Suppose you are given the utility function: where y = 5, y\' = 10, and the inter

Question

Suppose you are given the utility function:

where y = 5, y' = 10, and the interest rate r = 0.10.

a) What is the optimal value of current consumption c*?

b) What is the optimal value of future consumption, c'*?

Now suppose the consumer experiences a temporary rise in real income in period 1, so that y = 9.

c) What is the optimal value of current consumption c*?

d) What is the optimal value of future consumption, c'*?

Now suppose that y= 5 again, and there is only one consumer in the entire economy. If we add in government expenditures and taxation, the consumer’s budget constraint is now:

e) What is the value of second period taxation, t’?

f) What is the optimal value of current consumption c*?

g) What is the optimal value of future consumption, c'*?

Suppose that current taxation decreases decrease from t = 2 to t =1.5.

h) What is now the value of second period taxation, t’?

f) What is the optimal value of current consumption c*?

g) What is the optimal value of future consumption, c'*?

Suppose that current taxation decreases decrease from t = 2 to t =1.5.

h) What is now the value of second period taxation, t’?

i) What is the optimal value of current consumption c*?

j) What is the optimal value of future consumption, c'*?

Show transcribed image textU = VC+ Vc 1.10

need answers for E,F,G,h,i,j

Explanation / Answer

In the above question both utility function and budget constratint is given . So

utility function (U) = ?c + ?c' /1/10 and the

budget function is c+ c'/1+r = y+y/1+r - t-t/'1+r and

where y = 5, y' = 10, and the interest rate r = 0.10.

a) What is the optimal value of current consumption c*?

Ans:- Before you find out the Optimal value of Current consumption C* first we have to simplify the utility function

utility function (U) = ?c + ?c' /1/10  

= ?c + ?c' * 1.10-1

= c 1/2 + c' 1/2 * 1.10-1

= c 0.5 * c' 0.5 * 1.10-1

the budget constraint also give i.e.  c+ c'/1+r = y+y'/1+r

and Income (y) = 5, Absolute Income (y') = 10 and rate of interest (r) = 0.10

Put the value iin the budget constraints

    c+ c'/1+r = y+y'/1+r

     c+ c'/1+0.10 = 5+10/1+0.10

c  + c'  * 1.10-1 = 5 + 10/1+0.10

    c  + c'  * 1.10-1 = 5+ 9.09

    c  + c'  * 1.10-1 = 14.09

Lagragian function L = Utility +  ?( 14.09-  c- c'  * 1.10-1 )

=  c 0.5 * c' 0.5 * 1.10-1 +  ?( 14.09-  c- c'  * 1.10-1 )

1st order condition = dl/ dc = 0.5  c-0.5 -   ? = 0

0.5  c-0.5 =  ? .................................. (1)

dl/ dc ' =    1.10-1 *  0.5  c-0.5   - ? 1.10-1 = 0

   1.10-1 *  0.5  c-0.5 = ? 1.10-1 ( Cancel the  1.10-1 both left and right side)

0.5  c-0.5 = ? ............................ (2)

  dl/ d ? = 14.09 - c'  * 1.10-1

  14.09 = c'  * 1.10-1 c ' .......................... (3)

From (1) and (2) equation

0.5  c-0.5 =   0.5  c'-0.5

so c=c'

then put the value of c in the equation (3)

14.09 = c' + 1.10-1  c'

= c' + c'/1.10

14.09 = 1.10 c' +c'/ 1.10 = 2.10 c' /1.10

= 14.09 * 1.10/ 2.10 = c'

c' = 7.380 = c

a) What is the optimal value of current consumption c*?

Ans:- so optimal value of consumption is 7.380

b) What is the optimal value of future consumption, c'*?

Ans:- same as current consumption because c = c' so the answer is 7.380

Now suppose the consumer experiences a temporary rise in real income in period 1, so that y = 9.

c) What is the optimal value of current consumption c*?

ANs:- if y = 9 the the budget constraint is

c+ c'/1+r = y+y'/1+r

     c+ c'/1+0.10 = 9+10/1+0.10

c  + c'  * 1.10-1 = 9 + 10/1+0.10

    c  + c'  * 1.10-1 = 9+ 9.09

    c  + c'  * 1.10-1 = 18.09

Lagragian function L = Utility +  ?( 18.09-  c- c'  * 1.10-1 )

=  c 0.5 * c' 0.5 * 1.10-1 +  ?( 18.09-  c- c'  * 1.10-1 )

1st order condition = dl/ dc = 0.5  c-0.5 -   ? = 0

0.5  c-0.5  =  ? .................................. (1)

dl/ dc ' =    1.10-1 *  0.5  c-0.5   - ? 1.10-1 = 0

   1.10-1 *  0.5  c-0.5 = ? 1.10-1 ( Cancel the  1.10-1 both left and right side)

0.5  c-0.5 = ? ............................ (2)

  dl/ d ? = 18.09 - c'  * 1.10-1

  18.09 = c'  * 1.10-1 c ' .......................... (3)

From (1) and (2) equation

0.5  c-0.5 =   0.5  c'-0.5

so c=c'

then put the value of c in the equation (3)

18.09 = c' + 1.10-1  c'

= c' + c'/1.10

18.09 = 1.10 c' +c'/ 1.10 = 2.10 c' /1.10

= 18.09 * 1.10/ 2.10 = c'

c' = 8.614 = c

d) What is the optimal value of future consumption, c'*?

Ans:- same as current consumption because c = c' so the answer is 8.614

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