I. The Solow model Dynamic Macroeconomic Analysis Universidad Autónoma de Madrid Autumn 2014 Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 1 / 33
Objectives In this first lecture we will study the Solow growth model. The Solow model is the basis for modern growth theory. It offers a simplified representation of the mechanics of growth, placing particular emphasis on the role of capital accumulation. Households own the capital stock and save a fixed proportion of their income. In later themes we will construct a genuine dynamic general equilibrium models with almost identical predictions. Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 2 / 33
Proximate vs fundamental causes of growth The Solow model helps us understand how capital accumulation, population growth and technological change contribute to the growth of living standards. In the terminology of Acemoglu these are proximate causes of economic growth. They fail to explain the persistence of the pronounced differences in living standards. The study of the fundamental causes of these income differences, such as the quality of institutions, is left for future study. Even so, a thorough understanding of the mechanics of growth is essential. Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 3 / 33
Main assumptions A closed economy without a government Y t = C t + I t Y t C t S t = I t A neo-classical production function (see below) Y t = F (K t, L t, A t ) The final good can be consumed or used as capital. Technology, A t, is non-rival and non-excludable. Households save a fraction s (0, 1) of their income. S t = I t = sy t while capital depreciates at a constant rate δ so that D t = δk t Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 4 / 33
Technology A neo-classical production function F (K, L, A) with F (0, 0, A) = 0 has the following features: 1 Positive but decreasing marginal products F K = F K > 0 ; 2 F K 2 = F KK < 0 ; 2 Constant returns to scale in K and L F L = F L > 0 2 F L 2 = F LL < 0 F (λk, λl, A) = λf (K, L, A) 3 The function F satisfies the Inada conditions lim K 0 F K = lim L 0 F L = ; lim K F K = lim L F L = 0 Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 5 / 33
Production side of the economy Essentially, there are two alternative interpretations of the production side Autarky (each household owns its own firm) A representative firm hires capital and labour on a competitive labour market In the latter case, the representative s problem is given by max K,L Π = F (K t, L t, A t ) r t K t w t L t r t = F K (.,.,.) w t = F L (.,.,.) F (K t, L t, A t ) = r t K t + w t L t where the last line follows from CTS. Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 6 / 33
Continuous time For convenience we analyze the model in continuous time The instantaneous change in any variable X t is denoted by X t = dx ( ) t X = t+ X t lim 0 dt Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 7 / 33
Law of motion of the capital stock The change in the capital stock at t, K t, is the difference between gross investment I t, and depreciation, D t : K t = I t D t = sy t K t = sf (K t, L t, A t ) δk t Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 8 / 33
Units For future purposes, it is convenient to convert all aggregate variables in per capita quantities. We denote the quantities per capita by lower-case letters. For simplicity, let L t denote both the size of the population and the labour force. Then κ t = K t L t y t = Y t L t = 1 L t F (K t, L t, A t ) = F (K t /L t, 1, A t ) = F (κ t, 1, A t ) = f (κ t, A t ) Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 9 / 33
Benchmark In our benchmark, we abstract from population growth or technical progress. So, L t = L A t = A while κ t = δ δt ( ) Kt = L K t K t 0 K L L 2 = t L Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 10 / 33
The Fundamental Equation of the Solow Model Combining the expressions for K t and κ t we obtain: K t = sf (K t, L, A) δk t K t L = 1 L sf (K t, L, A) δ K t L κ t = sf (κ t ) δκ t Formally, this is a non-linear first-order differential equation. Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 11 / 33
Example: Cobb-Douglas production function The Cobb-Douglas production function is a convenient example of a neo-classical production function: Y = AK α L 1 α y = AK α L 1 α = AK α L α L = Aκ α = f (κ, A) The function f (.,.) is a strictly concave function of κ with δf (κ, A) = f (κ, A) = αaκ α 1 = αa δκ κ 1 α lim κ 0 f (κ, A) = lim κ f (κ, A) = 0 Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 12 / 33
Convergence to steady state Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 13 / 33
