第十四章 深圳大学 宏观经济 学 期末考试复习资料

Chapter 14 Solutions to the Problems in the Textbook:
Conceptual Problems:
1. Even if the economy has achieved the desired capital stock some (gross) investment still must take
place to keep the capital stock at this level. The level of investment has to be sufficient to cover depreciation (due to wear and tear or because capital becomes obsolete).
2. High-tech capital (such as computers) becomes obsolete at a very fast rate and therefore needs to be
replaced much earlier if firms want to stay competitive. Therefore the rate of depreciation will increase if more is invested in high-tech machines.
Human capital also depreciates since knowledge tends to become outdated (new theories are advanced and new discoveries are made continuously). Thus knowledge needs to be updated. Who for example, wants to be treated by a physician who hasn't kept up with new advances in medical technology? Similarly, since one can also think of health as human capital, we can see that, as we grow older, our stock of health tends to depreciate. But the more we invest in health, that is, the healthier we live and the more preventive measures we take, the slower this stock of human capital will depreciate.
3. The interest rate cannot simply be considered as the rental cost of capital but is also an opportunity
cost. Retained earnings can be used to invest in new machinery but also to make a loan to someone else. In other words, at any time retained earnings can be "financially invested," that is, given to someone in need of funds (the government, for example), in which case these funds would earn interest. For example, if the yield on a government bond or a commercial paper is much higher than the expected rate of return on an investment in real capital, a firm may not want to undertake this investment and "invest" in a government bond.
4. The price of a share of stock in a company should, in an efficient stock market, be equal to the price
of a claim on the capital in the company. Tobin’s q is an estimate of the value the stock m arket places on a firm’s assets relative to the cost of producing those assets. In other words, it can be thought of as the ratio of the market value of a firm to the replacement cost of capital. The replacement cost of capital is a measure for the marginal cost of capital. If q is greater than 1, then a firm should add physical capital, since for each dollar’s worth of new machinery the firm can sell stock for q > 1 dollars. But this means that the marginal product of capital exceeds its marginal costs.
5. A sudden increase in the demand for a firm's product will increase expectations of future sales and
induce a firm to increase its desired capital stock. This will require an increase in net investment. The speed with which the capital stock is increased to its new desired level depends on whether the firm believes that the increase in sales is permanent or temporary and on the cost to the firm of adjusting the capital stock to its new desired level quickly.
6. Large firms have easier access to credit than small firms, since large firms tend to have an established
and good credit rating. Therefore, small firms are often limited in their investment opportunities by
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their retained earnings. But if we have many more small firms than large firms, we may observe larger output fluctuations over the business cycle. This is due to the fact that small firms will invest less in a recession than they otherwise might have because credit is not available to them and profits are down. On the other hand, in a boom these small firms are likely to invest more than they would otherwise, since profits are high and they want to compensate for the lack of investment during the past downturn.
7.a. When profits are high, more internal funds are available for firms. Firms generally will use these
funds for financing new capital investments, even at times when outside funding is not easily obtainable. Higher profits may come from increased sales. This may make entrepreneurs more optimistic about future sales and encourage them to increase investment spending. Higher profits also mean higher dividends and thus higher stock values. But if stock values are high, firms are more inclined to raise new funds for additional investment. It should be noted, however, that higher profits and the availability of internal funds should not distract from the fact that the cost of capital still is an important factor in investment decisions. Any interest that could be earned on such funds has to be seen as an opportunity cost.
7.b. Credit rationing by banks occurs since banks realize that while a cautious entrepreneur may be
deterred from investing when interest rates are high, a more reckless entrepreneur may still invest in spite of high rates. Entrepreneurs who are more reckless are also more likely to fail and default on their loans. Therefore in times of tight funds, rationing credit may be a more effective way for banks to ensure profits than increasing the interest rate on loans. Credit rationing may also occur for other reasons. For example, the Federal Reserve can impose credit limits on financial institutions.
