Assumption 5. The technology is represented by a production function, F : R 3 + R +, F (K t, N t, A t )

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1 6. Economic growth Let us recall the main facts on growth examined in the first chapter and add some additional ones. (1) Real output (per-worker) roughly grows at a constant rate (i.e. labor productivity grows steadily). (2) Real capital has roughly the same constant rate of growth of output. (3) Labor input has a growth rate higher than those of real capital and output. (4) The rate of growth of profit on capital is roughly zero. (5) Output per-capita has a growth rate that varies a lot across countries. (6) The rate of growth is positively correlated with the saving rate (hence, with capital accumulation). As we previously discussed, 2) implies that the ratio of capital to output stays constant over time. Moreover, 2) and 4) imply that the ratio of capital-income, to total income is constant. Some facts are related to OECD cross-country comparisons. (1) Economies with a high profit/income ratio tend to have a high investment/output ratio. (2) The rate of growth differs across countries. (3) The average rate of growth of countries does not depend on the level of income. (4) The variance of the rate of growth of countries decreases with income Preliminary considerations. Assumption 5. The technology is represented by a production function, F : R 3 + R +, that satisfies the following properties. F (, N t, A t ) It is twice continuously differentiable in K, N and it is strictly increasing in K, N, F K, F N > 0, strictly concave in K, N, F KK, F NN < 0, and F (0,, ) 0. It has constant return to scale in the input factors K, N (CRTS). It satisfies Inada conditions: lim F x +, x 0 lim F x 0, for x {K, N} x The CRTS assumption, formally, says that the production function is omothetic in the input factors (implying homogeneous of degree one in K, N). Hence, by Euler s Theorem, F (, N t, A t ) F K (, N t, A t ) + F N (, N t, A t )N t, for all (, N t, A t ) and F K, F L are homogeneous of degree zero in K, N. Observe that A t represents the technological progress. Three typical specifications are, the Hicks-neutral, the Harrod-neutral and the Solow-neutral technological progress. In the first, A t shifts up and down isoquants, G(, N t ), that is F (, N t, A t ) A t G(, N t ). In the other two, A t enters as the multiplicand of N t and, respectively. For example, for a Cobb-Douglas function, F (, N t, A t ) A t Kt α Nt, F (, N t, A t ) Kt α (A t N t ), F (, N t, A t ) (A t ) α Nt 42

2 We say that the technological progress is exogenous (i.e. part of the economic fundamentals) when, as we do here, we assume that its dynamics is governed by a given, exogenous, model. Typically, one assumes that (the deterministic part) of this model has a constant per-period, growth rate, µ, µ : A t+1 A t A t, at all t T Hence, the dynamics of technological progress, at a given initial level A 0 : 1, is A 1 (1 + µ)a µ A 2 (1 + µ)a 1 (1 + µ) 2. A t (1 + µ) t A similar assumption is often used for demographics. The population, or work-force, grows at a constant, per-period rate n, according to the deterministic model, N t (1 + n) t N 0, t T Remark 6.1 (Production efficiency under CRTS). CRTS technologies have the property that the level of output that is productive-efficient is undetermined. Formally, this is just the consequence of the fact that a technology F (, N t ) is homogeneous of degree one in inputs. So that, proportionally increasing inputs one attains an equiproportional increase in output; hence, the scale of production does not affect the amount of input required to produce a unit of output. Indeed, by Euler s theorem, Y t α Y t + (1 α) Y t N t N t So if Ȳt is produced with a certain input combination (, N t ), we can expand it or reduce it by simply proportionally expand or reduce the initial input combination. This implies that, for each (, N t ), any production scale λȳt can be chosen keeping / N t, and thus Ȳt : κ t and Ȳ t N t η t, constant: λȳt ακ t (λ ) + (1 α)η t (λ N t ) For given input prices (taken as invariant with respect to the production scale) this says that CRTS technologies can accommodate any scale of production at the same unit (and marginal) cost; implying there is no efficient minimum unit cost scale. Going baco any undergraduate textbook in microeconomics, you can see that this is presented by saying that CRTS technologies have technical rates of substitutions, which are constant across isoquants if measured along any given ray from the origin of the Eucledean plane of coordinates (N t, ) (where rays have slope /N t ) The Solow s growth model. Solow s is a special case of the Ramsey-Cass-Koopmans growth model in which consumers behavior is not represented endogenously: households inelastically supply labor and have a constant saving rate s (i.e. also a constant propensity to consume, c 1 s). Hence, aggregate consumption is, C t (1 s)y t 43