Convergence to steady state The previous slide has demonstrated that the economy converges (monotonically) to a steady state with κ t = κ. The economy replicates itself because each agent (or household) saves a quantity i = sf (κ ) that exactly compensates for the depreciation of her capital d = δκ. Formally, for any initial capital stock κ 0 (0, κ ) the economy will experience a period of growth in which κ t and ẏ t are strictly positive. Once the economy reaches the steady state, growth comes to an end. Note, convergence to k will occur for any κ > 0, including the cases in which the economy starts out with a κ 0 > κ (see below). Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 14 / 33
What guarantees convergence? The assumption of a neo-classical production technology is a sufficient condition to guarantee convergence to a unique steady state: The Inada condition lim K 0 F K = guarantees that the curve sf (κ) is steeper than δκ near the origin; The diminishing marginal returns to capital guarantee that sf (κ) is a strictly concave function; The second Inada condition, lim K F K = 0, guarantees that the curve sf (κ) intersects deltaκ for some finite κ. The intersection is unique because the slope of sf (κ) is a strictly decreasing function of κ. Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 15 / 33
Relevant features of the steady state Our assumptions about technology guarantee that there is a unique steady state with a positive capital-labour ratio κ. There is also a trivial steady state with κ equal to zero, but we will ignore this unstable steady state. It is common to talk about steady-state equilibria. But formally the entire trajectory from κ 0 to κ should be part of any equilibrium definition. The unique positive levels of κ and y are strictly increasing in s and A, and decreasing in δ. By contrast, consumption per capita, c, is increasing in A but non-monotonic in s (lim s 0 c = lim s 1 c = 0). Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 16 / 33
The Golden Rule We cannot perform a formal welfare analysis as we have not specified agents preferences over consumption and labour. Nonetheless, we can ask ourselves whether the steady-state allocation generates the maximum feasible level of steady-state consumption. In any steady state, i = I t /L = δκ. Since c = f (κ ) i we can therefore define the golden-rule capital stock as argmax κ c = f (κ ) δκ The FOC that implicitly defines κ gold is: f (κ gold ) = δ Let s gold denote the savings rate required to attain a steady state with κ gold. There are no forces in the model to guarantee that s = s gold. Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 17 / 33
Steady-state consumption and the Golden Rule Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 18 / 33
Dynamic inefficiency The concept of the Golden Rule is not just useful to identify the allocation with the highest steady-state consumption level. It also allows us to recognize inefficient allocations. In particular, any steady state with κ > κ gold (or s > s gold ) is inefficient. If the agents were to reduce their savings rate to s gold consumption will converge to the maximum level c gold. And along the transition to steady state the agents will enjoy even higher consumption levels. In other words, if agents value consumption they must be unambiguously better off than under the initial steady state. Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 19 / 33
Intertemporal tradeoffs Now suppose the economy is located in a steady state with κ < κ gold (and s < s gold ). Once more c < c gold, but this time we cannot make unambiguous efficiency statements. To reach the Golden Rule capital stock, the agents have to raise their savings rate. This generates an immediate reduction in consumption from c = (1 s)f (κ ) to c = (1 s gold )f (κ ). Consumption will grow over time and will eventually exceed c. In other words, the agents have to trade off lower consumption today against higher consumption in the future. A formal analysis of the welfare consequences requires a fully specified model with preferences. Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 20 / 33
Population growth Once we allow for population growth, the economy converges to a steady state in which all aggregate variables grow at the same rate as L t, but again there is no growth in per capita variables. Constant population growth L t L t = n κ t = δ δt ( Kt L t = K t L t κ t ) = L t K t K t L t ( ) L t L t L 2 t = K t L t κ t n Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 21 / 33
The fundamental equation with population growth Recall that the evolution of the capital stock is governed by K t = sf (K t, L t, A) δ K t L t L t L t κ t = K t L t κ t n Combining these expressions, we obtain In steady state ( κ t = 0) κ t = sf (κ t ) (n + δ)κ t sf (κ ) = (n + δ)κ Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 22 / 33