8.a. Mortgage interest payments are an important consideration in buying a house. Even a small change in
mortgage interest rates can significantly affect homebuyers' monthly mortgage payments. Therefore most people will wait to buy a home until interest rates are fairly low. The purchase of a home is different from the purchase of a consumption good, since most people can delay purchasing a home for some time if market conditions are unfavorable. If interest rates are very low, many more prospective homebuyers will qualify for a mortgage. On the supply side, housing developers with large financing needs are more likely to undertake new construction if the cost of credit declines. But as the supply of new housing increases, the price for houses may drop, inducing more people to buy.
8.b. A state usury law prohibits banks or thrifts from charging mortgage rates above a certain maximum.
This in effect provides a price ceiling on mortgage rates. But when market interest rates go above this interest rate ceiling, the mortgage market cannot adjust to an equilibrium and there will be excess demand for mortgage funds. At such high interest rates banks may channel their funds away from mortgages and into other, higher yielding ventures. Thus the supply of mortgage credit may decrease.
If the inflation rate is high, the real interest rate that homebuyers actually pay may be quite low.
Therefore homebuyers may demand more mortgage funds, creating excess demand for mortgages while there is no opportunity for the market to clear due to the price ceiling. This explains why a low level of housing investment may exist even at low real interest rates.
9. The flexible accelerator model tries to explain the speed at which firms adjust their capital stock over
time. The larger the gap between the desired and actual capital stock is, the higher the level of the
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firm’s investment spending on machinery. Much, but not all, inventory investment can be explain ed with the help of this accelerator model. The level of inventory investment is based on changes in output and therefore sales expectations.
10. Unanticipated inventory accumulation occurs when the demand for a product is lower than was
expected. Firms generally respond by decreasing their level of production and a recession may be imminent. However, a planned inventory increase generally occurs when firms expect an economic upturn and want to be ready for the anticipated increase in the demand for their product.
11. The inventory-to-sales ratio did not increase in the 1990-91 recession, probably because of new and
better methods of management and firms keeping much tighter control over their inventories.
Advanced computer technology and the synchronization of shipments of material allow firms to operate with leaner, less costly inventories. Therefore the inventory-to-sales ratio is not only less likely to increase due to an undesired inventory increase at the beginning of a recession, but it is also less likely to increase due to a desired inventory increase at the beginning of a new boom. It appears that the role of inventory spending in a business cycle may have changed.
12. The level of net investment is the addition to the capital stock, that is, I N = ∆K. A low level of net
investment implies slow growth in the capital stock and hence future productive capacity. A low rate of capital accumulation generally implies lower future living standards.
13.In the long run (when the AS-curve is vertical), monetary policy will not affect the real interest rate.
However, it will affect inflation and therefore the nominal interest rate. The nominal interest rate (i n) is determined by the real interest rate (i r) plus the inflation rate (π), that is,
i n = i r + π
But if the nominal interest rate increases, the nominal cost of borrowing funds for investments rises and some borrowers may no longer qualify for loans because of perceived cash flow problems. Banks may actually limit credit when nominal interest rates increase, because they feel that borrowers still interested in loans may be high credit risks. This is particularly true for residential investment, since housing is very sensitive to real and nominal interest rates. One reason is that in the U.S. tax system nominal interest payments are tax-deductible, while nominal capital gains due to inflation remain untaxed. A higher rate of inflation may also have a negative impact on stock market values due to increased uncertainty about the future. But a decrease in stock values may make it more difficult for firms that want to raise funds for investment projects through issuing new stocks. Expectations about inflation also may affect the timing of investments. As a result, the level of investment may be affected by monetary policy despite the fact that real interest rates have not changed.
Technical Problems:
1. The rental cost of capital (r c) is equal to the real interest rate (r) plus the rate of depreciation (d), with
the real interest rate defined as the nominal interest rate (i) minus the expected inflation rate (πe):
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r c = = r + d = i - e+ d.
