Suppose we are in a two-period environment where the representative consumer has

Question

Suppose we are in a two-period environment where the representative consumer has a utilit;y function of the form: Let the discount factor, , represent the idea that the consumer values consumption at the future with some weight less than 1. Let initial assets, a 0 and the households income in the two periods be given as y,-5, y,-10. The real interest rate in this economy, r is equal to .1 (ie 10% return on any wealth saved). 1. Intuitively, why do we only need to consider the savings choice in period if this consumer lives for two periods? (Financial markets don't disappear when one is old, right?) 2. Solve for the consumer's allocations, (ie, ci, c2, a1). Is our consumer a saver or a borrower, and does this make sense intuitively? 3. Suppose now consumers are no longer allowed to be in debt following period 1 (ie a1 2 0 must hold). You may think of this either as banks imposing a strict limit on lending or government regulations prevent households from being leveraged. What are the new allocations? Explain 4. Is this borrowing limit welfare reducing, enhancing, or neutral? Explain intuitively, then compare utility levels in both environments.

Explanation / Answer

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

=> U = C1^0.5 + b*C2^0.5, "b=1/1.1".

Now, the budget constraint of the “period 1” is given by, “C1 = Y1+A1”, where “A1” be the saving. Similarly, the budget constraint of the “period 2” is given by, “C2 = Y2+A1*(1+r)”, where “A1” be the saving. Now, we have given the value of “Y1, Y2 and r”, => if we will be able to figure out the optimum saving, => the optimum consumption of “period 1” will be.

=> “C1 = Y1+A1 = 5+A1” and “C2 = 10+A1*(1.1)”.      

2).

So, here the utility function of the consumer is given by.

=> U = C1^0.5 + b*C2^0.5, => U = C1^0.5 + 0.91*C2^0.5. So, the marginal utility are given by.

=> MU1 = 0.5*C1^-0.5, and MU2 = 0.91*0.5*C2^-0.5.

So, the “MRS” is given by, “MRS=MU1/MU2”.

=> MRS = [0.5*C1^-0.5] / [0.91*0.5*C2^-0.5], => MRS = [C1^-0.5] / [0.91*C2^-0.5].

=> MRS = (1.1)*(C2/C1)^0.5.

Now, the intertemporal budget line is given by, C1+C2/1+r = Y1 + Y2/1+r. So, the absolute slope of the budget line is given by, “1+r=1.1”. So, at the optimum “MRS” must be equal to “1+r”.

=> MRS=1+r, => (1.1)*(C2/C1)^0.5 = 1.1, => C2 = C1.

So, at the optimum “C1” is same as “C2”. So, by putting this condition on the “intertemporal budget line” we will have the optimum choice.

=> C1+C2/1+r = Y1+Y2/1+r, => C1*(1+1/1+r) = Y1+Y2/1+r = 5+ 10/1.1 = 14.09.

=> 1.91*C1 = 14.09, => C1 = 14.09/1.91 = 7.38 = C1. So, from “C2=C1=7.38, and “A1=Y1-C1=(-2.38)”.

So, since “C1=7.38 > Y1 = 5”, => here the consumer is borrower not saver.

3).

Suppose consumers are no longer allowed to be in debt following “period 1”, => A1 > or equal to “0”. So, as the consumer is borrower with optimum consumption “C1=C2=7.38”, will reduce the “C1” to “Y1”, => “A1=0”. So, as the “A1=0”, => “C1=Y1=5” and “C2=9.09”.

4).

So, initially the consumer is getting utility “U = C1^0.5 + 0.91*C2^0.5.

=> U = 7.38^0.5 + 0.91*7.38^0.5, => U = 5.19.

Now, with the new allocation the utility is given by.

=> U = C1^0.5 + 0.91*C2^0.5 = 5^0.5 + 0.91*9.09^0.5, => U = 4.97 < 5.19.

So, here we can see that the utility of the consumer decreases. So it is reducing the welfare of the consumer.

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