The short run at a normal rate,and
Let's now turn to dynamics. Suppose that the economy is initially at its medium-run equilibrium: unemployment is at
unemployment is equal to the natural rate. Output growth is equal to the normal growth rate. The he naturr'1 rue of
. n . i . i i i i unemployment? To inflation rate is equal to the central banks target rate
answer, we need
Suppose, however, that the inllation rate and, by implication, the rate of nominal money growth are to discuss the costs of
high, and there is a consensus that inflation must be reduced. The central bank decides that the inflation inflation.We will do so
target must be lowered to 77"''. The question we now take up is what will happen along the way to the < in Chapters 24 ard 26. new medium-run equilibrium?
Just by looking at our three relations, we can tell the beginning of the story:
• Look at the dynamic aggregate demand relation: given the initial rate of inflation, a lower target " inflation rate leads to a higher interest rate, and thus to a decrease in output growth.
• Now look at Okun's law: output growth below normal leads to an increase in unemployment. 4 g. 1 => u
• Now look at the Phillips curve relation: unemployment above the natural rate leads to a decrease in % t =» n- i inflation.
So we have our first result. Tighter monetary policy (in the form of a lower inflation target) leads initially to lower output growth and lower inflation. If tight enough, it may lead to negative output growth and thus to a recession. What happens between this initial response after one year and the medium run (when unemployment returns to the natural rate)? The answer will depend on the path of monetary policy.
We can actually go a bit further in thinking about what happens to the economy over time:
• Look at the Phillips curve relation: so long as unemployment remains above the natural rate, inflation keeps decreasing. So, eventually, inflation approaches the lower inflation target, and as that
happens the central bank will be able to reduce the interest rate. * и > u„=> тг -l
You should remember three basic lacts about growth in rich countries since 1950:
• the large increase in the standard of living
• the decrease in growth since the mid-1970s
c. Suppose that the exchange rate is 0.1 (A$0.10 per kina i. Calculate PNG's consumption per capita in dollars.
d. Using the purchasing power parity method and Australian prices, calculate PNG consumption per capita in dollars.
4. Consider the production function Y = \ К \ К a. Calculate output when К - 49 and N = 81.
h. If both capital and labour double, what happens to output?
c. Is this production lunction characterised by constant returns to scale? Explain.
d. Write this production function as a relation between output per worker and capital per worker, c. Let K/N = 4. What s Y/W Now double K/N to 8. Does Y/N more than or less than double?
f. Does the relation between output per worker and capital per worker exhibit constant returns to scale?
g. Is your answer in (f) the same as your answer in i с)? Why or why not?
h. Plot the relation between output per worker and capital per worker. Does it have the same general shape as the relation in Figure 10.5? Explain.
Dig deeper
5. The growth rates of capital and output
Consider the production function given in problem 4. Assume that N is constant and equal to I. Note that if z = xa, then g. = a gx, where g- and gx are the growth rates of z and x.
a. Given the growth approximation here, derive the relation between the growth rate ol output and the growth rale of capital.
b. Suppose we want io achieve output growth equal to 2 per cent a year. What is the required rate of growth ol capital?
c. In (b), what happens to the ratio ol capital to output over time?
d. Is il possible to sustain output growth of 2 per ccnt lorever in this economy? Why or why not?
6. Between 1950 and 1973, Germany and japan experienced growth rates that were at least two percentage points higher than those in the United States. Yet the most important technological advances of that period were made in the United States. How can this be?
Explore further
7. In Table 10.1 we saw that the levels of output per capita in Australia, the United Kingdom, Germany, Trance, lapan and the United States were generally much closer to each other m 2004 than they were in 1950. llere we will examine convergence for another set of countries.
Go to the web address containing the Penn World Tables (see Table 10.1 and the focus box on the construction of PPP numbers) (http://pwt.econ.upenn.edu). Download the PW'I'6.2 bundle and
• Brad deLong, an economist at the Llniversity of California at Berkeley, has several fascinating articles on growth on his web page (www.j-bradford-delong.net). Read, in particular, Berkeley laculty lunch talk: Main themes ol twentieth century economic history, which covers many of the topics ol this chapter.
• A broad presentation ot tacts about growth is given by Angus Maddison in The World Economy. A Mill ennium Perspective Paris: OF.CD, 2001 ). The associated site, , has a large number of facts and data on growth over the last two millennia.
• Chapter 3 in Productivity and American Leadership, by William Baumol, Sue Anne Batev Blackman and Fdward Wolll (Cambridge. MA: MIT Press, 1989), gives a vivid description ot how life has been translormed by growth in the Llnited States since the mid- 1880s.
At the centre ol the determination of output in the long run are two relations between output and capital:
• The amount of capital determines the amount ol output being produced.