A car rental firm would want to know how fast its cars will depreciate (or what is needed to keep the
stock of cars at the original level) and what the costs are of having funds tied up in owning the cars.
In order to make a profit, a car rental firm would charge more per car than the interest it could get (or has to pay) on the funds tied up plus the depreciation rate times the value of the car. Since the rental firm charges a nominal price and is charged a nominal interest rate by a bank for any funds that are borrowed, the charge would be
P > (i + d)V, with V = the value of the car.
For example, if the current market interest rate is 10%, the rate of depreciation is 20%, and the value of the car is $10,000, then the car rental company would charge at least
(0.1 + 0.2)(10,000) = (0.3)(10,000) = 3,000
per year or roughly $8.22 dollars per day. If we further assume that the car is rented only half of the time, then the costs would go up to $16.44 per day. Since the $16.44 charge does not include any other costs to the firm or any profits, we should expect the actual price for a car rental to be higher. 2.a. If the net present discounted value (NPV) of a project is positive, the project is profitable. Assuming
that Year 1 is the present year, the net present value of the project can be calculated in the following way:
NPV = R1+ R2/(1 + r) + R3/(1 + r)2
At an interest rate of r = 5% we get
NPV = - 200 + 100/(1.05) + 120/(1.1025) = - 200 + 95.24 + 108.84 = 4.08 > 0.
This means that the project is profitable and should be undertaken.
2.b. Using the same equation as in 2.a. but with an interest rate of r = 10%, we get
NPV = - 200 + 100/(1.1) + 120/(1.21) = - 200 + 90.91 + 99.17 = - 9.92 < 0.
This means that the project is unprofitable and should not be undertaken.
3.a. A temporary investment tax credit should have a positive effect on investment for the period to which
it applies. In the long run, however, we should not see a significant effect on investment arising from
a temporary investment tax credit. The desired capital stock depends mainly on a firm's estimate of
future or permanent output and a temporary investment tax credit should not affect the firm's desired long-run capital stock.
3.b. A temporary investment tax credit will increase the number of current projects, since firms will invest
in projects that they otherwise might have delayed. Firms may also initiate some marginal projects
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that were unprofitable under previous conditions. Therefore the level of investment in the year that the tax credit is imposed will increase. Since so many projects are undertaken during the current year, fewer projects will be undertaken the following year, when the tax credit is no longer in effect. Thus the level of investment in the second year will be lower than it would have been otherwise.
3.c. While a permanent tax credit does not induce firms to accelerate investment projects, it may
encourage them to undertake marginal investment projects that might not previously have been profitable. We should therefore see an overall increase in the level of investment, both in the first year of the tax credit and beyond. The effect of a permanent tax credit on investment in the long run will be greater than that of a temporary tax credit.
4.a. The difference between output and final sales is a result of inventory adjustments. Before a recession,
actual sales may go down, while output (which is based on expected sales) is slow to respond.
Therefore we have an unanticipated inventory accumulation. This will cause firms to cut back their production, lowering output even more. On the other hand, if firms expect an increase in sales they will raise production to increase their inventories. This can increase output before sales pick up.
4.b. In 1981, GDP exceeded final sales, which meant that inventories increased. The initial inventory
increase may have been intentional as industries prepared for an upswing after the recession of 1980.
However, later in the year, as the economy headed into the recession of 1981/82, sales and GDP both decreased. Since GDP exceeded sales, an undesired inventory accumulation occurred. While output declined sharply, firms cut inventories. In the first quarter of 1982 there was a desired inventory decrease as sales exceeded output. In mid-1982, sales and GDP were almost equal, but by the fourth quarter sales started to increase faster than GDP. This meant that aggregate demand had picked up, a clear sign that the economy was recovering. This led to an undesired inventory decrease.
