Question

People live for2 periods. They work in 1period and retire in the other. They have U = 2ln(C1) + ln(C2) where C1 is consumption in the first and C2 consumption in the second period. They earn an income Y1 in the first and receive zero earnings after retirement. They can save as much as they like. The interest rate on savings is rm. There are no bequests and individuals cannot borrow money.

(a) What is their inter-temporal budget constraint? How much will these ippl consume and save in each period?

(b) A new prime minister implements a pension system. She mandates that each person pays P P1 into the system in the first period of their life and gets it back with the market interest rate rm interest in the second period. State the inter-temporal budget constraint under this new system, and derive optimal consumption and private savings in each period.

(c) Compare a policy where P P1 is smaller than the optimal savings derived in (a) with a policy where individuals are forced to save more than they would have without the existence of this policy, i.e. where P P1 is larger than optimal savings S1 in (a) would have been. What differentiates these two policies in terms of savings and consumption effects?

(d) Now think of a Pay-As-You-Go system. What determines the rate of return on pension contributions in this system? Why has the rate of return been decreasing over time in most developed countries?

(e) Assume China has a PAYG pension system. Would you expect the rate of return to change in reponse to the recent abolition of the one child policy? Discuss.

Answer 1

Consider an economy populated by identical people who live for two periods. So, th income of the 1st period is “Y1=$50” and in the 2nd period is “Y2=$55”. So, if the rate of interest rate is “r=10%”, => the following is their life time budget constraint.

=> C1 + C2/(1+r) = Y1 + Y2/(1+r), where Y1=50, Y2=55 and r=10%.

So, the slope of the budget line is, “dC2/dC1 = (-1)*(1+r)”.

b).

Here we have given the their utility function which is given by, “U=InC1 + b*InC2, where b=1.

So, the MU1=(1/C1) and “MU2=1/C2”. So the “MRS=MU1/MU2=(1/C1)/(1/C2)=C2/C1.

So, at the optimum the “MRS” must equal to the absolute slope of the budget line.

=> MRS = 1+r, => C2/C1 = 1+r, => C2 = (1+r)*C1………………(2).

Now, by substituting (2) into the budget line we will get the optimum solution of “C1” and “C2”.

=> C1 + C2/(1+r) = Y1 + Y2/(1+r), => C1 + [(1+r)*C1]/(1+r) = Y1 + Y2/(1+r).

=> 2*C1 = Y1 + Y2/(1+r), => C1 = (1/2)*[Y1 + Y2/(1+r)].

Now, by substituting the optimum value of “C1” into (2) we will get the optimum solution for “C2”.

=> C2 = (1+r)*C1 = [(1+r)/2]*[Y1 + Y2/(1+r)], => C2 =[(1+r)/2]*[Y1 + Y2/(1+r)].

So, given the income and the rate of interest the optimum value of “C1” and “c2” are given below.

=> C1 = (1/2)*[Y1 + Y2/(1+r)] = (1/2)*[50+55/1.1]=50.

Similarly, C2 =[(1+r)/2]*[Y1 + Y2/(1+r)] = 1.1*50=55.

=> C1=50 and C2=55.

So, here optimum saving is given by, “Y1-C1=50-50=0”, => the optimum saving is “0”, => the optimum saving rate is also “0”.

c).

Now, assume that the “Y1” is “100” and “Y2” is nothing, => “0”. So, under this situation the optimum consumption in both the period are given below.

=> C1 = (1/2)*[Y1 + Y2/(1+r)] = (1/2)*[100]=50 and C2=(1+r)*C1=1.1*50=55.

So, under this situation the optimum consumptions are same as before. So, here the “saving” is “100-50=50”.

So, the “saving rate” is “S/Y1=50/100=0.5”.

d).

So as we can see that according to the utility function they want to smooth their consumption, => if their income are significantly differ in both periods, => they will either borrow or save at the given rate of interest to make the “consumption” smooth to maximize the utility. So, there initially “Y1 + Y2/(1+r)” was “100” after the change in income the above expression remain same as before. So, that’s why the “Consumption” bundles are also same in both the cases.

So, we can see that the consumption path are same here in both the periods.