We have derived two relations:
• I ron the production side, equation (11.1) shows how capital determines output.
• I ron the saving side, equation (11.2) shows how output in turn determines capital accumulation.
When capital and output are low, investment exceeds depreciation and capital increases. When capital and output are high, investment is less than depreciation and capital decreases.
ol capital per worker, investment increases by less and less, while depreciation keeps increasing in pro¬portion to capital. For some level ol capital per worker, K*/N in Figure I 1.2, investment is iust enough to cover depreciation and so capital per worker remains constant. Го the left ot K*/N, investment exceeds depreciation and capital per worker increases. This is indicated by the arrows pointing to the right along the curve representing the production lunction. To the right ol K*/N, depreciation exceeds investment and capital per worker decreases. This is indicated by the arrows pointing to the left along the curve representing the production function.
Characterising the evolution ol capital per worker and output per worker over time is now easy. Consider an economy that starts with a low level ot capital per worker—say, KjN in Figure I 1.2. Because investment exceeds depreciation, capital per worker increases. And because output moves with capital, output per worker increases as well. Capital per worker eventually reaches К * / \ the level at which investment is equal to depreciation. Once the economy has reached the level ol capital per worker K*/S. output per worker and capital per worker remain constant at Y*/N and K*/N, their long- run equilibrium levels.
I hink tor example, ot a country that loses part ot its capital stock as a result ol bombing during a war. The mechanism we have iust seen suggests that if it has suffered much larger capital losses than population losses, it will come out ol the war with a low level ol capital per worker, so at a point to the left ol K*/S. The country will then experience a large increase in both capital per worker and output per worker for some lime. This appears to describe well what happened alter World War II to countries thai had proportionately larger destructions of capital than of human lives (see the locus box Capital accumulation and growth in France in the aftermath ot World War IF).
If a country starts instead from a high level ot capital per worker, from a point to the right ot K*/N, then depreciation will exceed investment and capital per worker and output per worker will decrease: the initial level ol capital per worker cannot he sustained given the saving rate. This decrease in capital per worker will continue until the economy again reaches the point where investment is equal to depreciation where capital per worker is equal to K*/N. From then on, capital per worker and output per worker will remain constant.
Steady-state capital and output
Lets look more closely at the levels ol output per worker and capital per worker to which the economy converges in the long run. I he state in which output per worker and capital per worker are no longer
1 in steady state hy definition, the change in capital per worker is zero thc steadv-state value ol capital per worker, K*/.X is given by
к*
(И.4)
Thc sleady-state value ot capital per worker is such that the amount of saving per worker (the left side) is just sullicieni to cover depreciation of the capital slock per worker (the right side1.
Given stcadv-siaic capital per worker (K*/N), the steady-state value ol output per worker (Y*/N > is given by the production function
• saving rate. 251
• steady state, 257
• golden-rule level ot capital. 261
• age pension, 262
• ageing population. 262
• pay-as-you-go social security system. 262
In 1928. Charles Cobb (a mathematician) and Paul Douglas (an economist, who went on to become a US senator) concluded that the following production function gave a very good description of the relation between output, physical capital and labour in the United States from 1899 to 1922:
Y = K" N]
with n being a number between zero and I.Their findings proved surprisingly robust. Even today, the production function (11 A.I), now known as the Cobb-Douglas production function, still gives a good description of the relation between output, capital and labour in the United States, and in many other countries, including Australia. And the Cobb-Douglas production function has become a standard tool in the economist's toolbox. (Verify for yourself that it satisfies the two properties discussed in the text: constant returns to scale, and decreasing returns to capital and to labour.)
The purpose of this appendix is to characterise the steady state of an economy when the production function is given by equation (I IA.I). (All you need to follow the steps is a knowledge of the properties of exponents.)
Recall that, in steady state, saving per worker must be equal to depreciation per worker. Let us see what this implies.
In an economy in which there is both capital accumulation and technological progress, at what rate will output grow? To answer this question, we need to extend the model developed in Chapter 11 to allow for technological progress. To introduce technological progress into the picture, wc must revisit ihe aggregate production function.
Technological progress and the production function
Technological progress has many dimensions:
• It may mean larger quantities of output for given quantities ol capital and labour. Think of a new type of lubricant that allows a machine to run at a higher speed and so produce more.
• It may mean better products. Think of the steady improvement in car safety and comfort over time.
• It may mean new products. Think ot the introduction of the CD player, the lax machine, mobile phones, llai screen monitors, the ifod.
Capitol per effective worker and output per effective worker converge to constant values in the long run.
• The relation between output per effective worker and capital per effective worker was derived in Figure 12.1. The relation is repeated in Figure 12.2. Output per effective worker increases with capital per effective worker, but at a decreasing rate.
