Question

Consider a model in which individuals live for two periods and have utility functions of the form ? = ??(?1) + .5??(?2). They earn income of $10,000 in the first period and save S to finance consumption in the second period. The interest rate, r, is 20.

a. Set up the individual’s lifetime utility maximization problem. Solve for the optimal C1, C2, and S. (Hint: ????1,?2 = −.5?2/?1 and the budget line is given by C2= (10,000-C1)(1+r) hence the slope of the budget line is -(1+r)). Sketch the optimal point in indifference curve and budget line diagram.

b. The government imposes a 9% tax on labor income. Solve for the new optimal levels of C1, C2, and S. Explain any differences between the new level of savings and the level in part (a), paying attention to any income and substitution effects.

c. Instead of the labor income tax, the government imposes a 9% tax on interest income. Solve for the new optimal levels of C1, C2, and S. (Hint: What is the new after-tax interest rate?) Explain any differences between the new level of savings and the level in part (a), paying attention to any income and substitution effects.

Answer 1

Given:

Utility function for a Consumption of 2 periods, U(C1,C2) = In(C1) + 0.5In(C2)

Income in period 1 in $ (Y1) = 10000

Interest rate in % (r) = 20

Saving in period 1 in $ (S) to finance C2

(a) Utility maximization problem of an individual is as follows:

Maximize U (C1, C2) such that C2 = (Y1-C1) (1+r)

Or, Max [In(C1) + 0.5In(C2)] such that C2 = (10000-C1) (1.2)

Then, slope of consumer’s indifference curve = MRS (C1, C2)                                   

= [(dU/dC1)/ (dU/dC2)]

= [(1/C1) / (-0.5/C2)]

= -0.5C2/C1

And slope of budget line = - (1+r) = -(1+0.2) = -1(1.2)

Then, at the optimal consumption point,

    slope of consumer’s indifference curve = slope of budget line

       or, -0.5C2/C1 = -(1.2)

       or, C2/C1 = 2.4

       or, C2 = 2.4 C1

then, from the budget line, we know C2 = (10000-C1) (1.2)

or, 2.4C1 = 12000 – 1.2C1

or, 3.6C1 = 12000

or, C1* = 3333.33

and C2* = 2.4 (3333.33) = 8000

and S*= (10000-3333.33) = 6666.67

(b) Now, government imposes 9% tax on labor income, then it will not affect consumer’s utility function but will change budget line as follows:

Let (t) be tax rate on labor income, such that new income of individual after tax (Y1t)            = Y1 – t%* Y1

= Y1- 9%*Y1

= 0.91Y1

And new budget line of individual = C2 = (Y1t-C1) (1+r)

                                                                 = (0.91Y1 – C1) (1.2)

                                                                 = 1.092Y1 – 1.2C1

                                                                 = 10920 – 1.2 C1

Hence, slope of budget line remains same as before in (a), then

At optimal point, from (a) we know,

                            C2 = 2.4 C1

                    And from new budget line, we know C2 = 10920 – 1.2 C1

Or, 2.4 C1 = 10920 – 1.2 C1

Or, 3.6C1 = 10920

Or, C1* = 3033.33

And C2* = 2.4(3033.33) = 7280

And S* = 0.91(10000) - 3033.33 = 6066.67

Now, comparing optimal C1, C2, S in (a) with (b), we find C1 reduces by 300, C2 reduces by 720 and S reduces by 600. Hence, there is negative income effect on C1 & C2, S .

(c) Suppose government imposes 9% tax on interest income instead of labor income tax as on (b), then it will also not affect consumer’s utility function but will change budget line as follows:

Let (t) be tax rate on interest income, such that new after-tax interest rate becomes r-t

And new budget line of individual = C2 = (Y1-C1) (1+r-t)

                                                                 = (Y1 – C1) (1 + 0.2 – 0.09)

                                                                       = (10000-C1) (1.11)

Thus, new slope of new budget line will be –(1+r-t) = -(1.11)

Then, at the optimal consumption point,

    slope of consumer’s indifference curve = slope of budget line

       or, -0.5C2/C1 = -(1.11)

       or, C2/C1 = 2.22

       or, C2 = 2.22 C1

then, from the budget line, we know C2 = (10000-C1) (1.11)

or, 2.22 C1 = (10000-C1) (1.11)

or, 3.33 C1 = 10000

or, C1* = 3003.003

and C2* = 2.22 (3003.003) = 6666.667

and S* = 10000 - 3003.003 = 6996.997

Now, comparing optimal C1, C2, S in (a) with (c), we find C1 reduces by 330, C2 reduces by 1333 and S in increases by 330. Hence, there is both negative income effect on C1 & C2 and positive substitution effect on S as increases