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Problem Set 1 Econ 101B Macroeconomics (Math Intensive) Fall 2024 Due at 10:00p...

Sep 19, 2024

Solution

Labor Demand Curve: In this economy, the labor demand curve can be derived from the production function

Y = A L

. The marginal product of labor (MPL) is given by

MPL = A

, and firms will hire labor until the wage

w

equals the MPL. Thus, the labor demand curve is

w = A

. This implies that as wages increase, firms will demand less labor

Household's Labor Supply Curve: The household's utility function is given by

U = \log C - \psi \frac{H^{1 + 1/\eta}}{1 + \eta^{-1}}

. The budget constraint is

C = (1 - \tau_l) w H + T

. By maximizing utility subject to the budget constraint, we derive the labor supply curve, which shows how hours worked

H

respond to changes in the after-tax wage

(1 - \tau_l)w

Expression for Hours Worked: Given

N

households, total tax revenue is

\tau_l w H N

. Using the budget constraint and the derived labor supply curve, we can express hours worked per person as

H = (1 - \tau_l)^{\frac{\eta}{\eta + 2}} \psi^{\frac{-\eta}{\eta + 1}}

. This curve illustrates how hours worked decrease as the tax rate increases

Tax Revenue Expression: Tax revenue can be expressed as

R = \tau_l w H N

. Substituting the expressions for

H

and

w

derived earlier, we can express tax revenue as a function of

\tau_l

and the exogenous variables

(A, N)

and parameters

(\eta, \psi)

Maximum Tax Revenue: To find the top of the Laffer curve, we differentiate the tax revenue function with respect to

\tau_l

and set it to zero. This yields the tax rate that maximizes revenue as a function of the parameters

(\eta, \psi)

Tax Rate Calculations: For the values

\eta = 0.5, 1, 3

, we substitute these into the expression derived in part (e) to calculate the corresponding tax rates that yield maximum tax revenue

Answer

The labor demand curve is derived from the production function, and the maximum tax revenue occurs at specific tax rates depending on the Frisch elasticity of labor supply.

Key Concept

The Laffer curve illustrates the relationship between tax rates and tax revenue, showing that there is an optimal tax rate that maximizes revenue.

Explanation

The analysis involves deriving labor demand and supply curves, expressing tax revenue in terms of tax rates, and finding the optimal tax rate for maximum revenue based on different elasticities.