The Solow Growth Model Lectures 5, 6 & 7 Topics in Macroeconomics Topic 2 October 20, 21 & 27, 2008 Lectures 5, 6 & 7 1/37 Topics in Macroeconomics
From Growth Accounting to the Solow Model Goal 1: Stylized facts of economic growth Goal 2: Understandingdifferencesover time and across countries Outline From Growth Accounting to the Solow Model 2 In growth accounting link of inputs in period t to output in period t no link of inputs or output across periods (t versus t + 1) Solow model links population/labor force, productivity and, in particular, capital stock in year t to labor force, productivity and capital stock in year t + 1 Solow (1956), Solow (1957) and Solow (1960) Lectures 5, 6 & 7 2/37 Topics in Macroeconomics
From Growth Accounting to the Solow Model Goal 1: Stylized facts of economic growth Goal 2: Understandingdifferencesover time and across countries Outline From Growth Accounting to the Solow Model 3 Solow s story about how the capital stock evolves over time Households save investment Households save a (constant) fraction s [0, 1] of their income every period/year Households consume the rest, i.e., fraction (1 s) of income Aggregate income : Yt Aggregate investment = It = sy t Law of motion of aggregate capital (δ [0, 1]) K t+1 = (1 δ)k t + I t Lectures 5, 6 & 7 3/37 Topics in Macroeconomics
From Growth Accounting to the Solow Model Goal 1: Stylized facts of economic growth Goal 2: Understandingdifferencesover time and across countries Outline Kaldor facts: Stylized facts of economic growth 4 1. The labor share and the capital share are almost constant over time. 2. The ratio of aggregate capital to output is almost constant over time. 3. The return to capital is almost constant over time. 4. Output per capita and capital per worker grow at a roughly constant and positive rate. 5. Different countries and regions within a country that start out with a different level of income per capita tend to converge over time. Lectures 5, 6 & 7 4/37 Topics in Macroeconomics
From Growth Accounting to the Solow Model Goal 1: Stylized facts of economic growth Goal 2: Understandingdifferencesover time and across countries Outline Understanding growth differences over time and across countries 5 Why do (developed) countries grow? Will developing countries catch up to developed countries? Solow model: a first attempt to explain the mechanics of growth Implications of Solow s theory: differences in initial condition, effectiveness of labor and population growth matter Lectures 5, 6 & 7 5/37 Topics in Macroeconomics
From Growth Accounting to the Solow Model Goal 1: Stylized facts of economic growth Goal 2: Understandingdifferencesover time and across countries Outline Outline 6 Assumptions Inputs Production function Depreciation Evolution of technology Evolution of population/labor force Consumption and savings Results Evolution of the capital stock Steady state Balanced Growth Lectures 5, 6 & 7 6/37 Topics in Macroeconomics
From Growth Accounting to the Solow Model Goal 1: Stylized facts of economic growth Goal 2: Understandingdifferencesover time and across countries Outline Further steps 7 Comparative statics Savings rate Population growth Technological change The Golden Rule Implications for Cross-country differences in GDP levels and growth rates Convergence across countries Lectures 5, 6 & 7 7/37 Topics in Macroeconomics
Assumptions Aggregation Firm s problem Law of motion of aggregate capital stock Assumptions of the Solow model 8 Assumptions Inputs: capital, Kt and labor L t Production function: neo-classical production function Depreciation: capital depreciates at rate δ [0, 1] from t to t + 1 Evolution of technology: A t+1 = (1 + g)a t, Evolution of population (labor force*): L t+1 = (1 + n)l t where δ, g and n are exogenously given parameters Lectures 5, 6 & 7 8/37 Topics in Macroeconomics
