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Peter Sinclair Econ 302: February 13 (Growth 1+2) and 20 (Growth 3+4), 2020, University of Birmingham PLEASE REMEMBER TO BRING YOUR LAPTOP OR A PRINTOUT OF THIS HANDOUT TO THE SECOND DOUBLE LECTURE Growth double-lecture 1, February 13: The Solow Growth Model and Beyond (SOLOW, Quarterly Journal of Economics 1956) 2 equilibrium concepts: steady state growth (SSG) and balanced growth (BG). SSG: all relevant variables are stationary in level or rate of change. BG: variables that grow, grow at a common rate. Equilibrium ideas in economics prompt 4 questions: a. existence? [is there an equilibrium?] b. Uniqueness? [if it exists, is there just one?] c. Stability? [does a chance divergence establish forces restoring it?] d. Optimality? [does it maximize welfare?] The assumptions of the simplest version of Solow’s model are: 1. population grows at n, exogenous and positive 2. savings (or, equivalently, investment) share of income is s, exogenous and positive 3. production function, y=f(k), has constant returns to scale, is twice differentiable with f ’>0>f’’, and a capital-output ratio, v, able to take on any positive value ( y and k are output per head and capital per head). 4. no depreciation 5. no technological progress 6. perfect competition (including marginal product factor pricing, full price flexibility and information, and market clearance). Assumptions 1-6 imply that existence, uniqueness and stability (but not optimality) are assured. Reasoning: the growth rate of capital is, by definition, the ratio of investment to capital, which may be written s/v. BG requires this to equal n. From 3, there exists some value of k at which v=s/n. Geometrically, n/s is the gradient of a ray from the origin depicting the average product of capital in balanced growth: this proves existence. Since the idea that f(k) could be zero at some positive k makes it possible that n/s could be so steep as to exceed the highest possible value of 1/v , f(0)=0, so that f’>0>f’’ bars multiple equilibria: equilibrium is unique. Stability is assured since, if k is below (above) its BG value,

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the average product of capital is above (below) n/s, so the capital labour ratio is rising (falling). More on optimality below. [Note that we can swap assumptions 1 and 4, and replace n by depreciation]. SEE DIAGRAMS IN LECTURE.

Consider now the balanced growth equilibrium (BGE) effects of a rise in n or fall in s. The ray with gradient n/s must steepen. So it will cut f(k) to the south west of the old BGE. So we infer at once that y, and k must fall. From concavity, v must fall, too, while the real interest rate (equals the marginal product of capital from assns 4 and 6, which is the gradient of the tangent to f(k)) must rise. The real wage rate falls ( dw / dk d{ f ( k )  kf ' ( k )} / dk  f ' ( k )  f ' ( k )  kf ' ' ( k )  kf ' ' ( k )  0. ) The profit share of national income is subject to conflicting forces: a higher rate of profit, but less capital per head. The first effect dominates the second if  , the elasticity of substitution between capital and labour in the production function is less than unity; they cancel if the production function is Cobb Douglas, where  1 ; and the profit share falls if   1 , so that capital and labour are “substitutes” rather than “complements”. A mnemonic for the impact of n on r: the late 1960s / early 1970s slogan, “Women in labour keep capital in power”.

In most cases, faster population growth is similar to a fall in the savings ratio. Here are two variables for which this is not true: the growth rate of output (call it g), and consumption per head. IN BGE, g=n. This means that faster population growth must imply faster long run output growth (one for one), but that a change in the savings ratio has no effect on the long run growth rate of output . Higher s raises the long run level of output per head, but the growth rate of capital gradually slips back, as v rises, to equal n in the long run. Consumption per head, c, equals f ( k )  nk in BGE. Differentiating this with respect to k gives f ' ( k )  n . So c reaches a maximum where the growth rate and the real interest rate are equal. (Prove it is a maximum by showing that d 2c / dk 2  f ' '  0). This is the GOLDEN RULE. But it is only by chance, with n and s given, that BGE satisfies this. Hence the answer to the “optimality” question is, generally NO. The Golden Rule condition f ' (k ) n implies that the savings ratio should equal the profit share of income in BGE (why? multiply both by v). The Golden Rule is straightforward and illuminating. But it has drawbacks. One is that it does not prescribe a path towards the best BGE if you are at an inferior BGE. (By contrast, Ramsey does). Further, it cannot properly handle extensions, such as to exhaustible natural resources or technological progress, the first of which would tend to make c drift downwards over time, and the second, upwards.

