### How income may change savings behavior

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Question 1: The lecture described how taxing income may change savings behavior. Suppose instead that the government taxed consumption.

To be specific, suppose we have a two-period model. An individual earns labor income Y0 =\$100k at time zero, and earns no labor income at time 1. The individual may consume or save that income. Savings grow at rate r=.03. For every dollar of consumption, the individual pays the tax rate τ=.30 to the government.

a. Graph the two-period budget constraint for consumption. What is the slope? Is this tax distortionary?

b. The government modifies the consumption tax somewhat so that the first \$20k of consumption in each period is tax free.  Now graph the budget constraint.

Question 2: Claim: "The Mortgage Interest Tax Deduction is a regressive policy.  A simpler, better policy that could achieve the same goals would be a home ownership tax credit that applies equally to all homeowners regardless of income or the value of the home."  In a mini-essay (300 words) state whether you agree or disagree with that claim and explain your reasoning.

Question 3: In lecture, we assumed that when a homeowner borrows, the entire the value of a home would be borrowed.  In fact, a borrower would need to put a "down payment" on the mortgage - a cash payment up front for part of the value of the home.  An additional simplification of the lecture was to not take into account the fact that for a mortgage you pay the interest + some fraction of the principal.  This is so at the end of the loan all the money borrowed will have been repaid. Take a look at the following mortgage calculator linked below to help answer the following.

The value of the property is \$500k, the interest rate is 3% (approximately the correct interest rate as of this writing), and the length of the loan is 30 years (360 months).   The marginal tax rate for the homeowner is 33%. Leave other values on the table at the default settings.  Assume the individual has \$500k cash on hand, and any of this money that remains after taking the mortgage/making mortgage payments is invested at the interest rate 3%.  The value of the property also grows at rate 3% per year, and this growth is not taxed.  This means that as the borrower repays the loan and starts building principal, the value of that principal goes up at the same rate as other investments.  Finally, assume that the year's mortgage payment is paid to the bank from cash on hand at the START of the year.  This turns out to be important if we want to make comparisons.

i. Suppose there is no MITD, and the homeowner borrows the full value of the property. For the first year:

a) How large is the annual mortgage payment? How much interest has been paid on the mortgage?

b) How much principal has been accumulated by the borrower?  What is the value of the principal at the end of the year?

c) The amount of the annual mortgage payment from part a) was paid at the beginning of the year.  That reduces the cash available to invest.  How much cash gets invested? What is the pre-tax value of the cash investment at the end of the year?  How much tax is owed on this investment?  What is the after-tax value of the investment after one year?

d) Add up the values of all the investor's assets at the end of year one.   How has this value changed over the year?

ii. Suppose there is a MITD, the homeowner borrows the full value of the property.

iii. Not surprisingly, you hopefully saw in ii that the deduction is a boon to the homebuyer.  Now, suppose the buyer makes a down payment of 20%, or \$100k.  Assume the MITD is not available. She invests the remaining cash at 3%.  Repeat a-d from part i. in this case.

iv. Once again, suppose the buyer makes a down payment of 20%.  Assume the MITD is available.

v. Comment on/compare your results for the different cases.

Question 4. A corporation produces output with a market price of \$200 per unit.  The marginal product of capital is 1/(2K), where K is units of capital, with each unit assumed to cost \$1.  (So when we talk about capital in this problem, units and \$ value are equivalent.)  The life span of the capital is 5 years, implying the straight line depreciation rate δ=.2.  The financing cost of capital is ρ=.05.

a. If depreciation and financing costs are not included in accounting costs, what is the optimal level of capital for the firm?

b. If the corporate tax is 35%, what is the optimal level of capital?

c. If depreciation at a rate δ=.2 is included in accounting costs, what is the optimal level of capital? [Hint: remember to calculate the present value of the deduction.  Use .05 for the discount rate.]

d. For c., what is the effective corporate tax rate?

e. The firm is going to borrow the money for its capital purchases.  The interest paid on the debt can be added to accounting costs.  Suppose it turns out that the present value of this expense is .10 for every dollar of capital purchased.  What is the optimal level of capital now?

f. For e., what is the effective corporate tax rate?

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