Assumptions Aggregation Firm s problem Law of motion of aggregate capital stock Assumptions of the Solow model 9 Last Assumption Consumption and savings: consumers save a constant fraction s of their income, y t, consume fraction (1 s) (s parameter) Per person income is: yt = r t k t + w t l t Labor is supplied inelastically & normalized to lt = 1 Savings per person are: syt Lectures 5, 6 & 7 9/37 Topics in Macroeconomics
Assumptions Aggregation Firm s problem Law of motion of aggregate capital stock Aggregating consumers 10 Savings per person are: sy t = s(r t k t + w t ) Multiplying by the number of people in period t pause Aggregate Savings/Investment = I t = L t sy t = L t s(r t k t + w t ) = s(r t K t + w t L t ) Lectures 5, 6 & 7 10/37 Topics in Macroeconomics
Assumptions Aggregation Firm s problem Law of motion of aggregate capital stock Firm s problem (lecture 2) 11 ] maxπ(k t, A t L t ) = max [F(K t, AL t ) r t K t w t L t Firms take prices as given and choose inputs K and L First order conditions Πt K t = F K (K t, A t L t ) r t = 0 Πt L t = F L (K t, A t L t ) w t = 0 Firm picks K t and L t such that FK (K t, A t L t ) = r t FL (K t, A t L t ) = w t Lectures 5, 6 & 7 11/37 Topics in Macroeconomics
Assumptions Aggregation Firm s problem Law of motion of aggregate capital stock Law of motion of aggregate capital stock 12 Using the solution to the firm s problem, we showed that r t K t + w t L t = F(K t, A t L t ) = Y t (lecture 2) Using the aggregation over consumers, we saw earlier I t = s(r t K t + w t L t ) Therefore, I t = sy t = sf(k t, A t L t ) Law of motion of aggregate capital K t+1 = (1 δ)k t + I t Consider K t+1 as a function of K t Lectures 5, 6 & 7 12/37 Topics in Macroeconomics
Law of motion (simple case) Steady state (simple case) Comparative statics (simple case) Comparative dynamics (for n = 0 and g = 0) The Solow Model (for n = 0 and g = 0) Lectures 5, 6 & 7 13/37 Topics in Macroeconomics
Law of motion (simple case) Steady state (simple case) Comparative statics (simple case) Comparative dynamics (for n = 0 and g = 0) Law of motion: simple case n = 0 and g = 0 14 Consider K t+1 as a function of K t : K t+1 = (1 δ)k t + I t K t+1 = (1 δ)k t + sy t K t+1 = (1 δ)k t + sf(k t, Ā L) Since marginal product of K positive, law of motion: increasing function Since marginal product of K diminishing law of motion: concave function Lectures 5, 6 & 7 14/37 Topics in Macroeconomics
Law of motion (simple case) Steady state (simple case) Comparative statics (simple case) Comparative dynamics (for n = 0 and g = 0) Solow s law of motion 15 50 45 40 35 30 K_t+1 25 20 15 10 5 Kt+1 = Kt (45 degree line) Kt+1 = (1-delta) Kt + s F(Kt,AL) 0 0 10 20 30 40 50 K_t Lectures 5, 6 & 7 15/37 Topics in Macroeconomics
Law of motion (simple case) Steady state (simple case) Comparative statics (simple case) Comparative dynamics (for n = 0 and g = 0) Solow s law of motion 16 50 45 40 Kt+1 = Kt (45 degree line) Kt+1 = (1-delta) Kt + s F(Kt,AL) K_t+1 35 30 25 20 15 10 5 0 0 10 20 30 40 50 K_t Lectures 5, 6 & 7 16/37 Topics in Macroeconomics
Law of motion (simple case) Steady state (simple case) Comparative statics (simple case) Comparative dynamics (for n = 0 and g = 0) Steady state 17 The state variable of this economy is capital K t We say that the economy is at a steady state if the state variable remains constant. That is capital is constant at K, K = K t = K t+1 Using the C-D production function, we get K t+1 = (1 δ)k t + sk α t (Ā L) 1 α K = (1 δ)k + s(k ) α (Ā L) 1 α Solving this equation for K yields* K = ( s δ ) 1 1 α A L Lectures 5, 6 & 7 17/37 Topics in Macroeconomics
Law of motion (simple case) Steady state (simple case) Comparative statics (simple case) Comparative dynamics (for n = 0 and g = 0) Comparative statics 18 K = ( s δ ) 1 1 α Ā L If s increases, K increases * If δ increases, K decreases* If A increases, K increases* If L increases, K increases* Lectures 5, 6 & 7 18/37 Topics in Macroeconomics