3 and aggregate savings, sy t, are, by assumption, invested. Definition 8 (Solow s Competitive Equilibrium). In an economy E (F, s, n, µ, K 0, N 0, A 0 ) a Solow s Competitive Equilibrium is a sequence of allocations (Y, C, K, N) and prices (w, ν) such that, at all t in T, Y t F (, N t, A t ) Y t C t + I t, C t (1 s)y t, I t sy t +1 I t + (1 δ) ν t F K (, N t, A t ), w t F N (, N t, A t ) N t (1 + n) t N 0, A t (1 + µ) t A 0 In this economy markets are competitive, implying that firms demand inputs up to the point at which theirs marginal productivity equate prices. Substituting this individual optimality condition into the Euler s formula, implies zero profits at all dates t in T, 23 Π t F (, N t, A t ) ν t w t N t 0 Since in the Solow s economy, labor supply is inelastic and corresponds to the whole labor force, the labor market equilibrium is attained at full employment, as the real wage rate adjusts so that the labor demand equals the whole labor force. Thus, the population (or workforce) dynamics fully translates into labor input dynamics, N t (1 + n) t N 0. Using the law-of-motion of capital, one finds that the equilibrium rate of growth of capital is, (11) +1 s Y t δ An equilibrium with balanced growth is one at which (Y, C, K) grow at the same constant rate g. If it exists, by the last equation, it implies that K/Y is also constant over time, s Y t g + δ To study the dynamics and establish the existence (or compute) balance growth, it is often useful to transform the original variables expressing them in per-capita terms or, when there is growth in labor input productivity, in efficiency units. We are going to show that such a transformation allows to easily compute the steady-state of the transformed economy and that this corresponds to the balanced growth equilibrium of the original economy. Indeed, let us consider the second case and, for any variable X t, let N t : A t N t and x t : X t / N t be that variable in efficiency units. In the case of a Harrod-neutral, Cobb-Douglas technology: y t : Y t N t Kα t (A t N t ) A t N t ( Kt A t N t ) α k α t 23 Because of CRTS, the optimal level of production of a firm is undetermined, in the sense that it can produce at any scale earning zero profits (see the discussion in the previous remark). 44

4 (12) +1/ N t+1 / N t 1 +1 Nt 1 N t+1 ( ) Kt+1 Nt N t+1 where in the second step we have simply transformed the first term in the right-hand-side, using the definition of growth rate of capital. Notice that, (1+η) {}}{ N t A t N t [(1 + µ)(1 + n)] t N 0 (1 + η) t N η : 1 + n + µ + nµ 1 + n + µ as nµ 0. Hence, the rate of growth of N is constantly equal to 1 + n + µ. Using this into (12), ( ) Kt η 1 1 ( ) Kt η 1 + η Hence, rearranging and using, η 0 (i.e. the products of the growth rate of capital in efficiency units with n and µ are zero). 24, the following is approximatively correct, (13) +1 η Thus, at a steady state of the model with variables in efficiency units, +1 n + µ : g K > 0 We are left to chechis is indeed a balanced growth equilibrium of the original model. By (11), g K s(y t / ) δ. Rearranging, we obtain that the capital-output ratio is constant, Y t s g K δ Denote the fraction on the right hand side z, and consider zy t at two consecutive periods. Subtracting, yields hence, +1 z Y t+1 Y t Y t+1 g K : +1 Y t+1 : g Y Y t We are left to chechat consumption also grows at the same rate. Since, C t (1 s)y t, g C : C t+1 C t C t (1 s) Y t+1 C t (1 s) Y t+1 (1 s)y t g Y 24 These type of approximations, taking the product of growth rates to be zero, tend to be exact as we reduce the time-interval and, in the limit, gives bache characterization obtained in continuous time models. 45

