Universidad de Montevideo Macroeconomia II. The Solow-Swan Growth Model

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1 Universidad de Montevideo Macroeconomia II Danilo R Trupkin Class Notes (very preliminar) The Solow-Swan Growth Model 1 Introduction This model is the starting point for almost all analyses of growth Understanding the Solow model is essential to understanding theories of growth By studying this model, in the end, we will arrive to the conclusion that the accumulation of physical capital cannot account for either the vast growth over time in per capita output or the vast geographic differences in per capita output In this sense, suppose that capital accumulation affects output through the standard channel of production, and it is paid its marginal product Then the Solow model implies that the differences in real incomes are too large to be accounted for by differences in capital inputs The model treats other potential sources as either exogenous (eg, technological progress) or just absent (eg, positive externalities from capital) Thus, to address the central questions of growth theory, we must go beyond but the departures from the standard model will be studied later in the course In essence, this is a model that specifes relations among aggregate variables, without individuals, then it neglects, among other things, the possibility of analyzing optimal decisions and welfare 2 The Environment The model focuses on 4 variables: output (Y), capital (K), labor (L), and knowledge (A) Then, the economy s production function, at any point in time t is: Y (t) = F (K(t), A(t)L(t)), (1) Notice that for given quantities of capital and labor, output may increase over time if the amount of knowledge grows over time (technologcial progress) The argument AL is called effective labor, and then techonological progress is assumed to be labor-augmenting or Harrod-neutral The production function is assumed to be of constant returns to scale in its two arguments That is, 1

2 F (κk, κal) = κf (K, AL) for all κ > 0 (2) From this assumption, we can work with the production function in intensive form Setting κ = 1/AL in (2), we have F ( ) K AL, 1 = 1 F (K, AL), and f(k) = y, (3) AL where k and y express capital and output per unit of effective labor, K/AL and Y/AL, and f(k) = F (k, 1) Interesting: now, the relevant measure of the economy is the amount of output per unit of effective labor, which depends only on the quantity of capital per unit of effective labor, and not on the size of the overall economy Assumptions on f(k) : f(0) = 0, f (k) > 0, f (k) < 0 Since F (K, AL) = ALf(k), then F (K, AL) K ( ) 1 = ALf (k) = f (k) (4) AL Therefore, given the assumptions on f(k), the marginal product of capital is positive, but it declines as k rises It satisfies the Inada conditions: lim (k) k 0 =, lim (k) k = 0 (5) (We will see later, that these conditions will help the economy not to diverge) An example is the Cobb-Douglas production function (a good approximation to actual production functions): F (K, AL) = K α (AL) 1 α, 0 < α < 1 (6) To obtain the intensive form of the C-D production function, just divide F by AL to 2

3 get: Then, F (K, AL) AL = ( ) K α = k α f(k) (7) AL f (k) = αk α 1 > 0, f (k) = α(α 1)k α 2 < 0, lim (k) k 0 = αk α 1 =, and lim (k) k = αk α 1 = 0 Assumptions on the evolution of inputs Taking as given the initial levels of capital, labor, and knowledge, the model assumes that labor and knowledge evolve according to: L(t) = nl(t), (8) A(t) = ga(t), (9) where n and g are the exogenous growth rates Remember that, for a given variable X, it is true that Ẋ(t)/X(t) = d ln X(t)/dt Then, (8) and (9) imply that ln L(t) = [ln L(0)] + nt, (10) ln A(t) = [ln A(0)] + gt (11) Applying exp() to both sides gives us L(t) = L(0)e nt, (12) A(t) = A(0)e gt (13) The model simply assumes that output is divided between consumption and investment It further assumes that the fraction of output devoted to investment, s, is exogenous and constant, and that capital depreciates at rate δ Thus, capital evolves according to 3

