Question

Consumption-Savings Consider a consumer with a lifetime utility function U = u(Ct) + βu(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. Now suppose Ctis taxed at rate τ so consumers pay 1 + τ for one unit of period t consumption.

(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 equation.

(c) Graphically depict the optimality condition. Carefully label the intercepts of the budget constraint. What is the slope of the indifference curve at the optimal consumption basket, (C∗t, C∗t+1)?

(d) Suppose the tax rate increases from τ to τ′. Graphically depict this.Carefully label the intercepts of the budget constraint. Is the slope of the indifference 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 effects. Using this intuition, can you definitively prove the sign of ∂C∗t∂τ and ∂C∗t+1∂τ ? It is not necessary to use math for this. Describing it in words is fine.

Answer 1

A).

Consider the given problem here the individual live in two period model “present(t)” and “future(t+1)”. So, here the budget constraint of the two periods are given by.

=> Ct+St=Yt for “period1” and “Ct+1+St+1=Yt+1+(1+r)*St” for “period2”.   

Now, since this is a two period model, => St+1=0.

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

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

b).

So, here the utility maximization problem is given by.

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

The lagrangian problem is given by.

=> L = U(Ct) + b*U(Ct+1) + c[Yt + Yt+1/(1+r) - Ct – Ct+1/1+r].

Now, the FOC are given by dL/dCt=dL/dCt+1 = 0”.

=> dL/dCt = 0, => u’(Ct) + c(-1) = 0, => u’(Ct) = c …………….(1).

=> dL/dCt+1 = 0, => b*u’(Ct+1) + c(-1)/1+r = 0, => b*u’(Ct+1) = c/1+r …………….(2).

Now, (1) divided (2) we have the following condition.

=> u’(Ct)/ b*u’(Ct+1) = (1+r) ………………………(3).

So, here the equation (3) is the “Euler Equation”. So, here this equation shows the MRS between consumption today and next period is equal to the relative price between the “today’s” consumption and the “tomorrow’s consumption”.