Question

Say the household has utility U (c, l) = c + log (l) and is endowed with h = 1 units of time and no units of capital. The government has planned expenditures of G = 1. The firm's production technology is Y = 4N.

Now assume that the government cannot finance G trough a lump-sum tax, but has to rely on a proportional tax on income.

1. Solve for the value (or values) of τ that allows the government to finance G.

2. Compare the equilibrium value for leisure to the solution for the planner problem. Discuss.

3. Next, let G = 1/2. Show that there are two values of τ for which the government manages to collect enough revenues to nance G. Comment, referring to the Laffer curve. Why at point 1 above you only found one value of τ?

Finally, keep Y = 4N but let G = 0 and assume that output production produces pollution P = 1/4Y . Utility decreases in pollution, that is U (c, l) = c + log (l) + 1/2log (1 − P )

1. Solve for the planner problem problem and show that the solution differs from the competitive equilibrium.

2. Now, let the government impose a tax on labor income equal to τ = 1/3. Solve for the competitive equilibrium and show that it is identical to the planner problem. Discuss.

Answer 1

Assuming that the government can not finance G through the lump sum tax, but through proportional tax .

1. In this case, the household consumption will be, instead of , as or . or or , and as w=4 again, . For , as the peice of leisure now includes the proportional tax. Hence, or or . Also, as , we have or . As , thus , and the output will be . The goods market equilibrium is at or , by putting the value of Y, c ang G (which is still 1), and hence or or or or , which is needed to finance the G.

Hence, and .

As proportional tax increased, leisure increased as price of labor, the after tax wage, decreased