4.c. In a period of slow and steady growth, firms anticipate an increase in sales and want to be prepared
by holding a large stock of inventory. However, since the growth is slow and steady, we should expect output to exceed sales only slightly and both should grow at about the same rate. Therefore we may see a fairly constant inventory-to-sales ratio.
$ output
sales
0 time
5.a. The total value of the outstanding shares is $25 million. Therefore Tobin’s q has the following valu e:
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q = (value of all shares)/(replacement cost of capital) = 25/18 > 1.
But a value of q that is greater than 1 implies that the value of the marginal product of capital is greater than its marginal cost. Therefore the firm should invest.
5.b. With a replacement cost of $25 million, q = 1. Therefore net investment should be zero. But there
should still be some gross investment to replace the capital stock that depreciates.
With a replacement cost of $28 million, q < 1. In this case, the firm should “disinvest,” that is, let its capital stock depreciate but not replace it.
6.a. From K* = (θY)/r c ==> K* = [(0.3)(5)]/(0.12) = 12.5
==> the desired capital stock is valued at $12.5 trillion.
6.b. From K* = [(0.3)(6)]/(0.12) = 15
==> the desired capital stock is now valued at $15 trillion.
6.c. From I1 = λ(K* - K) ==> I1 = (0.4)(15 - 12.5) = (0.4)(2.5) = 1
==> in the first year, net investment will be I1 = $1 trillion, and the new capital stock will be K1 = $13.5 trillion.
From I2 = (K* - K) ==> I2 = (0.4)(15 - 13.5) = 0.6
==> in the second year, net investment will be I2 =$0.6 trillion and the new capital stock will be K2 = $14.1 trillion.
6.d. As indicated above, the answers in 6.c. refer to net investment, that is, an addition to the capital stock.
Gross investment would include replacement of worn out capital.
7. The q-theory of investment predicts that high stock values will induce corporate managers to invest
more in real capital. In an efficient stock market, the price of a share of stock in a company should be equal to the price of a claim on the capital in the company. Tobin’s q can be thought of as the ratio of the market value of a firm to the replacement cost of capital, since it is an estimate of the value the stock market places on a firm’s assets relative to the cost of producing those assets. If the value of q is high, a firm wants to add physical capital and the level of investment rises. But this means that in periods in which stock prices rise rapidly, corporations will increase their investment spending.
Additional Problems:
1. Explain why the stock of existing housing would decline if no more new houses were built.
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There will always be some depreciation, as existing houses burn or are torn down. If no new houses are built (gross investment is zero), the housing stock will decline (net investment will be negative).
2. Low interest rates should encourage firms to invest. So why did the U.S. have a low level of
investment spending during the 1930s when interest rates were very low?
During the 1930s the economy was in a deep recession. Therefore, in spite of low interest rates, businesses were reluctant to invest. The level of investment depends not only on the interest rate but also on changes in income, that is, the sales expectations of businesses. In the 1930s, most businesses did not expect sales to increase.
3. "Firms will undertake investments as long as the value of the marginal product of capital is
below the rental cost of capital." Comment on this statement.
If this were true, then the last unit of capital employed would cost more than the value of the increase in output. Competitive firms minimize costs when the rental cost of capital is equal to the value of additional output produced as one more unit of capital is added. Otherwise it would pay the firm to use more (if the value of the marginal product of capital exceeded the rental rate) or less (if the value of the marginal product of capital were less than the rental rate) capital in the production process.
4. "Investment is pro-cyclical in the flexible accelerator model." Comment on this statement. Assuming away depreciation and the real time required to make adjustments to the capital stock, the desired capital stock (K*) is proportional to real output, where a is the capital-output ratio, that is, K*t= aY t .
If investment makes up part of the difference between the actual and the desired capital stock, then
I t = λ[K*t - K t-1] ==> I t = λa[Y t - Y t-1].
