Economics homework help
Economics homework help. 1
Short Questions [10]
- [5] King Shrek of Country Far Far Away is trying to build the largest royal swamp in the world, so he wants to raise the income tax rate on his citizens to collect the extra tax revenue needed to pay for this project. The income tax rate is already quite high at 80%. As King Shrek’s advisor on economic policy, how would you explain to him, using economics, that raising the tax rate might not help him collect more tax revenue? (King Shrek only cares about himself, and his swamp!)
- [5] Anna has been quite lucky recently. In August, she won a small lottery which gave her $1000. She spent $800 of that on a short trip to Orlando, and saved the remaining $200 in her banking account. In September, she got promoted to a new position at work which raised her monthly salary by $1000. She moved to a better apartment with $300 higher rent, but saved the remaining $700 in her banking account. Is Anna’s behavior qualitatively consistent with the permanent income hypothesis? Explain why.
The Two-Period Model of Households [70]
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Consider a household who lives for two periods: the current period and the future period. The income of the household is y in the current period and y′ in the future period. In both periods, the household has to pay lump-sum taxes to the government. The tax liability is t for the current period and t′ for the future period. A credit market is available to the household, so the household can lend (save) or borrow in the current period with an interest rate r. The household will receive or pay back the principal plus interest in the future period. The amount of savings in the current period is denoted by s. (A negative s implies the household is borrowing.) The household’s preferences over the current consumption c and the future consumption c′ are represented by a utility function U(c,c′), which is both monotonic and concave.
Suppose that the government introduces a proportional consumption tax on the household in both the current and future periods, and the tax rate is x ∈ (0,1). That is, for each unit of consumption purchased by the household, the household needs to pay 1 + x. Therefore, if the household’s current consumption is c, the household needs to pay (1 + x)c in the current period. Similarly, if the household’s future consumption is c′, the household needs to pay (1 + x)c′ in the future period.
You need to take the consumption tax into account when answering all the subquestions of Question 2. It may or may not affect your answers.
- [3] What is the current period budget constraint of the household? 1
- [3] What is the future period budget constraint of the household?
- [6] Derive the household’s lifetime budget constraint from the two period budget constraints in the last question. You need to show the derivation steps. And in the end, you should put all the terms about expenditures on one side and all the terms about income on the other side.
- [6] Write down the household’s optimization problem. Be clear about the following: (1) the objective function; (2) the choice variables; (3) the constraints.
- [6] Based on the household budget constraints and assume they hold with equality:
- (a) If the household does not save or borrow (i.e. s = 0), how much can the household consume in the current period and in the future period? (This is the household’s en- dowment point.)
- (b) How much the household can consume in the current period (i.e. the value for c) if the household choose to consume zero in the future period (i.e. c′= 0)?
- (c) How much the household can consume in the future period (i.e. the value for c′) if the household choose to consume zero in the current period (i.e. c = 0)?
- [8] Assume there is an interior solution to the household’s problem. In the (c,c′) space, use indifference curves and the set of feasible choices to identify the solution to the household’s optimization problem, i.e., the optimal choice of the household. Label clearly the follow- ing: (1) the slope of the budget line; (2) the intercepts of the budget line on the vertical and horizontal axes; (3) the optimal choice point H.
- [6] Based on your graph in the last question, write down the two optimality conditions that must be satisfied at the household’s optimal choice. Briefly explain why they must be satisfied.
- [10] Suppose both the current and future consumption goods are normal goods, use income and/or substitution effects to discuss without any graphs what will happen to the house- hold’s current consumption c, future consumption c′, and savings s if there is only (a) an increase in future lump-sum tax t′; (b) a decrease in interest rate r, assuming the household is a lender;
- [10] Suppose both the current and future consumption goods are normal goods, and the sub- stitution effect is stronger than the income effect. In the (c,c′) space, show graphically the effects of a decrease in interest rate r (assuming that the household is a borrower) to the household’s optimal choice. If applicable, be clear about which part of the change is due to the income effect and which part is due to the substitution effect. Also, label clearly the location of the endowment point E in your graphs when it is relevant.
