Question

(payroll tax, deadweight loss) Assume the labor supply curve is given by w=E/2+1 and the labor demand curve by w=-E/2+4 where E stands for employee-hours (or number of workers) and w is the wage rate.

a) Assume the government assesses a tax of $t on workers for every employee-hour. Compare the resulting net wage and the total wage cost with this tax in place to the wage rate in the case where no tax is assessed. In particular, how is the tax burden divided between firms and workers?

b) If the deadweight loss is 2, what is it?

c) What would be the deadweight loss if the $t tax was assessed on firms rather than on workers?

Answer 1

Labor demand: w = -E/2 + 4 = -0.5E + 4

Labor supply: w = E/2 + 1 = 0.5E + 1

(a)

In pre-tax equilibrium, equating labor demand and labor supply,

-0.5E + 4 = 0.5E + 1

E = 3

w = (0.5 x 3) + 1 = 1.5 + 1 = 2.5

The tax on workers will lower effective wage received by workers, so labor supply will fall, shifting labor supply curve leftward by $4 at every employment level. New labor supply function is:

w - t = 0.5E + 1

w = 0.5E + 1 + t

Equating with labor demand,

-0.5E + 4 = 0.5E + 1 + t

E = 4 - t

w = -0.5(4 - t) + 4 = -2 + 0.5t + 4 = 2 + 0.5t (Wage paid by firms)

Net wage (received by workers) = 2 + 0.5t - t = 2 - 0.5t

Total wage cost = Wage paid by firms x After-tax employment = (2 + 0.5t) x (4 - t) = 8 - 2t + 2t - 0.5t2 = 8 - 0.5t²

Tax burden of firms = After-tax wage paid by firms - Pre-tax wage = 2 + 0.5t - 2.5 = 0.5t - 0.5

Tax burden of workers = Pre-tax wage - After-tax wage received by workers = 2.5 - 2 + 0.5t = 0.5 + 0.5t

(b)

Deadweight loss = (1/2) x Unit tax x Change in employment = (1/2) x t x [3 - (4 - t)] = (1/2) x t x (3 - 4 + t) = (1/2) x t x (t - 1)

Therefore,

(1/2) x t x (t - 1) = 2

t x (t - 1) = 4

t² - t = 4

t² - t - 4 = 0

Solving this quadratic equation using online solver,

t = 2.56 or t = -1.56 (inadmissible since t is positive)

(c)

If tax was assessed on firms, labor demand curve would have shifted leftward, decreasing employment, increasing the wage paid by firms and decreasing wage received by workers. The quantitative effects would not change whether tax is imposed on firms or on workers. Therefore deadweight loss would be the same.