14.02 Principles of Macroeconomics Problem Set 6 Solutions Fall 2004

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1 Part I. True/False/Uncertain Justif our answer with a short argument. 4.0 Principles of Macroeconomics Problem Set 6 Solutions Fall 004. In an econom with technological progress, the saving rate is irrelevant in the long-run. False. It is true that the rate of capital and output growth in stead state is independent of the saving rate. However, the saving rate does affect the stead-state level of output per effective worer. Suppose there is an increase in the saving rate. Then, capital per effective worer and output per effective worer will increase for some time as the converge to their new higher level. Their growth rate will eventuall return to the sum of the population growth rate and the technological growth rate (g + g. (See pages The rate of technological progress declined in the mid-970s due to the rise of the service sector. Uncertain. Wh the rate of technological progress declined in the mid-970s is one of the unanswered questions in economics. The explanation that involves the rise of the service sector is onl one of a few different hpotheses that have been proposed. Other explanations include decreased R&D spending (a decline not in the amount, but rather in the fertilit of R&D or measurement error (that, in fact, there was no slowdown in the rate of technological progress at all. (See pages There is much theoretical and empirical support for the idea that faster productivit growth leads to higher unemploment. False. In the short-run, there is no reason to expect, nor does there appear to be, a sstematic relationship between movements in productivit growth and movements in unemploment. In the medium- /long-run, if there is a relation between productivit growth and unemploment, it appears to be an inverse relationship. Lower productivit growth leads to higher unemploment and vice versa. (See pages Increased globalization and technological progress can both be blamed, at least to some extent, for the increase in wage inequalit in the United States. True. n increase in globalization leads to an increase in international trade. Those firms that emplo higher proportions of low-silled worers are increasingl driven out of the maret b imports from similar firms in low-wage countries, according to this argument. The other line of argument focuses on sill-biased technological progress. ew machines and new methods of production require higher-silled worers. (See pages 80-8.

2 5. The Fisher hpothesis states that, in the short-run, the nominal interest rate increases one for one with inflation, while the real interest rate remains unchanged. False. In the short-run, the increase in nominal mone growth leads to an increase in the real mone stoc. This translates into a decrease in the real, as well as in the nominal, interest rate. (See page 30. The Fisher hpothesis states that in the medium-/long-run the nominal interest rate increases one for one with inflation, while the real interest rate remains unchanged. Part II. The Solow Model Revisited The Republic of Solowaia has the following production function: α α Y = F( K, = K, where α<. Let g be the growth rate of, g be the growth rate of, and δ be the rate of depreciation in this econom.. Interpret the parameter. Empiricall, what has happened to over time? is a technological parameter. It tells us how much output can be produced from given amounts of capital and labor at an time. (See page 44. lthough the boo does not directl provide us with data on the parameter, we see that the rate of growth of (the rate of technological progress given in Table - has been positive at least for ears for France, German, Japan, the UK, and the US. Over a longer period of time, the increase in has been even more dramatic, taing into account the gricultural Revolution and the Industrial Revolution.. Verif that the above production function has the propert of constant returns to scale. We must verif that f(λk, λ = λf(k,, that is, if ou multipl all inputs b a scalar, ou will end up multipling output b the same amount. f(λk, λ =(λk α (λ α =λ α+ α K α α =λk α α = λf(k,. 3. Verif that the above production function is concave in capital (which means that the second derivative is negative. What does that mean in economics terms? For this we must tae the partial derivative of F(. with respect to K. F( K, α α = αk K F( K, α α = α( α K < 0 because α<. Therefore, F(. is concave in K. K This means that this production function exhibits diminishing returns to capital (increases in capital lead to smaller and smaller increases in output as the level of capital increases. 4. What is effective labor in this econom? Let s rewrite the production function in the following fashion:

3 α α ( K, = K ( α F Let s redefine the technological parameter as is just a constant in this model. Then, effective labor is. α =. (ote that we can do that, because 5. t what rate do output and capital grow in Solowaia in the long run? (Hint: use g for the transformed g. We will first calculate the growth rates mathematicall (ou didn t have to do this. Let s α α start with the transformed production function, F( K, = K ( and tae logarithms of both sides: logy = α log K + ( α log + ( α log Taing the time derivative of the above equation, we get the following expression: logy log K log log = α + ( α + ( α logy log But is the growth rate of output (Y, g Y ; is the growth rate of, g ; log K log is the growth rate of capital (K, g K ; and is the population growth rate, g. So, we have the following decomposition of output growth into growth driven b inputs and growth driven b technolog: g = α g + ( α( g + g Y K Because in stead-state we are on a balanced-growth path, we now that g Y must equal g K. So we have, g = α g + ( α( g + g Y Y Y ( α = ( α( g Y = g g K = g g g + g g + g + Output and capital both grow at the rate g + g. Intuitivel, this is because the growth rate of effective labor equals g + g. It cannot be g + g, because we have transformed the parameter. In the long run stead state, in this econom, what is constant is not output but rather output per effective worer. This implies that output grows at the same rate as effective labor. Same logic applies to capital. (See pages