5 Thus, we have used the steady-state of the economy in efficiency units to compute the balanced growth path of the original economy g g K g Y g C. The Solow equilibrium represents some basic empirical facts: variables such as income consumption and capital tend to grow at the same rate and the ratio of capital stoco output is roughly constant. Moreover, the level of balanced growth g depends on demographics, the rate of growth of population (or labor force) and on the rate of technological progress. Studying the Solow s economy with variables expressed in per-worker terms (i.e. devided by N), one finds that the economy grows at the rate of technological progress; namely, the economy has no steady-state growth if there is no technological progress. In addition, notice that both of the growth determinants of Solow s model are exogenous. Modern growth theory has worked in the direction of explicitly representing (as equilibrium phenomena) both demographics and technological progress, providing models of endogenous growth. 25 Finally, observe that two economies (or countries) with different saving rates would have the same, steady state, rate of growth; and that, contrary to the empirical evidence, saving rates do not covariate with the growth rate in the medium- and short-run too. Instead, a change in the saving rate affects the steady-state, capital-output ratio. For completeness and for later use, we end this section computing the equilibrium of the economy in efficiency units. Recalling (13), use (11) to eliminate the growth rate of K, (14) s y t δ n µ skt α 1 δ n µ This is a non-linear difference equation of which we can determine the steady-state solution, ( k s δ + n + µ ) 1 > 0 iff s > 0 Problem 6. Consider the equilibrium dynamics of capital in efficiency units. Let γ( ) : 0 be the gross growth rate, so that, by equation (14), γ( ) s y t δ n µ (1) For g 0, draw the functions γ( ) g and γ( ) sk α t (δ n µ) in R 2 +, providing conditions such that they intersect twice, once at a > 0. (2) Explain why intersections, occurring at all t, identify balanced growth equilibria, one with no growth (a steady-state), the other exhibiting positive growth (in efficiency units!). (3) Show that the growth rate γ( ) decreases in. This first relationship is important since it explains why countries with identical fundamentals but different initial capital stock grow at different rates, with the poorer growing faster and catching up the richer. (4) Explain, also graphically, the effect of technological progress on /y t and, consequently, on the growth rate γ( ). (5) Show that the growth rate γ( ) increases in the saving rate s; then, analyze how your previous graphical representation changes with s. (6) Analyze the stability of the two equilibria in item 2), using both the graphical and the algebraic analysis. 25 See Acemoglu s (2009) textbook, for an extensive and updated analysis of neoclassical growth theories. 46

6 Problem 7. Consider the Solow model, for simplicity, without technological progress. The technology has CRTS, with f(k) : 1 N F (K, N) F ( K N, 1) F (k, 1), satisfying the assumptions in the text. (1) Give conditions under which a steady state exists. [Hint: f(k)/k (n + δ)/s must hold at steady state, thus the problem is to find a k such that the function g(k) (f(k)/k) satisfies g( k) (n+δ)/s. This can be proved by verifying the conditions for applying the intermediate value theorem, 26 then use Inada conditions to establish the limit values of g(k) as k approaches 0,.] (2) Can you prove that the steady state is unique when f(k) k α, 0 < α < 1 [Hint. start by plotting the graph of this function g.]? (3) Let the per-capita production function be f(k) 3 k, the depreciation rate δ is 10%, and the population growth rate n is 5%. Further, assume that individuals save 30% of their income. (a) What are the steady state, per-capita values of capital, output and consumption (, y t, c t )? (b) What happens when the saving rate is s.4? and when n.06 (and s.3)? (c) Suppose the production function is of the form F (K, N) KN.25N +.5K Give conditions under which a steady state exists.[hint. this depends on the fact that k > 0 and implies that the interest rate is high enough. This condition was unnecessary in the first item because the Inada conditions were satisfied; here they are not!? ] 6.3. The Ramsey-Cass-Koopmans growth model. Recall that in the Solow s economy the saving rate is given. We now endogenize consumption and saving decisions solving the Ramsey problem (P). Since we have studied that a solution to (P) is Pareto optimal, our analysis will obtain an efficient capital accumulation plan. As a result, a balanced growth path of a Ramsey economy defines an optimal, long-run, trend of the relevant economic variables: output, consumption, investment, employment. For expositional simplicity, consider the economy with inelastic labor supply. As for the Solow model, balanced growth with variables growing at an equal, positive rate is achieved introducing some growth in the fundamentals, population and technological progress (or factors productivity). Otherwise, we shall argue, the only possible balance growth is the steady state (i.e. zero growth of output, capital and consumption, with constant per period investment used to replace depreciated capital). To help the comparison with the Solow s model, we maintain our previous assumptions and consider the case of a Harrod-neutral, Cobb-Douglas technology. Thus, with a per-period, population growth n and an exogenous technological 26 This theorem says that if F ( ) is a real-valued, continuous function on the interval (a, b), and F (a) < c < F (b), then there is a z (a, b) such that F (z) c. In this setting, let (a, b) (0, ). 47

