Question

Two period saving model) Consider an economy populated by identical people who live for two periods. They have preferences over consumption of the following form: U=ln(c1) +βln(c2), where ct denotes the stream of consumption in period t. They also receive an income of 50 dollars in period 1 and an income of 55 dollars in period 2. They can use savings to smooth consumption over time, and if they save, they will earn an interest rate of 10% per period. Suppose also that people value the future and the present equally, i.e. β= 1, answer the following questions.

(a) Write down their life-time budget constraint.

(b) What is the individual’s optimal consumption in each period (c1*andc2*). What is the optimal saving rate? (Hint: the marginal utility of ln(c) is 1/c)

(c) Suppose now that these people receive an income of 100 in the first period, but nothing in the second, period. What is the individual’s optimal consumption in each period (c1* and c2*). What is the optimal saving rate?

(d) How does the new consumption path compare to the previous consumption path? Explain.

Answer 1

a).

Consider an economy populated by identical people who live for two periods. So, th income of the 1^st period is “Y1=$50” and in the 2^nd 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.