4 K(t) = sy (t) δk(t) (14) Finally, the model assumes that n + g + δ > 0 (As we will see later, once again this helps the economy not to diverge) This completes the description of the model 3 The Dynamics Now, we want to characterize the behavior of the model economy We know that the behavior of labor and knowledge is exogenous Then, to characterize the behavior of the economy, we need to analyze the dynamics of capital Since k = K/AL, we can use the chain rule to find 1 k = K AL K (AL) 2 [A L + ȦL] K = AL K L AL L + K A AL A = sy δk K AL AL n + K AL g (15) = sy δk kn + kg, (16) where, besides some algebraic steps, we use (14) in (15), and the definitions y = Y/AL and k = K/AL in (16) Finally, by using the fact that y = f(k) we obtain the key equation of the Solow model: k = sf(k) (n + g + δ)k (17) Equation (17) states that the rate of change of the capital stock per unit of effective labor is the difference between: (i) actual investment per unit of effective labor, sf(k), and (ii) break-even investment, (n + g + δ)k, the amount of investment needed to keep k at its existing level When actual investment per unit of effective labor exceeds both the the depreciation part, δk, and that part of capital accumulation required to follow the growth of effective labor, (n + g)k, then k will rise When the first term is lower than the second term, then k will fall When the two are equal, then k will be constant 1 I already described the variables and the parameters of the model in the preceding section Then, to simplify notation I will avoid writing the time arguments for the variables 4

5 Notice that the assumptions made explicit in the preceding section ensure the existence of a value of k > 0 such that k = 0 Of course, when k = 0 we already have k = 0 (remember that f(0) = 0) The Inada conditions imply that, for small values of k, actual investment will in fact be larger than break-even investment Those conditions also imply that f (k) falls toward zero as k becomes large At some point, the slope of the actual investment curve falls below the slope of the break-even investment line, which suggests that the two must eventually cross Finally, the fact that f (k) < 0 implies that the two functions intersect only once for k > 0 Let k be the value of k where actual investment and break-even investment are equal See the diagram below Notice that regardless of where k starts, it converges to k Of course, if k is initially zero, it will remain there; but we ignore this possibility (δ+n+g)k sf(k) k* k What is the behavior of the model s variables at the steady state? By assumption, labor and knowledge grow at rates n and g, respectively Aggregate capital, which is equivalent to ALk, evolves at the rate n + g because effective labor, AL, evolves at such a rate, while k = k does not changewith both capital and effective labor growing at the rate n+g, the CRS assumption implies that output, Y, must also be growing at that rate Finally, capital per worker, K/L, and output per worker, Y/L, evolve at the rate g As a conclusion, the Solow model shows that, regardless of its starting point, the economy converges to a balanced growth path a situation where each variable is growing at 5

6 a constant rate On the balanced growth path, the growth rate of output per worker is determined solely by the rate of technological progress Moreover, considering the variables in terms of effective worker, the model converges to a steady state a situation where the variables expressed in terms of effective worker (k, c, y) remain constant 2 4 The Impact of a Change in the Saving Rate The parameter of the Solow model that policy is most likely to affect is the saving rate (eg, government s expenditure, taxation, or public borrowing) We will consider an economy at the steady state, and suppose there is a permanent increase in s In general, we will see the model s properties when the economy is not at the steady state Effect on Output Whenever we observe a change in a parameter of the model, we will always look at the main equation, equation (17), which dictates the dynamics of the economy An increase in s shifts the actual investment curve upward, and so k rises Notice that k does not immediately jump to the new k, but what happens is that, at the old level k, actual investment now is larger than break-even investment, and so k is positive Thus, k begins to rise until it reaches the new value of k, at which point it remains constant Let us observe now the behavior of output per worker, Y/L Since Y/L = Af(k), then it will change depending on the changes in f(k), and so in k To sum-up, a change in s has a level effect but not a growth effect (indeed, in the Solow model only changes in the rate of technological progress have growth effects) Effect on Consumption At the end of the day, agents only care about consumption Let us analyze then the behavior of this variable Consumption per unit of effective labor is defined here as: c = (1 s)y = (1 s)f(k) Since s changes discontinuously at impact (while k still remains constant), then c jumps downward at impact Consumption then rises gradually as k rises Whether consumption eventually exceeds its level before the rise in s is not trivial At the steady state, consumption is equal to c = f(k ) (n + g + δ)k 2 From now on, we will refer both the balanced growth path and the steady state indifferently However, be careful not confuse these concepts in light of the measurement of each variable 6