7 progress µ, we now express all the variables in efficiency units (keep using lower case letters): y t f( ) : Y ( ) α t Kt kt N α t A t N t Accordingly, the economy resource constraint (in efficiency units) is, c t k α t + (1 δ) +1 (1 + η) where, as before, η : n + µ. Since c t : C t / N t, with N t (1 + η) t N 0, (P) can be rewritten as follows, max t T β t u ( c t Nt ) such that, for all t in T, c t f( ) + (1 δ) +1 (1 + η), k 0 > 0 given Forming the Lagrangian, at all t in T, we derive the following necessary conditions for an interior solution of (P), with (λ t ) t T being a sequence of Lagrange multipliers, From the first condition one obtains, L 0 c t β t u (c t Nt ) N t λ t 0 L 0 +1 λ t+1 ( f (+1 ) + (1 δ) ) λ t 0 L 0 λ t c t + +1 (1 + η) f( ) (1 δ) 0 λ t+1 λ t β u From the second condition one obtains, ( ct+1 Nt+1 ) N t+1 N t u ( (c t Nt ) ) β u c t+1 Nt+1 u ( ) (1 + η) c t Nt λ t+1 (1 + η) ( f (+1 ) + 1 δ ) 1 λ t Putting the two together yields the Euler equation, βu ( ) c t+1 Nt+1 u ( ( (E) ) f (+1 ) + 1 δ ) 1 c t Nt where, we can define an implicit interest rate, r t : f ( ) δ. From section 5.2 we can approximate marginal utility and write explicitly the consumption dynamics as, ( ) C t+1 ct+1 (1 + η) 1 1 ( 1 1 ) C t c t σ(c t ) βr t+1 where σ(c t ) (u (C t )/u (C t )) C t is the elasticity of substitution of the per-period utility. This provides an important insight on the existence of a steady state in the economy in efficiency units (and of a balanced growth in the original one). Indeed, suppose a steady state exists, then (c t, ) are constant and the real interest is also constant. But then, the latter equation 48

8 holds provided that σ(c t ) is also constant over time. A steady state exists in the special case of a iso-elastic preferences, with σ(c t ) equal to a constant, σ > 0. Thus, also to ease the exposition, assume the per-period utility is iso-elastic, u(c) C 1 σ /(1 σ), σ > 0. First order conditions imply, ( ) ( λ t+1 β u ct+1 Nt+1 N t+1 λ t u β (1 + η) c ) σ t+1 (1 + η) (c t Nt ) N t c t Hence, (1 + r t+1 ) 1 (1 + η) β(1 + η) 1 σ }{{} β(1+η) yielding a new version of the Euler equation (E), ( ) σ ct+1 β (1 + r t+1) 1 c t ( ) σ ct+1 where β : β(1 + η) σ. For capital, as the condition (13) applies to this economy too, This yields an approximate relationship, (1 + η) +1 η +1 η that is, the rate of growth of the original economy differs from the one in efficiency units by η n + µ. The resource constraint can be rewritten as follows, simply adding and subtracting η, which, using the approximation, yields, (1 + η) kt α (δ + η) c t k α t (δ + η) c t Moreover, we can use the law of motion of capital to find the relationship between savings and growth. Since, s t : 1 c t y t (1 + η) y t (δ + η) c t y t (δ + η) (1 s t )y t s t y t (δ + η) c t That is, long-run excluded, the saving rate is positively correlated with the growth rate of capital. 49

