The Consumers Preferences Over The Consumption Assignment Paper
Question
Consider an economy with the representative consumer, representative firm, and government. Suppose that . The consumer s preferences over the consumption good and leisure are given by the utility function . Let . The firm’s production function is . Suppose that the government imposes a proportional income tax on the representative consumer. The government uses the tax revenue to finance its expenditures as well as to give give each consumer a lump-sum social security transfer . There are no lump-sum taxes. That is, the consumer’s budget constraint is . (a) (5 points) Write down the consumer’s optimality condition. (b) (5 points) Set up the firm’s optimization problem. What is the equilibrium wage rate equal to? What are the firm’s profits equal to? (c) (10 points) Using your answers to parts (a) and (b) solve the consumer’s problem to find the labor supply as a function of t. Take into account that the worker’s labor supply cannot be lower than 0 ! (d) (10 points) Use your answer to part (c) to derive an expression for the government tax revenue as a function of the tax. Using Excel, plot the tax base (labor income) and the tax revenue (Laffer curve) against a grid of the tax rate . (e) (5 points) Provide an intuitive explanation for the shape of the Laffer curve that you plotted in part (d). In particular discuss the effects of the tax rate on the tax revenue through its effect on the different components of the tax revenue The Consumers Preferences Over The Consumption Assignment Paper.
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Step-by-step
a) The consumer’s optimization problem can be formulated as:
max U(C, l) subject to C = wh + π + 1 – twl
where w is the wage rate, π is the firm’s profit, t is the proportional income tax rate, and S is the lump-sum social security transfer.
To solve this problem, we can use the Lagrangian method:
L = U(C, l) + λ[C – wh – π – 1 + twl – S]
where λ is the Lagrange multiplier associated with the budget constraint.
The first-order conditions for this problem are:
λ∂L∂C=12C−12−λ=0
λ∂L∂l=12C12tw−λ=0
From the first equation, we get:
λC−12=2λ
which implies:
λC=(2λ)−2
From the second equation, we get:
λtw=2λC12
Substituting the expression for C from above, we get:
λλλtw=2λ(2λ)−1=λ
Therefore, we have:
C12l=1t
which is the consumer’s optimality condition.
- Explanation for step 1
This condition states that the marginal rate of substitution between consumption and leisure should be equal to the wage rate adjusted for the proportional income tax. That is, the consumer should allocate their time between work and leisure so that the additional utility gained from an extra unit of leisure (measured by the marginal utility of leisure) is equal to the opportunity cost of that leisure (measured by the after-tax wage rate).
Step 2/4
b) To set up the firm’s optimization problem, we first need to write down the profit function for the representative firm.
The firm’s production function is given by Y=N, where Y is output and N is the amount of labor hired by the firm. The firm’s cost function is given by wN, where w is the wage rate. The firm’s revenue function is given by PY, where P is the price of the output.
Therefore, the firm’s profit function is: The Consumers Preferences Over The Consumption Assignment Paper
π = PY – wN = PN – wN
To find the equilibrium wage rate, we need to maximize the firm’s profit function with respect to N.
Taking the derivative of the profit function with respect to N, we get:
dπ/dN = P – w
Setting this derivative equal to zero, we get the first-order condition for the firm’s optimization problem:
P = w
This means that in equilibrium, the price of the output is equal to the wage rate.
To find the firm’s profits in equilibrium, we can substitute the equilibrium wage rate into the profit function:
π = PN – wN = P(N – w)
Since we know that P = w in equilibrium, we can substitute w for P to get:
ππ=w(N−w)=wN−w2
We can simplify this expression by substituting the firm’s production function:
ππ=wY−w2
Since Y = N in the firm’s production function, we can substitute N for Y to get:
ππ=wN−w2
Using the first-order condition, we know that in equilibrium, P = w. Therefore, we can substitute P for w to get:
ππ=PW−w2=wN−w2
Finally, we can substitute the equilibrium condition P = w into the expression to get:
ππ=wN−w2
So the equilibrium wage rate is equal to the price of the output, which is P = w. The firm’s profits in equilibrium are given by ππ=wN−w2.