• for more on growth, both theory and evidence, read Clharles Jones Introduction to Economic Growth New York: Norton. 2002, 2nd edn). Jones's web page. , is a useful portal to the research on growth.
• For more on patents, see the Economist survey on Patents and Technology. 20 October 2005
• For an evaluation ot thc contribution of IT to the growth in thc standard ol living 'compared with other great inventions in the past see Robert Cordon, Does the "New Economy" measure up to the great inventions ol the past? , journal of Economic Perspectives. Fall 2000.
• For an in-depth analysis of the role ol R&D lor growth in Australia, read the 2008 Cutler report Venturous Australia .
Readings on two issues we have not explored in the text:
We have looked so lar at the short-run effects of a change in productivity on output, employment and unemployment. In thc medium mn, we know the economy returns to the natural level of output—the level ol output consistent with the natural rate ol unemployment. Now we must ask: Is the natural rate of unemployment itself allectcd by changes in productivity?
Recall Irom Chapter 6 that the natural rate of unemployment is determined hy two relations, the price-setting relation and the wage-setting relation. Our first step must be to think about how changes in productivity allcct each of these two relations.
Price setting and wage setting revisited
Consider price setting first.
• From equation 13.1), each worker produces A units ol output; equivalently, producing I unit of output requires \IA workers.
• If thc nominal wage is equal to IV, thc nominal cost of producing 1 unit of output is therefore equal
to (1/A)W= W/A.
• If firms set their pricc equal lo I + fx times cosl (where /л is lhe markup), lhe price level is given by:
W
P = (j + fJLV
Thc only diflcrcncc between this equation and equation (6.3) is lhe presence ol the productivity term, A (which we had implicitly set to I in Chapter 6). An increase in productivity decreases cost which decreases lhe price level given the nominal wage.
Turn to wage setting. The evidence suggests that, other things being equal wages are typically set to rcllect lhe increase in productivity over time. It productivity has been growing at 3 per cent a year on average lor some time, then wage contracts will build in a wage increase of 3 per cent a year. This suggests thc lollowing extension ol our earlier wage-setting equation:
W = A'PTiu.z)
Look at the three terms on the right side of equation (1 3.4).
• Two of them, P1' and T:u,z), are familiar Irom equation (6.1). Wages depend (negatively) on the unemployment rate, u, and on institutional factors captured by the variable z. And workers care about real wages, not nominal wages, so wages depend on the expected) price level, P1'.
• Thc new term is A''—wages now also depend on the expected level of productivity, A'\ If workers and firms both expect productivity to increase, they will incorporate those expectations into the wages set in bargaining.
The natural rate of unemployment
We can now characterise the natural rate of unemployment. Rccall that the natural rate of unemploy¬ment is determined by the price-selling and wage-setting relations, and the additional condition that expectations be correct. In this case, this condition requires lhat expectations of both prices and productivity be correct, so P1' = P and A' - A.
• For more on the process of reallocation that characterises modern economies, read The Churn: The Paradox of Progress, a report by the Federal Reserve Bank of Dallas, 1993.
• For a fascinating account ol how computers arc transforming labour markets, read Frank Levy and Richard Murnane, The New Division of Labour: How Computers are Creating the Next lob Market (Princeton, NJ: Princeton University Press, 2004).
• For the role of institutions in growth, read Abhijit Banerjee and Esther Duflo, Growth theory through the lens of development economics', in Handbook of Economic Growth (Amsterdam: North Holland, 2005) (read sections 1 to 4 of Chapter 7).
• For more on institutions and growth, read Daron Acemoglu, Llnderstanding institutions', Lionel Robbins Lectures, 2004, .
• For a detailed analysis of growth in China, read the OF.CD's Economic Survey of China, published in 2005, .
I .et's now turn to the second key concept introduced in this chapter that of expected present discounted value.
To gain an understanding of this concept, let's return to the example of the manager considering whether to buy a new machine. On the one hand, buying and installing the machine involves a cost today. On thc other, the machine allows lor higher production, higher sales, and thus higher prolits in the future. The question lacing the manager is whether the value of these expected profits is higher than the cost ot buying and installing the machine. This is where the concept ot expected present discounted value comes in handv. The expected present discounted value ot a sequence ot future payments is the value today ol this expected sequence ol payments. Once the manager has calculated the expected present discounted value ot the sequence ot profits, her problem becomes simple. Il this value exceeds the initial cost, she should go ahead and buy the machine. Il it doesn't she should not.
As in the case ot the real interest rate in Section 14.1 thc practical problem is that expected present discounted values aren t directly observable. They must he constructed Irom inlormation on the sequence ot expected payments and expected interest rates. Let's first look at the mechanics of construction.
