Question

Consider an individual who allocates his total hours of work per week between the household and a competitive market. The available total number of hours of work is fixed at 50 per week. In the market the wage (w) is equal to the value of marginal product of labour. The value of marginal product of labour in the market (VMPM) and the value of marginal product in the household (VMPH) are given by the following equations:

VMPM = 100 - LM

VMPH = 80 - 2LH

where LM is the number of hours worked in the market and LH is the number of hours worked in the household

a) Find the number of hours worked in the market and the household per week, in absence of any taxes.

b) Suppose that a 20% tax is levied on wage in the market. Find how the hours worked in the market and the household change after the tax. What is the excess burden of the tax?

Answer 1

Solution :-

(a) :-

The available total number of hours of work is fixed at 50 per week.

The value of marginal product of labour in the market (VMPM) and the value of marginal product in the household (VMPH) are given by the following equations:

VMPM = 100 - LM

VMPH = 80 - 2LH

where LM is the number of hours worked in the market and

LH is the number of hours worked in the household.

VMPM = VMPH

100 - LM = 80 - 2LH

100 - 80 = LM - 2LH

20 = LM - 2LH......(1)

50 = LM + LH.......(2)

Substracting equation ( 1) from (2) we get

50 - 20 = LM + LH - ( LM - 2LH)

30 = LM + LH - LM + 2LH

30 = 3LH

LH = 30/3

[ LH = 10 ]

Then, put the value of LH = 10 in equation (2)

We get,

50 = LM + LH

50 = LM + 10

50 - 10 = LM

[ LM = 40 ]

VMPW = 60

(b) :-

Suppose that a 20% tax is levied on wage in the market.

So, new VMPM = 0.8 VMPM

0.8 VMPM = VMPH

0.8 ( 100 - LM) = 80 - 2LH

80 x 100 - 0.8LM = 80 - 2LH

80 - 80 = 0.8LM - 2LH

0 = 0.8LM - 2LH

0.8LM - 2LH = 0

Now,

LH + LM = 50

LH = 50 - LM

0.8LM - 2LH = 0

Put LH = 50 - LM in above equation

0.8LM - 2 ( 50 - LM) = 0

0.8LM - 100 + 2LM = 0

2.8LM - 100 = 0

2.8LM = 100

LM = 100/2.8

[ LM = 35.71]

Now, for LH

LH + LM = 50

LH + 35.71 = 50

LH = 50 - 35.71

[ LH = 14.29 ]

VMPH = 80 - 2LH

= 80 - 2 ( 14.29)

= 80 - 28.58

= 51.42

DWL = 1/2 x 8.571428 x 4.28

= 8.571428 x 2.14

= 18.3428