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For Question 10 to 11 only, let the utility function of the household be U(c,c′) = ln(c)+ln(c′).
To simplify the notations, assume that both the current and future lump-sum taxes are zero, i.e., t = t′ = 0 (and hence you can omit t and t′ in your derivations). Note that the consumption tax is still there.
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- [10] Assume that there is an interior solution to the household’s problem, solve the house- hold’s optimization problem with the consumption tax. That is, find the household’s optimal (i) current consumption c, (ii) future consumption c′, and (iii) savings s in terms of (y,y′,r,x). (Recall that 0 < x < 1 is the consumption tax rate.)
- [2] How does an increase of the consumption tax rate x affect the household current con- sumption c, future consumption c′, and savings s?
The Two-Period Model of Firms [20]
Consider a firm that produces the consumption good in two periods: the current period and the future period. The production function in the current period is
Y =zF(K,N)
where Y is the current output, z is the current total factor productivity, K is the current capital stock,
and N is the current labor hired. The production function in the future period is Y′ =z′F(K′,N′)
where Y ′ is the future output, z′ is the future total factor productivity, K′ is the future capital stock, and N′ is the future labor hired.
The firm is a price-taker, and the real wage of labor is w in the current period and w′ in the future period. The interest rate in the credit market is r.
The capital stock in the current period K is given (i.e., not chosen by the firm), but the firm can invest I units of output (i.e., the consumption good produced by the firm) in the current period to
′
K′ =(1−d)K+I.
At the end of the future period, all the capital left after the production (1 − d)K′ can be converted one-for-one back into the consumption good, which can be sold in the same way as the output produced.
Suppose that the government introduces a proportional corporate income tax on the firm in
both the current and future periods, and the tax rate is x ∈ (0, 1). That is, for each unit of profit
made by the firm, the government will take away x, and the firm is left with 1 − x. Therefore, if the
firm’s before-tax profit in the current period is π, the firm can only keep (1−x)π, and the remaining
increase its capital stock in the future period K . Capital depreciates after being used in production, and the depreciation rate is d. So the future capital stock is determined by the following law of motion for capital.
xπ of before-tax profit is taken away by the government. Similarly, if the firm’s before-tax profit ′′
in the future period is π , the firm can only keep (1−x)π .
You need to take the corporate income tax into account when answering all the subques-
tions of Question 3. It may or may not affect your answers.
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- [6] Write down the firm’s optimization problem with this corporate income tax. Be clear about the following: (1) the objective function; (2) the choice variables; (3) the constraints if any. You should show clearly that the objective function is a function of choice variables. For example, if applicable, you should replace the output Y and Y ′with the corresponding production functions.
For Question 2 to 4 only, let the production functions in the current and future periods be Y = zK0.5,
′ ′′0.5 Y=zK.
These production functions imply that no labor is used in the production, and hence there is no labor cost for the firm in either period.
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- [2] Write down the new objective function of the firm’s problem with these production func- tions and the corporate income tax.
- [10] Solve the firm’s problem with the these production functions and the corporate income tax. That is, find the firm’s optimal (i) investment I and (ii) future capital stock K′in terms of (z,z′,K,r,d,x). (Recall that 0 < x < 1 is the corporate income tax rate.)
- [2] How does an increase of the corporate income tax rate x affect the firm’s investment decision I and future capital stock K′?
Bonus Question [6]
(Warning: DO NOT try this question before you finish all the other parts of the exam.)
Consider the two-period model of the representative household as described in Section 2 (The Two-Period Model of Households). Suppose now, in addition to consumption and savings, the household can also spend some of its current income on education, which will increase the house- hold’s future income. In particular, if the household spends e (in units of the current consumption good) on education in the current period, the household’s income in the future period y′ is given by
′ eα y=A,
α
where α ∈ (0, 1) and A > 0 are parameters. Therefore, the higher is e, the higher is the household’s
′
an increase of the interest rate r affect the household’s education decision e? You need to justify your answers with details. (Hint: How does the education spending e affect the household’s set of feasible choices in the (c,c′) space?)