Therefore, we can see that if output is growing, the level of investment is positive. But if output is falling, investment is negative. This means that investment in the accelerator model is pro-cyclical, that is, it follows output. Note, however, that as long as output is increasing at an increasing rate, investment will increase. On the other hand, if the increases in output become smaller, the level of investment will decline.
5. Why is the level of investment inversely related to the level of the interest rate? Can you think
of any consumer spending that may also depend on the interest rate? Explain your answer. When calculating the present discounted value of an investment, the interest rate is in the denominator. Thus a decrease in the interest rate increases the net present value of an investment. As more investment projects become profitable, the level of investment rises. Also, the neoclassical theory predicts that the desired capital stock increases as the interest rate, and therefore the rental cost of capital decreases, that is,
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K* = Y/r c.
Durable consumption goods, such as cars and appliances, are often purchased on credit. Therefore, spending on durable goods is influenced by interest rate changes, which affect the overall costs of purchasing these goods. But, as we have seen in the previous chapter, one can interpret the purchase of a durable good as an “investment.”
6. "Restrictive monetary policy has a negative effect on investment." Comment on this statement. If inflationary expectations are held constant, monetary restriction will raise both real and nominal interest rates, at least in the short run. Since the level of investment is inversely related to the real interest rate, investment will fall. Business fixed investment and housing investment will fall as the returns generated by projects have to compete with the much higher costs of funds. Inventory investment will also fall as firms find it more expensive to warehouse raw materials and finished goods.
In the long run, restrictive monetary policy will not have any effect on real interest rates, but will lower nominal interest rates due to a lower inflation rate. When nominal interest rates are lower, more people may qualify for mortgage loans and therefore housing investment will increases. A lower inflation rate may also have a positive impact on stock values. But higher stock values may enable more corporations to issue new stocks to finance more investment projects. Thus monetary policy may affect investment even if it does not affect the real interest rate.
7. Which of the following two policy measures would affect GDP more and why: a temporary
investment tax credit for businesses or a temporary income tax cut for individuals? Explain your answer.
A temporary investment tax credit should have a positive effect on investment for the period in which it applies. Firms may decide to accelerate projects that they would have undertaken later. They also may initiate some marginal projects that would otherwise not have been profitable. But in the long run, we should not see a significant effect on investment from a temporary investment tax credit. On the other hand, if the government temporarily decreases income taxes, consumption will not be significantly affected according to the permanent income hypothesis (unless liquidity constraints exist). Therefore the economy will not be stimulated at all by a temporary income tax cut.
8. Which would stimulate the economy more: a $50 billion tax cut financed by a $50 billion
spending cut, or a temporary tax credit of 15% on each productive investment undertaken next year? Explain your answer.
According to the balanced budget theorem, a decrease in taxes and government spending by $50 billion would actually decrease national income. A temporary investment tax credit, which has a positive effect on the economy, is therefore much more beneficial. A temporary tax credit would make marginal investment projects profitable. Some firms would also undertake projects earlier to take advantage of the tax credit. Temporary investment tax credits can be used successfully to stimulate economic activity.
9. “Firms are more likely to invest if current GDP increases.” Comment on this statement.
The level of desired capital stock generally depends on the level of expected future (or permanent) output. But if the desired capital stock is above the actual capital stock, firms will invest. Entrepreneurs will
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invest more if they feel optimistic about the future and expect their sales to increase. If they are pessimistic about the future and have low sales expectations, they will invest relatively little. Current output affects capital demand only to the extent that it affects expectations about future output.
10. For a given nominal interest rate, how does an increase in the expected rate of inflation affect
the level of investment?
The rental cost of capital is defined as r c = i - πe + d. This means that an increase in the expected rate of inflation (πe) decreases the rental cost of capital (r c). But a decrease in the rental cost of capital increases the level of investment, as more investment projects become profitable. A decrease in r c also increases the desired capital stock K* = (θY)/r c and therefore encourages investment.
