Problem #2 Assume that the preferences of a consumer can be represented with U(C

Question

Problem #2 Assume that the preferences of a consumer can be represented with U(Ci,Ca, Cs,Ca) -log(C1) + log(Ca)+ log(Cs)+ log(Ca). Furthermore, assume that the real interest rate is equal to 0 and that the income profile of the consumer is given by Yi 20, y.-a, y's m 60, and Y 20. Finally, assume that the consumer starts her life with no initial assets. i) Find the optimal consumption profile of the consumer ii) Imagine that in the second period the consumer learns that her income in period 2 is 1 rather than 0. How much will the consumer consume in the second period in this case? Hint: Do not forget about debts/assets carried from the first period ili) Now, continue with assumptions of ii), imagine that in the third period the consumer learns that she cannot borrow froely. What will her consumption be the third period in this case? iv) Now, continue with assumptions of ii) and ilil), assume that the consumer learns in the fourth period that the interest rate is 10% rather than 0. what will her consumption in the fourth period be in this case?

Explanation / Answer

Consider the given problem here the utility function is given by.

=> U = logC1+logC2+logC3+logC4, and the as the rate of interest is “0”, => the budget line is given by.

=> C1 + C2/1+r + C3/(1+r)^2+ C4/(1+r)^3 = Y1 + Y2/1+r + Y3/(1+r)^2+ Y4/(1+r)^3.

So, here the optimization problem is to maximize, “U = logC1+logC2+logC3+logC4”, subject to

“C1 + C2/1+r + C3/(1+r)^2+ C4/(1+r)^3 = Y1 + Y2/1+r + Y3/(1+r)^2+ Y4/(1+r)^3”.

So, here the lagrage function is given by.

=> L = logC1+logC2+logC3+logC4 + c*[Y1 + Y2/1+r + Y3/(1+r)^2+ Y4/(1+r)^3 - C1 - C2/1+r - C3/(1+r)^2- C4/(1+r)^3].

So, here the FOC for maximization require, “dL/dC1 = dL/dC2 = dL/dC3 = dL/dC4 = 0”.

So, here “dL/dC1 = 0”, implied “1/C1 + c*(-1)=0”, => 1/C1 = c ……………………(1).

Now, by “dL/dC2 = 0”, => 1/C2 + c*[-1/1+r] = 0, => 1/C2 = c………………(2), since “r=0”.

Similarly, dL/dC3 = 0, => 1/C3 = c, and “dL/dC4 = 0”, => 1/C4 = c. So, from here we can see that “C1=C2=C3=C4”. So, by substituting this relation into the budget line we have.

=> Y1 + Y2/1+r + Y3/(1+r)^2+ Y4/(1+r)^3 = C1 + C2/1+r + C3/(1+r)^2 + C4/(1+r)^3.

=> Y1+ Y2+ Y3 + Y4 = C1 + C2 + C3 + C4, as “r=0”. Now, as “C1=C2=C3=C4”, => the optimum consumption is given by, “C1 = C2 = C3 = C4 = (Y1+Y2+Y3+Y4)/4 = 100/4 = 25”.

b).

now, as “Y2” increases to “15” from “0”, => from the above equation we have.

=> C1 = C2 = C3 = C4 = (Y1+Y2+Y3+Y4)/4 = 115/4 = 28.75”. So, as the “Y2” increases to “15”, => the consumption also increases to “28.75”.

c).

Now, in the previous part we got that the consumption is each period is “28.75”. Now, “Y1 = 20 < C1 = 28.75” and “Y2 = 15 < C2 = 28.75”, => in “1st” and the “2nd” period consumer is consuming more than its income, => the consumer is borrower. Now, “Y3 = 60 > C3 = 28.75”, => in the 3rd period the consumer is lender, => if the consumer can’t borrow freely, => this will not affect the optimum consumption choice. So, the optimum consumption choice will remain same as before.

d).

Now, if the consumer learn in the fourth period the “interest rate” is “10%” rather than “0”, => “C4 = 8.75*1.1 + Y4 = 29.625”, => the consumption of the 4th period increases to “29.625”. Since we can see that consumer is lender in the 4th period, => here the consumer will save “8.25” for the 4th period after paying the debt for “1st “ and for the “2nd” period, => the saving will be “8.75*1.1” in the 4th period and the consumer will get “Y4”. So, the total consumption will be “C4 = 8.75*1.1 + Y4 = 29.625”.

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