I. The Solow model Dynamic Macroeconomic Analysis Universidad Autónoma de Madrid September 2015 Dynamic Macroeconomic Analysis (UAM) I. The Solow model September 2015 1 / 43
Objectives In this first lecture we study the Solow(-Swan) growth model. The Solow model is the basis for modern growth theory. It offers a concise representation of the mechanics of growth, placing particular emphasis on the role of capital accumulation. Households own the capital stock and save a fixed (and arbitrary) proportion of their income. One of the central predictions shows that the economy tends to converge to a (dynamically) inefficient allocation. In later themes we will derive similar predictions in an OLG model in which agents have finite lives. Dynamic Macroeconomic Analysis (UAM) I. The Solow model September 2015 2 / 43
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 September 2015 3 / 43
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. Households save a fraction s (0, 1) of their income. S t = I t = sy t Capital depreciates at a constant rate δ so that D t = δk t Dynamic Macroeconomic Analysis (UAM) I. The Solow model September 2015 4 / 43
Technology A neo-classical production function F (K, L, A). Technology, A, is non-rival and non-excludable. 1 F (0, 0, A) = 0 2 Decreasing marginal product of capital and labor F K = F K > 0 ; 2 F K 2 = F KK < 0 ; 3 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) 4 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 September 2015 5 / 43
The production side of the economy There are two alternative interpretations of the production side Autarky each household owns its own firm Competitive product markets with a representative firm and household In the latter case, the problem of the representative firm 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 ) = F K K t + F L L t = r t K t + w t L t where the last line follows from Euler s Theorem Dynamic Macroeconomic Analysis (UAM) I. The Solow model September 2015 6 / 43
Continuous time The standard law of motion of capital in discrete time is K (t + 1) = (1 δ)k (t) + I (t) Here we opt for an analysis in continuous time using a linear approximation: K (t + 1) K (t) = δk (t) + I (t) = I (t) δk (t) K (t + t) K (t) = t(sy (t) δk (t)) K (t + t) K (t) Kt = lim t 0 t = sy (t) δk (t) = sf (K (t), L(t), A) δk (t) Dynamic Macroeconomic Analysis (UAM) I. The Solow model September 2015 7 / 43
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 September 2015 8 / 43
Per capita variables For future purposes, it is convenient to convert all aggregate variables in per capita quantities. 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 September 2015 9 / 43
Benchmark specification In our benchmark, we abstract from population growth or technical progress. So, L t = L A t = A Consequently, we can derive the following result: 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 September 2015 10 / 43
The Fundamental Equation of the Solow Model The evolution of all relevant variables is defined by the evolution of the capital stock. 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 The fundamental equation is a non-linear first-order differential equation. Dynamic Macroeconomic Analysis (UAM) I. The Solow model September 2015 11 / 43
Example: Cobb-Douglas production function The Cobb-Douglas production function is the most well-known example of a neo-classical production function: F (K, L, A) = AK α L 1 α F (K, L, A) L = AK α L 1 α L = Aκ α = f (κ, A) = AK α L α The function f (.,.) is a strictly concave function of κ with f (κ, A) = αaκ α 1 = αa κ 1 α lim κ 0 f (κ, A) = lim κ f (κ, A) = 0 Dynamic Macroeconomic Analysis (UAM) I. The Solow model September 2015 12 / 43
Convergence to steady state The evolution over time of our economy is divided in two phases: An initial period of growth in all (per capita) variables Steady state A steady state allocation is defined as an allocation in which all relevant variables remain constant over time. The capital stock remains constant because savings are equal to depreciation. κ t = sf (κ t ) δκ t = 0 Dynamic Macroeconomic Analysis (UAM) I. The Solow model September 2015 13 / 43
Convergence to steady state Dynamic Macroeconomic Analysis (UAM) I. The Solow model September 2015 14 / 43
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 September 2015 15 / 43
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 September 2015 16 / 43
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. Comparative statics: κ and y are strictly increasing in s and A, and decreasing in δ. By contrast, 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 September 2015 17 / 43