7 that In fact, k is determined by s and the other parameters of the model Then, we have c s = [f (k ) (n + g + δ)] k s We know that k / s > 0 Thus, the long-run effect on consumption will depend on whether the marginal product of capital per unit of effective worker is more or less than the break-even investment If they are equal, then the marginal effect of the change in s will be null In this case, consumption is at its maximum possible level among steady states This value of k is known as the golden-rule level of the capital stock The figure below shows three values of k, for which only k g is the one maximizing consumption among all possible steady states (notice that both k l and kh provide lower consumption at the steady state) f (k g *) f(k*) c g * s h f(k*) s g f(k*) s l f(k*) k l * k g * k h * k* 5 On the Speed of Convergence In practice, we are interested not only in the eventual effects of some change, but also in how rapidly those effects occur In general, for any dynamic model, among the first tasks one is called to take is to check the existence of a steady state, and whether it is unique Then, the next task is to check for stability of the system (either local or global) That is, if the system is slightly moved from its original position, then we are interested in knowing 7

8 whether it goes back to origin, it approaches to a steady state, or it just diverges Even then, one might also be interested in calculating the speed of adjustment or convergence of the model being studied This is highly important on many topics of macroeconomics, and it is related to the issue of persistence For example, in monetary policy a vast literature intends to address the issue of inertia in inflation and persistence in output after a monetary shock The evidence suggests that prices adjust slowly to changes in the growth rate of money, generating persistent deviations in output from its long-run equilibrium For those dynamicgeneral-equilibrium models trying to capture this fact, the hardest part is to account for the speed of convergence of output that is sufficiently slow to produce such a persistence Now, going back to Solow, we are interested in the speed of convergence to the balanced-growth path (remember, the steady state of the variables in units of effective worker), because it will help us find answers to the cross-country convergence problem In pursuing this issue, we will use approximations around the long-run equilibrium, and will focus on the behavior of k Our goal is then to determine how rapidly k approaches k Remember that the behavior of k is determined solely by k Then, we can write k = k(k) A first-order Taylor approximation of k(k) around k = k yields where we use the fact that k(k ) = 0 Let λ denote k(k)/ k k=k Then, we have [ ] k(k) k (k k ) (18) k k=k k(t) λ[k(t) k ] We know that k > 0 when k < k, and viceversa Thus, k(k)/ k k=k < 0 and λ > 0 As a result, k moves toward k, in the vicinity of the latter, at a speed approximately proportional to its distance from k That is, the growth rate of k(t) k is approximately constant and equal to λ This implies k(t) k + e λt [k(0) k ] Notice that the dynamic behavior of k follows from the fact that the system is stable (k converges to k ) and that we are linearizing the equation for k around k = k Now, to find λ we proceed by differentiating the key Solow equation wrt k, and evaluates it at k = k By applying a number of substitutions, we obtain the following results: 8

9 λ k(k) k k=k = [sf (k ) (n + g + δ)] (19) = [1 α k (k )](n + g + δ), (20) where we use the fact that sf(k ) = (n+g+δ)k to substitute for s, and α k = k f (k )/f(k ) is the elasticity of output with respect to capital at k We can calibrate this result in order to see how actual economies converge For instance, n + g + δ is about 6% per year, and capital s share is about one-third, then the growth rate of k k is about 4% per year, which implies approximately 18 years to get halfway to their steady-state values A quick conclusion regarding the Solow model is that if the returns to capital in the market roughly explain its contributions to output, then variations in the accumulation of physical capital cannot account for a significant part of either worldwide economic growth or cross-country income differences 6 Growth Accounting In the Solow model, long-run growth of output per worker depends only on technological progress But short-run growth can result from either technological progress or capital accumulation Consider the production function Y = F (K, AL) This implies Y = Y K K + Y L + Y A (21) L A Dividing both sides by Y and rewriting the terms on the RHS yields Y Y K = K Y Y K K + L Y Y L K = α K K + α L L L + R L L + A Y Y A A A Finally, rearranging terms we have Y Y [ L L = α K K K L L ] + R, where the α s are the elasticities of output wrt the factors of production, and R is the so-called Solow residual Then, the contributions to the growth of output per worker can 9

10 be accounted for by the growth rates of the factors of production (they can be measured straightforwardly from data), the elasticity of output wrt capital (measured by using data on the shares of income going to capital, provided capital is paid its marginal product), and the Solow residual In fact, the last term can be measured by ultimately using data on the growth rate of output This equation provides a way of decomposing the growth of output per worker into the contribution of growth of capital per worker and a remaining term The Solow residual, although many times interpreted as techonologcial progress, reflects all sources of growth other than the contribution of capital accumulation via its private return 10