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Technology trends can be incorporated provided technical progress is Harrod-neutral (equivalently, labour augmenting). So if Y, K and N are aggregate output, capital and labour, the production function is written Y=Y(K,BN) where B advances at a given, constant rate (say x). Since returns to scale are constant, we may write (using the same notation as in ~ ~ y (t )  f (k (t )) ; in a BGE, k (t ) is stationary, and the the Ramsey model) ~ relevant geometry of a BGE requires that 1/v = (n+x)/s. Faster technical progress raises the rate of profit and reduces the long run value of capital per human efficiency unit, k~ . The BGE growth rate of output is now n+x, and both output per head, and consumption per head, climb at x. There are numerous other possible extensions to Solow’s model (eg, additional sectors – for example for capital goods; an endogenous savings ratio, displaying sensitivity to n, or factor shares of national income, or to the real interest rate). It is a very powerful and versatile model. One controversial aspect is that, like the Ramsey model, it cannot incorporate short-run Keynesian problems, such as may arise from price rigidities and either deficient or excessive aggregate demand: it simply assumes such problems away, invoking assumption 6. A more serious criticism, perhaps, is that the savings ratio is not derived from optimization conditions. It is just imposed. Responding to that criticism is simple – switch to the Ramsey model (1928). But a still more powerful criticism is that is not a model of long run growth at all. Solow’s model is really a set of relationships explaining how y, k, v, r, w, c and so on vary with the fundamental parameters n, s and x. Long run growth, n+x, is simply exogenous, and unexplained. To rectify this, we need to explore models of ENDOGENOUS GROWTH. We shall consider 5 such models: endogenous population growth; exhaustible natural resources (oil), training, externalities and invention. Endogenous population growth: invoke Malthus’s argument that population tends to expand (contract) when the real wage exceeds (falls short of) subsistence level. Write n=n (w), whereas before w=f(k)-kf’(k), the real wage rate. Malthus would predict n’>0; let us assume this (but imagine the effect is small). Now consider the impact of a (permanently) higher savings ratio. The Solow model showed that the BGE value of the real wage rate must climb, in response to the additional capital per head. If n were to rise as a result of this, we would find that the BGE growth rate of output went up. Note the contrast with the standard Solow model, where it remained unchanged. (Note too that, if this Malthusian relationship is strong enough, we might imperil the existence, or uniqueness, or stability of BGE). Evidence from the 19 th and earlier centuries supports Malthus; but recent decades, especially in rich countries, much less so. Indeed dn / ds may be taken as broadly negative for much of the 20 th century, chiefly because of the labour market opportunity cost effects of bearing children for

4 married women, plus direct child rearing costs. If Malthusian effects hold at low and high w, but anti-Malthusian effects at intermediate ones, note that n/s curve could be at first convex, then concave, possibly (but not necessarily) intersecting f(k) three times. That could generate a purely demographic explanation of a poverty trap for some countries, where a stable BGE was established at low k. Much richer countries could have a stable BGE with higher k and higher output per head (and lower n). It is not only an endogenous birth rate that could lead to multiple equilibrium. It could arise with an endogenous value of the savings ratio, s. And of course it could well be that persistent technology level differences might lead to big persistent gaps in national living standards (per head). On the other hand, there has been some tendency in recent decades for many poorer countries to tend to catch up gradually with richer ones. A very interesting paper on this subject, which looks at the size of this effect and many other possible influences on long term economic growth, is Doppelhoefer, Miller and Sala-i-Martin, American Economic Review, 2004.

Growth 2: Growth with an Exhaustible Natural Resource (302 Growth double lecture 2, 13.2.2020) This double lecture examines the dynamics of the most disturbed macroeconomic variable in the post-war period – the price of oil. It also provides a second endogenous growth model. Let aggregate production of final output be described by the Cobb Douglas production function: Q(t )  TK (t )  ( B( t) N ( t ))  E (t )1    

(1)

where E(t) denotes inputs of oil, extracted at that date. (Note that the sum of the 3 exponents is unity – implying that competitive factor rewards add up to the value of final output; each of the 3 exponents is strictly  positive). So if S(t) is the stock of unextracted oil at date t, E (t )   S (t ) and let us define the proportionate rate of extraction,  x (t ) E (t ) / S (t )   S (t )/ S (t ) . B(t) advances at rate b, and N(t) at rate n. There is perfect competition, the savings share of income is s and, as again in Solow, no depreciation. So r(t) = Q(t ) / K (t ) .

