = Yt + Yt+1/(1+r)……………………(3).

Here by (1) divided by (2), we get the following condition.

=> [du(Ct)/dCt]/ [du(Ct+1)/dCt+1] = λ/[ λ/b*(1+r)]

=> [du(Ct)/dCt]/ [du(Ct+1)/dCt+1] = b*(1+r), => Ut/Ut+1 = b*(1+r), be the Euler condition of the given problem.

Here the “Ut/b*Ut+1” is the absolute slope of the indifference curve and (1+r) is the absolute slope of the budget line. At the equilibrium the MRS must be equal to (1+r). The consumer is indifferent between consuming one more unit in “period t” on the one hand and saving that unit and consuming in period t+1 on the other hand.

c, d).

U2 41 → 92 B1 B2

Here A1B1 be the intertemporal budget line and B1 is the horizontal intercept shows the amount of consumption if the consumer totally spends in “period t”. Similarly, A1 is the vertical intercept shows the amount of consumption if the consumer totally spends in “period t+1”. The equilibrium is E1, where MRS is equal to the absolute slope of budget line. So, the optimum consumption of both periods are “Ct1” and “Ct+11”.

Now, let’s assume the income of period t+1 increases, => the budget line will shift right side parallelly to A2B2 and the new equilibrium is E2, where MRS is equal to the absolute slope of budget line. So, the optimum consumption of both periods increases to “Ct2” and “Ct+12”.