Question

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)S

(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.

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”.