Example: Cobb-Douglas Remember, in the case of the Cobb-Douglas production function we can write: The Fundamental Equation: f (κ) = Aκ α t κ t = saκ α t δκ t Steady state: Steady-state capital stock saκ α = δκ κ = ( ) 1 sa 1 α δ Dynamic Macroeconomic Analysis (UAM) I. The Solow model September 2015 18 / 43
Welfare We cannot perform a formal welfare analysis since we have not specified agents preferences over (current and future) consumption and labour. Nonetheless, we can ask ourselves whether the steady-state allocation generates the maximum feasible level of steady-state consumption. The steady-state allocation that does so is called the Golden Rule allocation Dynamic Macroeconomic Analysis (UAM) I. The Solow model September 2015 19 / 43
The Golden Rule 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 September 2015 20 / 43
Example: Cobb-Douglas production technology The Golden Rule capital stock is defined by f (κ GR ) = δ In the case of the Cobb-Douglas production function this leads to: Hence, αa(κ GR ) α 1 = δ κ GR = ( ) 1 αa 1 α δ while the steady state capital stock for any given s is defined by: κ = ( ) 1 sa 1 α δ Dynamic Macroeconomic Analysis (UAM) I. The Solow model September 2015 21 / 43
Predictions There is exactly one savings rate, s = α for which the decentralised steady state allocation coincides with the Golden Rule. For all s = α, c(s) < c GR There are no forces in the model that align the savings rate, s, and the capital share, α In steady state, the private agents may save too little or they may over-accumulate capital. In both cases, they consume less than c GR. Dynamic Macroeconomic Analysis (UAM) I. The Solow model September 2015 22 / 43
Steady-state consumption and the Golden Rule Dynamic Macroeconomic Analysis (UAM) I. The Solow model September 2015 23 / 43
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 September 2015 24 / 43
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 September 2015 25 / 43
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 September 2015 26 / 43
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 September 2015 27 / 43
Steady state or balanced growth path with population growth Dynamic Macroeconomic Analysis (UAM) I. The Solow model September 2015 28 / 43
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 September 2015 29 / 43
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 September 2015 30 / 43
Growth and convergence Correct interpretation: g=0 Dynamic Macroeconomic Analysis (UAM) I. The Solow model September 2015 31 / 43
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 September 2015 32 / 43
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 September 2015 33 / 43
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 September 2015 34 / 43
(Un)conditional convergence Source: Acemoglu (2008) Dynamic Macroeconomic Analysis (UAM) I. The Solow model September 2015 35 / 43
(Un)conditional convergence Source: Acemoglu (2008) Dynamic Macroeconomic Analysis (UAM) I. The Solow model September 2015 36 / 43
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 September 2015 37 / 43
Decentralized equilibrium So far we have abstracted from markets, assuming that agents operate a backyard technology. It is straightforward to relax this assumption and assume that households offer their capital and labour services on competitive markets The representative firm maximizes profits so that w t = F L (K t, L t ) R t = F K (K t, L t ) All factor markets clear at all dates Dynamic Macroeconomic Analysis (UAM) I. The Solow model September 2015 38 / 43
Equilibrium Consider the version with population growth: For a given sequence of {L(t), A t } t=0 and and initial capital stock K 0 an equilibrium path is a sequence of capital stocks, output levels, consumption levels, wage rates and rental prices {K t, Y t, C t, w t, R t } t=0 such that K t = sf (A t, K t, L t ) (n + δ) Y t = F (A t, K t, L t ) C t = (1 s)y t R t = F K () = f (κ t ) w t = F L () = f (κ t ) f (κ t ) Dynamic Macroeconomic Analysis (UAM) I. The Solow model September 2015 39 / 43
Dynamic inefficiency In this economy, the agents receive steady state net-interest rate r = R δ = f (κ ) δ. The Golden Rule capital stock is defined by f (κ GOLD ) = n + δ Thus for dynamically inefficient allocations with k > k GOLD, it must true that f (κ ) = r + n < n + δ r < n In later themes we will see that this condition implies that a PAYGO pension scheme improves welfare Dynamic Macroeconomic Analysis (UAM) I. The Solow model September 2015 40 / 43
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 September 2015 41 / 43
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 September 2015 42 / 43
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 September 2015 43 / 43