11. Comment on the following statement:
“The discounted cash flow analysis is inconsistent with the neoclassical theory of investment.”The neoclassical theory of investment states that the level of desired capital stock increases with an increase in expected output (sales) and a decrease in the rental cost of capital. In practice, firms base their investment decisions on the discounted cash flow analysis. An investment project is evaluated by calculating its net present discounted value. If a firm expects an increase in sales, the cash flow associated with any investment project will be larger and the net present discounted value of the investment will rise. At the same time, if the rental cost of capital decreases, then the net present discounted value of the project increases and the firm is more likely to undertake the investment. The decision making of firms using this discounted cash flow analysis is therefore consistent with the neoclassical theory of investment.
12. Assume your firm buys a new computer system at a cost of $12,600 that will save you $5,600
after one year, $4,840 more after the second year and another $4,000 after the third year. Then the computer will become outdated and no further saving will accrue. Is this a worthwhile investment if we assume that there is no inflation and that the market rate of interest remains at
i = 10% over these three years?
The net present discounted value of your investment is
NPV = - 12,600 + 6,600/(1+ 0.1) 1 + 4,840/(1+ 0.1)2 + 4,000/(1+ 0.1)3
= - 12,600 + 6,000 + 4,000 + 3,000 = + 400 > 0.
Since the net present discounted value of your investment is positive, it is profitable and you should undertake this investment.
13.Evaluate the following project, assuming that the market interest rate remains at i = 10%.
Year 0 Year 1 Year 2 Year 3
costs $3,700 $605 $363 $0
revenues $0 $2,640 $2,420 $400
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The investment should be undertaken, since the net present discounted value of this investment is positive. It can be calculated as follows:
NPV = -3,700 + (2,640 – 605)/(1 + .1)1 + (2,420 - 363)/(1 + .1)2 + 400/(1 + .1)3
= - 3,700 + 1,850 + 1,700 + 300 = 150 > 0
Since NPV > 0, this investment is profitable and should be undertaken.
14. Would you undertake the following investment project? Why or why not?
Year 0 Year 1 Year 2 Year 3 costs $3,993 $4,840 $3,630 $0
revenues $0 $3,630 $4,840 $3,993
This problem does not list an interest rate so, under normal circumstances, we cannot solve the problem. But if we simply add up the costs and the benefits, we can see that both sums are equal. Since the costs occur earlier than the benefits, we can conclude that the project is unprofitable, since we could have invested the funds we spent at any positive interest rate. For example, if the market interest rate was i = 10%, then the net present value would be:
NPV = - 3,993 + (3,630 - 4,840)/1.1 + (4,840 - 3,630)/1.21 + 3,993/1.331
= - 3,993 - 1,100 + 1,000 + 3,000 = - 1,093
Since NPV < 0, this investment is not profitable and should not be undertaken.
The investment project is unprofitable at any positive real interest rate. But it is conceivable that the real interest rate is actually negative, if the rate of inflation is greater than the nominal interest rate. In this case, the project would be profitable. At high rates of inflation it makes sense to borrow money and pay it back later with money that has lost some of its purchasing power.
15. "Credit rationing reinforces monetary policies." Comment on this statement.
Credit is rationed when lending institutions limit the amount that firms or consumers can borrow. Tight money policy raises the cost of borrowing and banks may be concerned that irresponsible borrowers will continue to borrow at the higher rates while more conservative borrowers will be deterred. To avoid the higher risks incurred with irresponsible customers, banks often limit the amount of credit given to any one customer. As fewer funds are made available, less investment takes place. Thus the rationing of credit reinforces restrictive monetary policy.
However, in the early 1990s the Fed tried to stimulate the U.S. economy by repeatedly lowering the discount rate. Banks nonetheless chose to increase their holdings of Treasury bills rather than to expand their lending, since they were still recovering from huge losses encountered in the real estate market. In this case, the banks, in effect, rationed credit and worked against the expansionary monetary policy of the Fed.
16. Is inventory investment pro-cyclical? Why or why not?
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