Calculating expected present discounted values
If the one-year nominal interest rate is ;',, lending one dollar this year yields I + i, dollars next year. Equivalently, borrowing one dollar this year implies paying back I + i, dollars next year. In that sense, one do.lar this year is worth I + t dollars next year. I his relation is represented graphically in the first line of Figure 14.3.
Turn the argument around and ask: One dollar next year is worth how many dollars this year? The answer, shown in the second line ol Figure 14.3, is 1/(1 + i',) dollars. Think of it this way: if you lend 1/(1+ i,) dollars this year, you will receive l/( I + /,} times (I + t,) - I dollar next year. Equivalentlv il you borrow l/( I + i, dollars this year you will have to repay exactly one dollar next year. So, one dollar next year is worth 1/(1 • /,) dollars this year.
More formally, we say that l/( I + i.) is the present discounted value ot one dollar next year.
The word present comes from the fact lhat we are looking at the value ol a payment next year in terms of dollars today.
The word discounted comes Irom the fact lhat the value next year is discounted, with l/l 1 + /,) being the discount factor. (The one-year nominal interest rate, /',, is sometimes called the discount rate.)
Because the nominal interest rate is always positive, the discount factor is always less than I: a dollar next year is worth less than a dollar today. The higher the nominal interest rate, the lower the value
today ot a dollar next year. It i 5 per cent, the value this year of a dollar next year is l/l .05 «= 05 cents. II i - 10 per cent, the value today of a dollar next year is 1/1.10 = 91 cents.
Now apply the same logic to the value today ol a dollar two years Irom now. For the moment, assume that current and future one-year nominal interest rates arc known with certainty. Let i, be the nominal one-year interest rate lor this year, and he the one-year nominal interest rate next year.
Il today, you lend one dollar tor two years, you will get • I + i,)( 1 + /,.,) dollars two years from now. Put another way, one dollar today is worth (I + /, м I •/,,,) dollars two years Irani now. This relation is represented in the third line ol Figure 1-13.
What is one dollar two years from now worth today? I!y the same logic as before, the answer is l/( 1 + I. I + /,. | dollars: it you lend l/( 1 + /,)( I + i,_, dollars this year, you will get exactly one dollar in two years. So. the present discounted value of a dollar two years from today is equal to I/(1 + /,) I +/'..,) dollars. T his relation is shown in the fourth line of Figure 14.3. II, for example, the one-year nominal interest rate is the same this year and next, and equal to 5 per cent, so i, - /,., = 5 per cent, then the present discounted value ol a dollar in two years is equal to I I .05)-, or about 91 cents, today.
A general formula
I laving gone through these steps, it is easy to derive the present discounted value for the general case.
Consider a sequence ol payments in dollars, starting next year and continuing into the future. Assume tor the moment that these future payments arc known with certainty. Denote the lirst year's payment by the payment the next year by $z,.lf the payment three years from today by and so on.
The present discounted value ol this sequence ol payments that is the value in today's dollars ol
the sequence ot payments which wc will call $V't, is given by

1 + /', ~ (I + /,)(1 + iM)
Each payment in the future is multiplied by its respective discount factor. The more distant the payment, the smaller the discount factor and thus the smaller today's value of that distant payment. In other words, future payments are discounted more heavily, so their present discounted value is lower.
We have assumed so far that future payments and future interest rates were known with certainty. Actual decisions, however, have to be based on expectations of future payments rather than on actual values for these payments. In our earlier example, the manager cannot be sure of how much profit the new machine will actually bring.- nor can she be sure what interest rates will be in the future. The best she can do is get the most accurate forecasts she can. and then calculate the expected present discounted value ot prolits, based on these forecasts.
How do wc calculatc the expectcd present discounted value when future payments or interest rates arc uncertain? This is done in basically the same way as before, but replacing the known future payments and known interest rates by expected future payments and expected interest rates. Formally, denote expected payments next year by expected payments two years from now by and so on.
Similarly, denote the expected one-year nominal interest rate next year by /',',,, and so on. The one-year
I
Ж - $=h—r + 7ГТ
l + t
We will spend the next three chapters using the tools we have iust developed. In the rest ot the chaptcr we take a first step, introducing thc distinction between real and nominal interest rates in the IS-LM model and then exploring the relation between money growth, inflation, and real and nominal interest rates.
In the IS-LM model we developed in The Core i Chapter 5), the interest rale entered in two places: it allecied investment in the IS relation, and ii allecled the choice between money and bonds in the LM relation. Which interest rate—nominal or real—were we talking about in each ease"
The RBAs decision to allow for easier monetary conditions is the main lactor behind the decline in interest rates in the last six months' Imaginary quote, circa carlv 2002).