4 6. Rewrite the production function in terms of onl capital per effective worer. α α α K K F ( K, = = K Y Let = and =. Define f so that f ( F(,. Then, = f( = α. 7. Draw the diagram that plots required investment and investment for this econom. Label the stead-state level of capital per effective worer *. Label the stead state as point on the diagram. Required Investment ( δ + g + g Investment s α *

5 8. Suppose the rate of population growths falls. Use a diagram to analze what happens in Solowaia. Label the new stead state capital per effective worer *. Label the new stead state as point B. δ + g + g ( ( δ + g + g B s α * * s g decreases (g <g, less capital is required per effective worer, and therefore we achieve a higher stead state capital per effective worer *. 9. Starting at the stead state in part 7 (point, suppose the saving rate increases and the depreciation rate decreases at the same time. Use a diagram to analze what happens in Solowaia. Label the new stead state capital per effective worer 3 *. Label the new stead state as point C. ( δ + g + g δ + g + g C ( s α s α * 3 * Here we have a double shift. higher saving rate shifts the investment relation up. It follows that the stead-state level of capital per effective worer increases. lower depreciation rate shifts out the required investment relation, which also leads to a higher stead-state capital per effective worer.

6 0. Starting at the stead state in part 7 (point, suppose a neighboring countr gives Solowaia a gift of new capital equipment. Use a diagram to analze what happens in the long run. Label the new stead state capital per effective worer 4 *. Label the new stead state as point D. If we were given a gift of capital, capital would (immediatel jump up to a level such as in the short-run. ( δ + g + g s α * However, in the long run, the dnamics will bring the econom bac to the original steadstate level of capital per effective worer, *. Wh? When we receive the gift of capital, we =D ( δ + g + g D s α OD end up at point D, where investment per effective worer equals the vertical distance OD. The amount of investment required to maintain that level of capital per effective worer is clearl greater than the amount OD. (ote that for all > * the line ( δ + g + g is above the curve s α. Because investment that is * = 4 * required to maintain the existing level of capital per effective worer exceeds actual investment at D, decreases. Hence, starting from, the econom moves to the left, with the level of capital per effective worer decreasing over time. This continues until investment per effective worer is just sufficient to maintain the existing level of capital per effective worer, that is until we return to the initial stead-state,. (See page 48. So, the effect of the gift will onl be temporar (because nothing happened to investment or required investment.

7 Part III. Expected Present Discounted Values. Suppose that the interest rate is 5% toda and is expected to sta at 5% for the next three ears. Calculate the present discounted value of a securit that pas $0,500 in one ear, $,05 in two ears, $3,5.5 in three ears, and nothing thereafter. PDV 0,500,05 = ( ,5.5 + ( = 3 40,000 It is not 0,500 +,05/.05 + because the securit pas $0,500 not toda, but a ear from now. ote also that the fact that this securit stops maing paments after the first three ears means that its maturit is 3 ears.. Suppose the securit in part sells for $38,34.08 toda. Would its discount rate need to be smaller or higher than 5%? nswer without computing it first. Then calculate the discount rate. ssume again that the interest rate is expected to remain the same for the next three ears. If the securit is sold at $38,34.08 (which is less than $40,000, the discount rate is higher than 5%. Recall that the price of a bond (which is basicall the same as present discounted value and the interest rate move in opposite directions. 0,500,05 3,5.5 $ V = i ( + i ( + i i=7% = $38,34.08 ote that this discount rate is called ield to maturit. It is the rate of return on cash flow of a securit that is held until it matures (3 ears. 3. What is the constant pament required to achieve the same present discounted value over three ears as in part (i.e. $38,34.08, at the same constant interest rate as ou found in part? $ 38,34.08 $ z ( $ z 4, ( = 3 4. Suppose that the securit pas the annual pament ou calculated in part 3 forever at the constant interest rate from part. Calculate the present discounted value of this securit. 4,60.3 $ V = = $08,