9 In this economy there is no balanced growth, except for the steady state. y t kt α, ( ) y α t+1 kt+1 y t Indeed, since and a balanced growth g would require, 1 + g (1 + g) α, which holds if and only if g 0. Thus, we now compute the steady state, (y, c, k ). Thus, we now compute the steady state, (y, c, k ). From the Euler equation, αk α 1 β ( αk α δ ) 1 + η δ η β 1 + η δ 1 + β(1 + η) 1 σ δ 1 + (1 + θ)(1 + η) σ δ 1 + (1 + θ)(1 + ση) δ + θ + ση where the penultimate condition exploited a first-order approximation and the last assumes θση Solving for k, [ ] 1 α k [ ] α y k α α Implying, k α y Next, we solve for c, given y k α, and finally we compute the steady-state saving rate s. From the feasibility constraint, c y + (1 δ)k (1 + η)k y (δ + η)k From the law of motion of capital, at steady state, the saving rate is, s : k y (δ + η) α(δ + n + µ) The saving rate is higher the more patient is the household (θ close to zero). If, after a permanent shoco one of the parameters, the saving rate increases, capital increases only temporarily, adjusting the capital-output ratio. In fact, in the long-run the steady state equilibrium predicts no capital accumulation. 27 To clarify, suppose we approximate around a point in which there is neither population nor productivity growth, η n + µ 0. Then, at the first order, yielding, (1 + η) σ 1 + ση. (1 + η) σ (1 + η) σ η0 +σ(1 + η) η0 (η 0) 50

10 Finally, going baco the original variables, we can derive a balanced growth equilibrium. g : +1 η n + µ The ratio of capital stoco output is, Y t ( kt y t ) ss k α Hence, the ratio of capital to output is constant; this is so because they grow at the same rate (the balanced growth g). Indeed, at a balanced growth, [ ] α Y t y Nt k α (1 + n + µ) t α N 0 (1 + n + µ) t N 0 [ k Nt α ] 1 (1 + n + µ) t N 0 C t c Nt [y (δ + n + µ)k ] (1 + n + µ) t N 0 Thus, the model predicts that, an economy experiences a higher balanced growth the higher is technological progress and population growth, g η : n+µ; implying that without (exogenous) technological progress the per-capita variables display zero growth, in the long-run. capital-output ratio tends to be constant in the long-run, and it is lower the higher is g, σ, θ, δ (and s ) and the lower is α. A permanent shoco any of the parameters (δ, θ, σ, α) that makes the saving rate increase, will also determine a temporary adjustment of the great ratios; namely, it will increase the capital-output ratio and decrease the consumption-output ratio; this adjustment is such that the balanced growth remains constant at g (remember g n+µ is independent of the saving rate). The steady-state value of capital is exactly as in the Solow model (just use s to substitute into k and find), [ k s δ + n + µ ] 1 with the caveat that k is always below the golden rule capital stock k G, for β < 1 (or θ > 0). Therefore, the Solow model and the Ramsey-Cass-Koopmans optimal growth model predict the same steady state level of capital. However there are two major differences. First, in RCK the saving rate is endogenously determined, increasing in the share of capital income to income f (k )k /f(k ) α, decreasing in θ, possibly decreasing in n + µ. Second, in the RCK model the golden rule is never an equilibrium (for β < 1 or θ > 0), because it would violate optimality; conversely, it could be an equilibrium in the Solow model, for a sufficiently low (exogenous) saving rate. Moreover, there is a clear difference of the two models outside of the steady state, as in the optimal growth model the saving rate adjusts in response to economic shocks, and so do the great ratios. This is important as it raises the speed of adjustment of the economy when a permanent shoco fundamentals defines a new steady state. 51

11 A further interesting prediction of the model concerns cross-countries analysis. If two countries have the same economies except for the levels of capital stock, the country that has lower capital has to grow at a faster rate. Indeed, since the two economies have the same steady state, the transitional dynamics of the one with lower capital is characterized by a higher growth, in the transition phase to the steady state. This helps capturing why, for example, some developing countries, which roughly have the same technological opportunities and progress of some developed ones, tend to grow at much higher rates than richer, developed countries. You should try some more comparative statics using both the graphical and the algebraic analysis proposed in section 5. 52