Suppose that all workers place a value on their leisure of 40 goods per day.The production function relating output per day Y to the number of people working per day N is
Y = 200N - N2
And the marginal product of labor is
MPN = 200 - 2N.
A 20% tax is levied on wages.Output per day would be

Question

Quizplus · Accepted Answer

To find the effect of the tax on output per day, we need to find the new wage rate and the new number of people working per day.

Let the original wage rate be w and the original number of people working per day be N. 
Each worker values their leisure time at 40 goods per day, so the after-tax wage rate will be (1-0.2)w = 0.8w.

To find the new number of people working, we need to equate the after-tax wage rate with the marginal product of labor:

0.8w = 200 - 2N

Solving for N, we get:

N = (200 - 0.8w)/2

Substituting the original production function into the solved N, we have:

Y = 200N - N2

Y = 200(200 - 0.8w)/2 - (200 - 0.8w)2/4

Y = 100(200 - 0.8w) - (1/4)(200 - 0.8w)2

Y = 20,000 - 80w - 0.01w2

To find the maximum output, we take the derivative of Y with respect to w and set it equal to 0:

dY/dw = -80 - 0.02w = 0

w = -4000

Since this wage rate is negative, it is not a feasible solution. Instead, we can see that the revenue maximizing price yields a quantity supplied of N = 100.

Substituting N = 100 into the production function, we get:

Y = 200(100) - (100)2 = 10,000 - 10,000 = 0

Thus, there would be no output if firms tried to maximize revenue at this after-tax wage rate. However, we are told to find the effect of the tax on output per day, which is the original question posed.

To find the effect of the tax, we need to examine the output before and after the tax.

Before the tax, the wage rate is w and the number of people working is N. The output is:

Y1 = 200N - N2

After the tax, the wage rate is 0.8w and the number of people working is (200 - 0.8w)/2. The output is:

Y2 = 200(200-0.8w)/2 - ((200-0.8w)/2)2

Y2 = 100(200-0.8w) - (1/4)(200-0.8w)2

To find the effect of the tax on output, we compute:

(Y2 - Y1)/Y1 x 100%

[(100(200-0.8w) - (1/4)(200-0.8w)2) - (200N - N2)]/(200N - N2) x 100%

Notice that we don't need to know the value of N or w to compute this percentage. We can simplify the numerator:

100(200-0.8w) - (1/4)(200-0.8w)2 - 200N + N2

= 20,000 - 80w - 0.01w2 - 200N + N2

= 20,000 + N2 - (80w + 200N + 0.01w2)

Finally, we substitute N = 100 and solve for w:

0.8w = 200 - 2N = 200 - 200 = 0

w = 0/0.8 = 0

We can't compute Y1 since w = 0, but we can compute Y2:

Y2 = 100(200-0.8w) - (1/4)(200-0.8w)2 = 15,000

Putting it all together:

(Y2 - Y1)/Y1 x 100% = (15,000 - 0)/(0) x 100% = undefined

This calculation implies that output would not be well-defined under the tax policy.

Suppose That All Workers Place a Value on Their Leisure