- Explanation for step 2
To find the equilibrium wage rate, we need to maximize the firm’s profit function with respect to N.
Step 3/4
c) To solve the consumer’s problem, we need to maximize the utility function subject to the budget constraint. We can write the Lagrangian as: The Consumers Preferences Over The Consumption Assignment Paper
λπL=C12l12+λ[w(h−l)+π+1−C]
where λ is the Lagrange multiplier.
Taking partial derivatives with respect to C and l, we get:
λ∂L∂C=12C−12l12−λ=0
λ∂L∂l=12C12l−12−λw=0
Solving for C and l, we get:
λC=(2λ2w)2
λl=(2λ2w)2
Substituting these into the budget constraint, we get:
λπ(2λ2w)2=w(h−l)+π+1
Solving for λ, we get:
λλ=w4t2
Substituting this back into the expressions for C and l, we get:
C=l=w216t4
The labor supply is given by h – l, so we have:
h−l=24−w216t4
We want to find the labor supply as a function of t, so we substitute the equilibrium condition w = P into the expression for labor supply:
h−l=24−P216t4
Therefore, the labor supply as a function of t is:
L(t)=max{0,24−P216t4}
where we take the maximum with 0 to ensure that labor supply cannot be negative.
- Explanation for step 3
to solve the consumer’s problem, we need to maximize the utility function subject to the budget constraint which we do by using the lagrangian method
Step 4/4
d) λπL=C12l12+λ[w(h−l)+π+1−C]
Taking first-order conditions with respect to C and l, we obtain:
λC−12l12=λw
and
λC12l−12=λw
Dividing the two equations, we obtain: The Consumers Preferences Over The Consumption Assignment Paper
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lC=1t
Substituting this into the budget constraint, we get:
Cπ=w(h−(1t)C)+π+1
Solving for C, we get:
πC=w(h+π+1)1+wt
Substituting this into the labor supply equation, we get:
πl=(tw2)(wh+π+11+wt)2
The labor supply function is increasing in t, which means that the higher the tax rate, the more the consumer will supply labor. This is because the consumer needs to work more in order to maintain their desired level of consumption.
To find the government tax revenue, we need to multiply the tax rate by the tax base (labor income), which is given by w(h – l). Substituting the expression for l, we get:
Tax revenue = πtw(h−(tw2)(wh+π+11+wt)2)
Simplifying this expression, we get:
Tax revenue = πt(hw−th+π+11+wt)
- Explanation for step 4
The labor supply function is increasing in t, which means that the higher the tax rate, the more the consumer will supply labor. This is because the consumer needs to work more in order to maintain their desired level of consumption The Consumers Preferences Over The Consumption Assignment Paper.
Final answer
e) Using Excel, we can plot the tax base (labor income) and the tax revenue against a grid of tax rates t. The resulting graph is the Laffer curve, which shows the relationship between the tax rate and the tax revenue.
The Laffer curve is typically U-shaped, which means that there is an optimal tax rate that maximizes the tax revenue. At low tax rates, the tax revenue is low because the tax base is small. As the tax rate increases, the tax base also increases because consumers supply more labor. However, at some point, the tax rate becomes too high and consumers start to reduce their labor supply because the marginal benefit of working is lower than the marginal cost of paying taxes. At this point, the tax revenue starts to decrease because the tax base is shrinking faster than the tax rate is increasing The Consumers Preferences Over The Consumption Assignment Paper.
The shape of the Laffer curve is determined by two opposing effects: the income effect and the substitution effect. The income effect causes consumers to work less as their disposable income decreases due to taxes. The substitution effect causes consumers to work more as the relative price of leisure increases due to taxes. The Laffer curve is upward-sloping at low tax rates because the substitution effect dominates the income effect, and it is downward-sloping at high tax rates because the income effect dominates the substitution effect. The peak of the Laffer curve occurs where the two effects exactly offset each other. The Consumers Preferences Over The Consumption Assignment Paper