(2)

Differentiating this last equation totally with respect to time implies ˆ (t )  Kˆ (t ) Qˆ(t )  sQ(t ) / K (t ) Qˆ (t )  sr (t ) / . rˆ (t ) Q

(3)

Equation (3) is a law of motion for the real rate of interest, which we can relate to the other key dynamic variable, x(t), a little below.

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We now invoke the HOTELLING RULE (Hotelling, JPE 1931) to explore the dynamics of the price of oil, call it P(t). Assume (strong assumptions) that oil exporters are neutral to risk, perfectly competitive, not subject to tax, and free of any extraction costs. These assumptions imply that the price of oil should be expected to rise at the real rate of interest. Given foresight, and shocks apart, actual and expected paths of oil prices should coincide, so: ˆ( t) r( t) P

.

(4)

Perfect competition among final output producers implies P(t )  (1     )Q (t ) / E (t )

(5)

Note that E should normally be falling over time (and must be in a steady state), and also that Q will presumably be rising – so the price of oil should exhibit an upward trend over time. So, from (4) and (5) and the definition of x(t), we have r (t ) Qˆ( t )  x(t )  xˆ (t ).

(6)

Next, it helps to eliminate the growth rate of final output,Qˆ (t ) . From (1), we have ˆ ( t) K ˆ ( t)  ( b  n)  (1     )( x ˆ (t )  r (t ) ˆ ( t)  x( t)) sr( t)  ( b  n) (1     )( Q Q

using (6) and (2). Simplifying: ˆ ( t)(  )  ( b  n)  r ( t )[1      s] . Q

(7)

BY SUBSTITUTING (7) INTO (3) AND (6) WE CAN NOW FIND 2 KEY LAWS OF MOTION, for the extraction and interest rates, in terms of these variables alone: rˆ( t)(  )  (b  n)  r (t )[1      s / ] xˆ (t )(   )  x (t )(   )   (b  n )  r (t )(1 s ) .

(8) (9)

(8) says that the stationarity locus for r is independent of x and always attracts r (a standard feature of Solow-type and many Ramsey-type models). In other words, the long run real interest rate is determined by the production function characteristics, plus the parameters b, n and s. The savings ratio tends to lower r; b and n to raise it. So r is stable – the dynamics of r are stabilizing. Diagram 2.1 (drawn in lecture) refers.

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(9) says that the stationarity locus for x is linear in r and x, which are related positively. Intuition: higher r implies, from Hotelling, faster rate of ascent of the price of oil, and therefore faster extraction. The dynamics of x are unstable. Diagram 2.2 (drawn in the lecture) refers. Putting (8) and (9) together, into Diagram 2.3, shows a steady state at point E, where the 2 stationarity loci intersect – and that there is a unique saddle path towards E from either side. Given foresight, and knowledge of the model and information about the parameters, agents should be able to plot the evolution of r and x. At point E, the steady state, the long run values of r and x (and also the growth of final output) will be: r  ( b  n) A; x  ( b  n)(1  s / ) A; g  A

1

bn (1    ) , 1 s

where

1      s  / .

Many points of interest are apparent. One is that the (long run) growth rate now INCREASES with s, the savings ratio. Reason: over time it raises capital, reduces the rate of interest, and implies slower oil extraction and therefore higher SUSTAINABLE growth. Note that the faster oil is extracted, the faster oil inputs into production will decline, and the slower output growth must be (eventually, at least). (Note too that if the role of fossil fuels in production were to disappear suddenly, 1     would vanish, and g would be the sum of population growth and labour augmenting technical progress). Another is that sensible results require the profit share to exceed the savings ratio (otherwise we get a nonsensical, negative solution for extraction). The most interesting feature of all, perhaps, is how we can now use the model to try to understand the dynamics of oil prices. An unexpected permanent rise in the savings ratio would flatten the stationarity locus for x, and push the stationarity locus for r leftwards, with the combined long run effect of lowering both r and x. The saddle path to the new long run equilibrium would show x slipping very slightly, and r slipping faster. Diagram 2.4 illustrates. So the impact effect would be a big fall in x. Counterpart: a big jump in the spot price of oil (which would thenceforward increase more slowly). And other asset prices would also go up – equities, gold, copper, real estate. Similar consequences would ensue from unanticipated permanent falls in b or n. (1970s? 2003-mid 2008? After 2009?). And falls in expected s would have the opposite

