Lecture notes on modern growth theory

Transcription

1 Lecture notes on modern growth theory Part 2 Mario Tirelli Very preliminary material Not to be circulated without the permission of the author October 25, 2017 Contents 1. Introduction 1 2. Optimal economic growth: the Ramsey-Cass-Koopmans model 1 3. Qualitative predictions 5

2 1. Introduction We are now ready to analyze the balanced growth equilibrium of a Ramsey-Cass-Koopmans RCK economy and to compare it with Solow-Swan s. The main difference in the two contributions is that, unlike in the Solow s economy, in RCK consumption and saving decisions are determined endogenously. More precisely, we find the optimal allocation as a solution of the Ramsey problem P; then, by the II Welfare Theorem, we conclude that this allocation can be decentralized as a competitive equilibrium of a Walrasian economy i.e. one with a complete set of future commodity markets, or with spot commodity markets and a complete set of asset markets allowing for sequential trade. Finally, we discuss how accounting for consumption and saving decisions might sharpen Solow s qualitative predictions, and why some early criticisms remain valid. 2. Optimal economic growth: the Ramsey-Cass-Koopmans model For expositional simplicity, consider the RCK economy with inelastic labor supply. As for the Solow model, balanced growth with variables growing at an equal, positive rate is achieved introducing some exogenous 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 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, 1 = k α t + 1 δ η where, as before, η := n + µ. Since := / N t, with N t = 1 + η t N 0, P can be rewritten as follows, max t T β t u Nt such that, for all t in T, = f + 1 δ η, k 0 > 0 given 1

3 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 = β t u Nt N t λ t = 0 L 0 +1 = λ t+1 f δ λ t = 0 L 0 λ t = η f 1 δ = 0 λ t+1 λ t = β u From the second condition one obtains, λ t+1 λ t ct+1 Nt+1 N t+1 N t u Nt = β u +1 Nt+1 u 1 + η Nt = 1 + η f δ 1 Putting the two together yields the Euler equation, βu +1 Nt+1 u E f δ = 1 Nt where, we can define an implicit interest rate, r t := f δ. As before, we can approximate marginal utility and write explicitly the consumption dynamics. First-order approximation of marginal utility around, for a sufficiently small length period to make +1 small, yields: u +1 = u + u +1. Dividing through by u, u +1 u = 1 + u u C +1 t Using the elasticity of substitution of the per-period utility, σ = u /u, β u +1 u = β 1 σ C t+1 Using this to rewrite the Euler equation gives, E +1 = 1 σ where, by definition, +1 / = 1 + η+1 / 1 1 β1 + r t+1 +1 η = +1 + g η +1 and the following can be easily derived, This provides an important insight on the existence of a balanced growth. Indeed, by E, grows at a constant rate g C := η only if σ is also constant over time. Equivalently, a balanced growth exists only if preferences are iso-elastic, with σ = σ > 0. Thus, we the 2

4 aim of studying efficient dynamics entailing balance growth, we let the per-period utility be, uc = C 1 σ /1 σ, σ > 0, with associated intertemporal marginal rate of substitution, β u +1 u = β Ct+1 σ = β1 + η σ ct+1 σ Substituting into E, we obtain a new version of the Euler equation, Eσ σ ct+1 β 1 + r t+1 = 1 where β := β1 + η σ. Letting consumption growth, in efficiency units, be measured by gt c := +1 / 1, we can rewrite Eσ as, 1 + gt c = βr t+1 1/σ the optimal consumption growth is increasing in the return to investing in physical capital. The resource constraint 1 can be rewritten as follows, simply adding and subtracting η from both sides, dividing through by and using η +1 / 0, +1 = y t δ + η We can use the law of motion of capital to find the relationship between savings and growth. Since, s t := 1 y t +1 = y t δ + η = y t δ + η y t 1 s t = s t δ + η or +1 = 1 s t δ + η This implies that the saving rate is positively correlated with the growth rate of capital. At balance growth i.e. in the long-run, when +1 = 0, income per capita is negatively correlated with savings, as in the Solow s model: yields, y = δ + ηk /s. Steady-state. For the exact same reasons as in the Solow s economy, the Ramsey s in efficiency units has no balanced growth too, only a steady state might exists. Thus, assuming that a steady state exists, we now compute it, letting y t,,, r t = y, c, k, r at all t in T. From the Euler equation Eσ, β αk α δ = 1 3

5 αk α 1 = δ β 1 = δ 1 + β1 + η σ = δ θ1 + η σ δ θ1 + ση δ + θ + ση where the third condition uses the definition of the individual discount rare θ = β 1 1, the penultimate one exploits a first-order approximation and the last condition assumes σθη 0. 1 Solving for k, α k = δ + θ + ση y = k α = ] 1 α δ + θ + ση ] α Implying, k α = y δ + θ + ση 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 δk 1 + ηk = y δ + ηk From the law of motion of capital, at steady state, the saving rate is, s := k y δ + η = αδ + η δ + θ + ση 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. Balanced growth of the original economy. Finally, going baco the original variables, we can derive a balanced growth equilibrium. For capital, as the condition found studying the Solow economy applies here too; hence, at steady state, g := K t+1 K t kt+1 ss + η = η that is, the rate of growth of the original economy g is exogenously given by η = n + µ. The ratio of capital stoco output is, K t kt = = k α = Y t δ + θ + σn + µ y t ss 1 To clarify, suppose we approximate around a point in which there is neither population nor productivity growth, η = 0. Then, at the first order, yielding, 1 + η σ 1 + ση. 1 + η σ 1 + η σ η=0 +σ1 + η η=0 η 0 4

6 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 δ + θ + σn + µ K t = k Nt = α δ + θ + σn + µ ] n + µ t N 0 = c Nt = y δ + n + µk ] 1 + n + µ t N 0, Y t, K t have a growth rate that equals η, the growth rate of N t. Aggregate savings do also grow at η, simply because S t = s Y t = I t. 3. Qualitative predictions We can now use the results from our previous analysis to summarize the qualitative predictions obtained with the RCK model and compare them with Solow s. The RCK 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. This is again as in the Solow model. 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 have very similar qualitative predictions. However there are two major differences: 1 In RCK the saving rate is endogenously determined, increasing in the share of capitalincome f k k /fk = α, decreasing in θ impatience and in n + µ. 2 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 high 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 5

7 great ratios. This is important as it raises the speed of adjustment of the economy when a permanent shoco fundamentals leads to a new steady state. In particular, it changes considerably the response of consumption. You should try some more comparative statics using both the graphical and the algebraic analysis. 6