Consider the following version of our one period macro model. The representative household’s preferences are defined over consumption C and labor N.

Their preferences are given by the utility function U(C, N) = C −( N^(1+ (1/ν))/( 1 + (1/ν) where ν > 0. The idea of this utility function is that, rather than enjoying leisure, as the household usually does, it dislikes working. The government in this model, instead of simply demanding that the household pay a fixed amount of taxes T, charges a proportional tax on labor income at a rate τ where 0 ≤ τ ≤ 1. This means that if the household works N hours at a wage w it must pay taxes equal to τwN.

The household’s budget constaint is thus

C = (1 − τ )wN + π 1 where π is firm profits.

The government’s budget constraint is G = τwN .

Finally, there is a representative firm. This firm operates a constant return to scale technology that uses labor N to produce output Y according to Y = N There is no capital. Answer the following questions.

i) For a given tax rate τ define a competitive equilibrium for this model.

ii) Argue that, in a competitive equilibrium, the firm must earn zero profits (π = 0). Show that in any competitive equlibrium the wage rate w must equal 1.

iii) Show that household labor supply in a competitive equilibrium is given by N = (1 − τ ) ^ ν Use this equation and the budget constraint to solve for household consumption C. Show that the household’s utility in equilibrium is decreasing in the tax rate (HINT: Show that for two tax rates τ1 < τ2 the household can afford its equilibrium consumption under τ2 when facing the lower tax rate τ1 while working less than its equilibrium labor supply under τ2).

iv) We are now ready to define the Laffer curve for this model. As stated in the introduction to this problem, the Laffer curve is a plot of government tax revenue as a function of the tax rate. Government tax revenue in this model is τwN. From ii.) we know that w = 1 and from iii.) we know that N = (1 − τ ) ν . This implies that the Laffer curve is given by R(τ ) = τ (1 − τ ) ν. Notice that R(0) = R(1) = 0. Show that there is a tax rate 0 < τ ∗ < 1 that maximizes the Laffer curve, and show that the Laffer curve is decreasing for tax rates greater than τ ∗ and increasing for tax rates less than τ ∗ . Give a formula for τ ∗ in terms of ν.

v) Now pretend that you’re a member of the Reagan administration and you want to know if lowering tax rates will increase tax revenue. From iv.), this means that the previous administration would have had to set a tax rate larger than τ ∗ . Use your last result from iii.) (that welfare is decreasing in the tax rate) to make an argument that no government would choose a tax rate greater than τ ∗ (HINT: It might be helpful to try plotting the Laffer curve for some values ν in Excel). What does this tell you about the claims of supply-side economics?