Law of motion (simple case) Steady state (simple case) Comparative statics (simple case) Comparative dynamics (for n = 0 and g = 0) Comparative dynamics 19 Suppose the level of the capital stock in some economy (country) in year t is at its steady state level K t = K = ( s δ ) 1 1 α Ā L That is, there is no more growth, i.e. K t+1 = K t. In t + 1, s suddenly increases to s > s, sf(k t, AL) increases to s F(K t, AL) K increases to K > K On the graph, we can see that now, the economy starts growing again, i.e. K t+2 > K t+1 (drawn in class)*...until the capital stock reaches the new steady state...k Lectures 5, 6 & 7 19/37 Topics in Macroeconomics
Law of motion (simple case) Steady state (simple case) Comparative statics (simple case) Comparative dynamics (for n = 0 and g = 0) Homework 20 Derive the same reasoning for * If δ decreases or increases* If A decreases or increases* If L decreases or increases* Lectures 5, 6 & 7 20/37 Topics in Macroeconomics
Law of motion (simple case) Steady state (simple case) Comparative statics (simple case) Comparative dynamics (for n = 0 and g = 0) Steady state comparative statics: savings rate s 21 50 45 40 35 30 K_t+1 25 20 15 10 5 Kt+1 = Kt (45 degree line) Kt+1 = (1-delta) Kt + s F(Kt,AL) 0 0 10 20 30 40 50 K_t Please complete as drawn in class. Lectures 5, 6 & 7 21/37 Topics in Macroeconomics
Law of motion (simple case) Steady state (simple case) Comparative statics (simple case) Comparative dynamics (for n = 0 and g = 0) Steady state comparative statics: deprec. rate δ 22 50 45 40 35 30 K_t+1 25 20 15 10 5 Kt+1 = Kt (45 degree line) Kt+1 = (1-delta) Kt + s F(Kt,AL) 0 0 10 20 30 40 50 K_t Please complete as drawn in class. Lectures 5, 6 & 7 22/37 Topics in Macroeconomics
Law of motion (simple case) Steady state (simple case) Comparative statics (simple case) Comparative dynamics (for n = 0 and g = 0) Steady state comparative statics: productivity A 23 50 45 40 35 30 K_t+1 25 20 15 10 5 Kt+1 = Kt (45 degree line) Kt+1 = (1-delta) Kt + s F(Kt,AL) 0 0 10 20 30 40 50 K_t Please complete as drawn in class. Lectures 5, 6 & 7 23/37 Topics in Macroeconomics
Law of motion (simple case) Steady state (simple case) Comparative statics (simple case) Comparative dynamics (for n = 0 and g = 0) Next steps 24 What happens if there is exogenous technological progress? What if there is population growth? Ramsey Model What if people explicitly choose how much to save? Does the savings rate depend on the rate of technological progress, the rate of depreciation, preferences, labor s share in output, taxes..., and if so, how? Endogenous Growth What if there is no steady state? can there be endogenous growth forever? Lectures 5, 6 & 7 24/37 Topics in Macroeconomics
Law of motion (general case) Balanced Growth (general case) Goals? The Solow Model with population growth (n 0) and technological progress (g 0) Lectures 5, 6 & 7 25/37 Topics in Macroeconomics
Law of motion (general case) Balanced Growth (general case) Goals? Outline 26 Introduce n 0 and g 0 Derive growth rate of capital stock per worker on balanced growth path (BGP) Derive growth rate of GDP (output) per capita on BGP Derive growth rate of wages Compare results to Kaldor stylized facts of growth Comparative statics of growth rates on BGP Lectures 5, 6 & 7 26/37 Topics in Macroeconomics
Law of motion (general case) Balanced Growth (general case) Goals? Balanced growth: n 0 and g 0 27 Evolution of technology: A t+1 = (1 + g)a t, Evolution of population (labour force*): L t+1 = (1 + n)l t Law of motion of aggregate capital K t+1 = (1 δ)k t + sf(k t, A t L t ) Want to find growth rate of capital per worker, k t = K t L t GDP per capita y t = Y t L t and Lectures 5, 6 & 7 27/37 Topics in Macroeconomics
Law of motion (general case) Balanced Growth (general case) Goals? Redefine variables per unit of effective labour 28 With technological progress and population growth (g 0, n 0) Law of motion: K t+1 is not a stable function of K t transform law of motion to get stable function Let ŷ t = Y t A t L t output per unit of effective labour ˆk t = K t A t L t capital per unit of effective labour Then we can write ŷ t A t L t = Y t ˆk t A t L t = K t Lectures 5, 6 & 7 28/37 Topics in Macroeconomics