Steady state with population growth Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 23 / 33
Characteristics of the steady state In steady state all per capita variables, such as κ t, y t, i t or c t, remain constant over time. By contrast, all aggregate variables grow at the same rate as the population. For example, K t = κ L t log(k t ) = log(κ ) + log(l t ) Taking derivatives with respect to t yields K t K t = 0 + L t L t = n From CRTS, it follows that Y t and hence C t and I t also grow at rate n. Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 24 / 33
Transitional dynamics Since y t is stricly increasing in κ t, the growth rate of y (ẏ t / t ) is proportional to the growth rate of κ γ κ κ κ = s f (κ, A) κ (n + δ) The concavity of f (.,.) implies that the average output per unit of capital is decreasing in κ. Hence the growth rate of κ is strictly decreasing in κ with γ κ > 0 for all κ t < κ and γ κ < 0 for κ t > κ. This combined with lim κ 0 (f (κ, A)/κ) = guarantees existence and uniqueness. Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 25 / 33
Growth and convergence Correct interpretation: g=0 Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 26 / 33
Technological progress In the basic Solow model we can only have sustained growth if there is sustained technological progress. Moreover, balanced growth is only possible if technological progress is labour-augmenting. Formally, we will assume Y t = F (K t, A t L t ) Ȧ t A t = x Next, we let ˆL t = A t L t denote the efficiency units of labour and ˆκ t = K t ˆL t = K t A t L t yˆ t = F (K t, ˆL t ) = f ( ˆκ t ) ˆL t Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 27 / 33
Balanced growth with exogenous technological change Following the same procedure like before, we arrive at ( δ Kt δt A t L t ) = K t ˆL t ˆκ t (x + n) K t ˆL t = sf ( ˆκ t ) δ ˆκ t Combing the above equations we arrive at the fundamental equation with technological growth δ ˆκ t δt = sf ( ˆκ t) (n + δ + x) ˆκ t Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 28 / 33
Interpretation of the balanced growth path Once the economy reaches the steady state, All quantities per efficiency unit (ŷ, ĉ and ˆκ) are constant over time All quantities per capita (y, c and κ) grow at the rate n. All aggregate variables (Y, C and K) grow at the rate n + x. However, it should be reminded that the growth of A t is generated exogenously. Eventually, we would like to understand the drivers behind technological progress (investments in human capital, R&D, etc. ). In order to generate endogenous growth with investment in R&D we need to abandon the assumption of perfect competition. Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 29 / 33
Conditional convergence The model predicts that the growth rate of capital and income per capita are falling in the level of capital. Hence, if the level of capital is the ONLY difference between two countries, then: The poor country should grow at a faster rate than the rich country Both countries should eventually converge to the same steady state. The conditional convergence of income levels is a prediction of the model that is verified by the data. Similarly, nothing prevents the divergence between countries with different fundamentals. Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 30 / 33
Sustained growth The AK model In order to generate sustained growth without exogenous technological change, we need to relax some of the assumptions of the Solow model. A logical candidate is to consider alternatives to the assumption of a neo-classical production technology In the Solow model growth vanishes due to the decreasing marginal product of capital We can show that perpetual growth is feasible if output is linear in K so that Y t = AK t This interpretation is acceptable if we use a broad concept of capital that includes both human and physical capital. Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 31 / 33
Sustained growth The AK model In the AK-model, per capita output is equal to y = AK /L = Aκ. In other words, the fundamental equation remains valid κ = sy (n + δ)κ = saκ (n + δ)κ and so the growth rate of κ can be written as κ κ = sa (n + δ) which is constant and independent of κ. Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 32 / 33
Distinctive features of the AK model 1 The model produces sustained growth without sustained growth in an exogenous variable like A 2 The growth rate is increasing in the savings rate 3 No transitional dynamics the growth rate is constant and equal to sa (n + δ) 4 No convergence 5 Recessions produce permanent effects on living conditions 6 The economy is never dynamically inefficient Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 33 / 33