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effect (mid 1980s?). As would the end of traditional communism, adding large population, but little extra useful capital, to the prospective advanced trading world’s endowments (late 1980s, early 1990s?). The big early 21C jump in oil prices may also reflect (in our model) a phased, unexpected, set of jumps in parameter T, which could represent the effect of rapid growth in China and India, and the increased demand for natural resources entailed by this. There could also be sudden changes in beliefs about the stock of oil as yet not extracted, S(t). Another possible culprit is increased life expectation, which, in the context of Blanchard’s model (JPE 1985) would also lead to forecasts of lower real interest rates. But why, despite some blips, have oil prices tumbled so far since 2013-4? Unexpectedly large discoveries, the result of exploration initiated earlier when oil was dear? Fracking? Growth slowdown news (equivalent to forecasts of lower future T levels)? Monetary policy normalizing? Fiscal challenges? Falling costs of solar and other renewable technologies? Oil sellers pumping out more now, fearing their assets could be stranded in the future, as a result of action against global warming? Global warming and climate change are surely the most serious policy problem now facing the planet. How can economics contribute to guiding us to the best way forward?

Note: If you want to read more on these issues, Hotelling JPE 1931 is a classic reference, and a symposium in the Review of Economic Studies 1974 is also very good. So is Dasgupta and Heal, Economic Theory and Exhaustible Resources, CUP, 1979. On climate change, Nordhaus AEJ applied economics 2018 and AER 2019 are useful; Sinclair OEP 1994 and maybe either Singapore ER 2015 or MS 2019 (on uncertainty Pascal’s wager) might also interest. On oil prices, recent papers include Baumeister + Kilian JEP 2016 and Bjornland + al AEJ macro 2018.

QUESTIONS on RAMSEY for open house discussion (scheduled for February 13) 1. “Inflation is bad for welfare and for output”. Discuss 2. What are the benefits and costs of adopting an expenditure tax? 3. If y (t ) T k (t ) , show how, in the simplest Ramsey model, all macro variables react to a surprise doubling of T, assuming a=0.5. Describe what could happen if a>m; and if news of the T shock were made public V years earlier.

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4. Suppose you learn that T will jump now for a while, and then revert to its previous value after five years. Sketch the likely consequences, first when a=m, and then when a>m. 5. Derive the Ramsey steady state savings ratio in a growing economy. Is there any ambiguity about how parameters affect it, and if so how might the ambiguity be resolved? 6. How would agents in a simple Ramsey model react to a permanent reduction in the rate of population growth? 7. How would consumption per head and capital per head react to a surprise permanent change in the rate of depreciation? 8. Does Britain save too little or too much? (you might look at first section of Sinclair MS 2019)

QUESTIONS on GROWTH for open house discussion (scheduled for February 27 or possibly March 5): 1. Suppose output per worker ( y) is related to capital per worker ( k) by y=2k/(1+k). If k is now 3, find factor prices, y, and labour’s share of income. If s/n = 4, will k be stationary, rising or falling? What is labour’s share in balanced growth, and why? 2. Suppose y is the cube root of k, that n=1% and s=9%. If k is now 8, will it stay there, grow or slip as time goes by? What will happen to the wage rate and the rate of profit? Consumption per head? 3. Beginning in steady state, explore the likely effects of unexpected permanent jumps in the savings ratio, the technology parameter T and the volume of oil believed to lie in the earth’s crust. 4. In question 3, which of these changes alters the long run values of the real interest and extraction rates? In which cases, if any, will the oil price witness a kind of overshooting? 5. Why might financial frictions impair long run growth? . 6. How will changes in relative risk aversion affect steady state values of growth and real interest? In Romer’s 1990 model, what would happen if patents became finite rather than infinite? 7. What economic policies might help to check climate change?

© Peter Sinclair 2020...