1. Ventilators cost $40,000 each. The local hospital is trying to decide how many of these machines to buy. Local hospital expects to save the following numbers of lives each year for each level of capital stock shown.  Wow!  Economics can be cold.  Saving lives can be looked at like production, but it is our method.

                Number of Ventilators                                                                 Number of Saved People

0                                                                                                                                                                     0

1                                                                                                                                                                     100

2                                                                                                                                                                     150

3                                                                                                                                                                     190

4                                                                                                                                                                     220

5                                                                                                                                                                     240

6                                                                                                                                                                     250

Now is the very difficult part for an economist and the hospital.  I need the value of saving the life.  Obviously hospitals charge a cost plus markup because they are nonprofit.  A court case would value a life at expected lifetime earnings.  I digress and this is just a practice exercise so a saved person will be valued at $150.

a.       Find the expected future marginal product of capital (in terms of dollars) for each level of capital. The MPK^f for the third ventilator, for example, is the real value of the extra "output" obtained when the third fabricator is added

b.      If the real interest rate is .001% per year and the depreciation rate of capital is .001% per year, find the user cost of capital (in dollars per ventilators per year). How many ventilators should the hospital buy?

c.       Repeat Part (b) for a real interest rate of .002% per year

d.      A technical innovation doubles the number of lives saved. How many ventilators should the hospital buy buy when the real interest rate is .001% per year?

2. An economy has full-employment output of 6000. Government purchases, G, are 1200. Desired consumptions and desired investment are

                                C^d =3600 – 4000r + 0.10Y, and

                                I^d = 1200 – 8000r,

Where Y is output and r is the real interest rate.

a.       Find an equation relating desired national saving, S^d, to r and Y

b.      Find the real interest rate that clears the good market. Assume that output equals full-employment output.

c.       Government purchases rise to 1500. How does this increase change the equation describing desired national saving? Show the change graphically. What happens to the market-clearing real interest rate?

3. Suppose that the expected future marginal product of capital is MPK^f = 20 – 0.02K, where K is the future capital stock. The depreciation rate of capital, d, is 20% per period. The current capital stock is 900 units of capital. The price of a unit of capital is 1 unit of output. Firms pay taxes equal to 50% of their output. The consumption function in the economy is C= 100 + 0.5Y-200r, where C is consumption, Y is output, and r is the real interest rate. Government purchases equal 200, and full-employment output is 1000.

                a. suppose that the real interest rate is 10% per period. What are the values of the tax-adjusted user cost of capital, the desired future capital stock, and the   desired level of investment?

                b. Now consider the real interest rate determined by goods market equilibrium. This part of the problem will guide you to this interest rate.

                                i. Write the tax-adjusted user cost of capital as a function of the real interest rate r. also write the desired future capital stock and desired investment as functions of r.

                                ii. Use the investment function derived in Part (i) along with the consumption function and government purchases, to calculate the real interest rate that clears the goods market. What are the goods market-clearing values of consumption, saving, and investment? What are the tax-adjusted user cost of capital and the desired capital stock in this equilibrium?

4.  Use the saving-investment diagram to analyze the effects of the following on national saving, investment, and the real interest rate. Explain your reasoning.

a.       Consumers, scared over the affects of the coronavirus, decide to self-quarantine, consume less, and, therefore, save more.

b.      The government announces a $1200 coronavirus payment. Let us assume this payment will be financed by additional taxes levied on the wealthiest Americans.

c.       The government introduces an investment tax credit (offset by other types of taxes, so total tax collections remain unchanged).

d.      The discovery of a vaccine for the corona virus also advances cancer research thereby increasing the marginal product of labor.  (This is not just an attempt to make you feel good.  The current reports indicate that must young people do not die of the virus, but rather how their immune system fights the virus.  I can see a small possibility of a spill over in research.)