Suppose that all workers place a value on their leisure of 75 goods per day.The production function relating output per day Y to the number of people working per day N is Y = 500N - 0.4N² and the marginal product of labour is MPN = 500 - 0.8 N.A 25% tax is levied on wages.In terms of lost output,what is the cost of the distortion introduced by this tax?

Question

Quizplus · Accepted Answer

To calculate the lost output we need to find the difference in output before and after the tax is imposed.

Before the tax: 
Total wage bill = N*w where w is the wage rate. 
But there is no tax yet so the workers keep all the wages. 
So, the cost of labor = N*w = (MPN)*Y/0.8

Substituting the values: 
N*w = (500-0.8N)*(500N-0.4N²)/0.8

N*w = (500N-0.4N²)*(625-N)/1000

N*w = 312500N - 250N² - 250N³/1000

Taking the derivative w.r.t N 
w = 312500 - 500N - 750N²/1000

w = 312.5 - 0.5N - 0.75N²

Now, we can calculate the output using the production function: 
Y = 500N - 0.4N²

Putting the value of w in terms of N in the production function: 
Y = 500N - 0.4N² - 0.5Nw - 0.75N²w

After the tax:
The wage rate now is (1-0.25)w = 0.75w 
Putting this value in the production function:

Y' = 500N - 0.4N² - 0.5N(0.75w) - 0.75N²(0.75w)

Y' = 500N - 0.4N² - 0.375Nw - 0.5625N²w

The lost output = Y - Y'

= 0.125Nw + 0.1875N²w

Substituting the value of w in terms of N:

= 0.125(312.5 - 0.5N - 0.75N²)N + 0.1875N²(312.5 - 0.5N - 0.75N²)

= 2734.375

Hence, the answer is C) 2734.

Suppose That All Workers Place a Value on Their Leisure