Question

Consider an individual that lives for two periods. She only works in the first period and receives a labor income equal to 200 Euros. Additionally, this individual receives a non-labor income equal to 20 Euros in each period. The interest rate in the economy is 10 %. She can consume in period 1 (c1) and in period 2 (c2). The price of the consumption good is equal to 1 in both periods. The individual has a Cobb-Douglas utility function of the following form: u(c1, c2) = (c1)1/2 (c2)1/2.

Find c1, c2, and private savings in the following cases:

a) A proportional labor income tax of 10%.

b) A proportional income tax of 10% (the same in each of the two periods).

c) A proportional income tax of 10%, in each period, with an exemption in the base of each period equal to 100 Euros (the first 100 euros are not taxed)

d) A labor income tax, in each period, with an increasing rate such that (the first100 euros pay no taxes, the euros between 100 and 150 pay 10%, and the euros above 150 pay 20%):

Base Tax rate

[0 , 100) 0%

[100 – 150) 10%

[150, ∞] 20%

e) A consumption tax, in each period, of 10%

  

Answer 1

This problem pertains to intertemporal choices in economics. It implies that a consumer has a choice to decide to consume today or in the future. The choice is driven by the interest rates that are offered during the period.

a.

Case 1	Consumption	Price	Tax	Income (After tax- M)	Tax
Period 1	C1	1	10%	200	20
Period 2	C2	1	10%	200	20 (Only on labour income)

Utility function is = C11/2C21/2

Consumption in period 2 is driven by the saving or overconsumption in period 1. This can be formed into the following equation-

C2= M2 + (M1 - C1) +r(M1 - C1)

C2= M2 + (1+r) (M1 - C1)  ---- (1)

dC2/ dC1 = -(1+r)

For every dollar saved in period 1 amount of composite good purchased in period 2 has increased by (1+r).

MU C1= 1/2(C1C2)1/2 * C2

MU C2= 1/2(C1C2)1/2 * C1

MRS= C2/C1

Equating MRS to budget line

C2/C1 = 1+r

C2= 1.1 C1 ----- (2)

Plugging (2) in (1)

C1 = 191 and C2= 210

b.

Case 1	Consumption	Price	Tax	Income (After tax- M)	Tax
Period 1	C1	1	10%	198	22
Period 2	C2	1	10%	198	22 (On total income)

Utility function is = C11/2C21/2

Consumption in period 2 is driven by the saving or overconsumption in period 1. This can be formed into the following equation-

C2= M2 + (M1 - C1) +r(M1 - C1)

C2= M2 + (1+r) (M1 - C1)  ---- (3)

dC2/ dC1 = -(1+r)

For every dollar saved in period 1 amount of composite good purchased in period 2 has increased by (1+r).

MU C1= 1/2(C1C2)1/2 * C2

MU C2= 1/2(C1C2)1/2 * C1

MRS= C2/C1

Equating MRS to budget line

C2/C1 = 1+r

C2= 1.1 C1 ----- (4)

Plugging (4) in (3)

C1 = 189 and C2= 208

c.

Case 1	Consumption	Price	Tax	Income (After tax- M)	Tax
Period 1	C1	1	10%	208	12
Period 2	C2	1	10%	208	12 (On total income)

Utility function is = C11/2C21/2

Consumption in period 2 is driven by the saving or overconsumption in period 1. This can be formed into the following equation-

C2= M2 + (M1 - C1) +r(M1 - C1)

C2= M2 + (1+r) (M1 - C1)  ---- (5)

dC2/ dC1 = -(1+r)

For every dollar saved in period 1 amount of composite good purchased in period 2 has increased by (1+r).

MU C1= 1/2(C1C2)1/2 * C2

MU C2= 1/2(C1C2)1/2 * C1

MRS= C2/C1

Equating MRS to budget line

C2/C1 = 1+r

C2= 1.1 C1 ----- (6)

Plugging (6) in (5)

C1 = 198.54 and C2= 218

d. Increasing tax rate on labour income of 200.

Slab	Taxes
0-100	0% ~0
100-150	10% ~ 5
150-200	20% ~ 10

Total tax paid is 15

Case 1	Consumption	Price	Tax	Income (After tax- M)	Tax
Period 1	C1	1	Increasing	205	15
Period 2	C2	1	increasing	205	15 (On labour income)

Utility function is = C11/2C21/2

Consumption in period 2 is driven by the saving or overconsumption in period 1. This can be formed into the following equation-

C2= M2 + (M1 - C1) +r(M1 - C1)

C2= M2 + (1+r) (M1 - C1)  ---- (5)

dC2/ dC1 = -(1+r)

For every dollar saved in period 1 amount of composite good purchased in period 2 has increased by (1+r).

MU C1= 1/2(C1C2)1/2 * C2

MU C2= 1/2(C1C2)1/2 * C1

MRS= C2/C1

Equating MRS to budget line

C2/C1 = 1+r

C2= 1.1 C1 ----- (6)

Plugging (6) in (5)

C1 = 19 and5.68 C2= 215.25