Law of motion (general case) Balanced Growth (general case) Goals? Law of Motion 29 Law of motion becomes K t+1 = (1 δ)k t + sy t or, ˆk t+1 A t+1 L t+1 = (1 δ)ˆk t A t L t + sŷ t A t L t or, ˆk t+1 (1 + g)a t (1 + n)l t = (1 δ)ˆk t A t L t + sŷ t A t L t ˆk t+1 (1 + g)(1 + n) = (1 δ)ˆk t + sŷ t Lectures 5, 6 & 7 29/37 Topics in Macroeconomics
Law of motion (general case) Balanced Growth (general case) Goals? Law of Motion 30 Law of motion for capital per unit of effective labour ] 1 ˆk t+1 = (1+g)(1+n) [(1 δ)ˆk t + sŷ t Note that ŷ t = Y t A t L t = F(K t,a t L t ) A t L t = ˆk t α Lectures 5, 6 & 7 30/37 Topics in Macroeconomics
Law of motion (general case) Balanced Growth (general case) Goals? Steady state in units of effective labour (=BGP) 31 Using ŷ t = ˆk t α, law of motion for ˆk t [ ] 1 ˆk t+1 = (1+g)(1+n) (1 δ)ˆk t + sˆk t α Law of motion: ˆk t+1 is now a stable function of ˆk t (none of the parameters depends on t) Show that ˆk t+1 is an increasing and concave function of ˆk t if α,δ [0, 1], g, n [ 1, 1] Lectures 5, 6 & 7 31/37 Topics in Macroeconomics
Law of motion (general case) Balanced Growth (general case) Goals? Solow s law of motion (capital per u. of eff. labour) 32 50 45 40 k(hat)_t+1 = k(hat)_t (45 degree line) k(hat)_t+1 = (1-delta) k(hat)_t + s k(hat)t^alpha 35 k(hat)_t+1 30 25 20 15 10 5 0 0 10 20 30 40 50 k(hat)_t Lectures 5, 6 & 7 32/37 Topics in Macroeconomics
Law of motion (general case) Balanced Growth (general case) Goals? Per capita/worker variables along BGP 33 Steady state ito capital per u. of eff. labour [ 1 ˆk t+1 = (1+g)(1+n) (1 δ)ˆk t + sˆk t α This again can be solved for ˆk, the value for which capital per unit of effective labour does not change anymore, i.e. ˆk t = ˆk t+1 = ˆk ( * ˆk = s g+n+ng+δ ] ) 1 1 α Lectures 5, 6 & 7 33/37 Topics in Macroeconomics
Law of motion (general case) Balanced Growth (general case) Goals? Capital per worker and GDP per capita 34 When the capital stock per unit of effective labour, ˆk t, reaches its steady state level ˆk, we get: Growth rate of capital per worker: k t+1 k t = K t+1 L t+1 K t L t = A t+1 K t+1 A t+1 L t+1 A t K t = A t+1 ˆk A t L t A t ˆk = (1 + g) Growth rate of output per capita: y t+1 y t = kα t+1 A1 α t+1 kt α A 1 α t = ( kt+1 k t ) α ( At+1 ) 1 α = (1+g) α (1+g) 1 α A t y t+1 y t = (1 + g) Lectures 5, 6 & 7 34/37 Topics in Macroeconomics
Law of motion (general case) Balanced Growth (general case) Goals? The wage rate and rental rate 35 Growth rate of wages w t+1 = F L(t + 1) (1 α)k α t+1 = (A t+1l t+1 ) α A t+1 w t F L (t) (1 α)kt α (A t L t ) α A t (ˆk ) α ( At+1 ) = = (1 + g) ˆk A t Show that the rental rate on capital, r t, is constant along the BGP* Lectures 5, 6 & 7 35/37 Topics in Macroeconomics
Law of motion (general case) Balanced Growth (general case) Goals? Kaldor facts: Stylized facts of economic growth 36 1. The labor share and the capital share are almost constant over time. 2. The ratio of aggregate capital to output is almost constant over time. 3. The return to capital is almost constant over time, while wages grow at a roughly constant rate. 4. Output per capita and capital per worker grow at a roughly constant and positive rate. 5. Different countries and regions within a country that start out with a different level of income per capita tend to converge over time. Lectures 5, 6 & 7 36/37 Topics in Macroeconomics
Law of motion (general case) Balanced Growth (general case) Goals? Predictions of the Solow model 37 1. What does the Solow model have to say about the growth experience in Chad versus the UK, for example? 2. What is the Solow model missing relative to theories coming up later in the course? (keep this question in mind for later) 3. What is the Solow model missing according to Romer, Barro and Sala-i-Martin (textbooks)? 4. What is the Solow model missing according to Easterly (book)? 5. What is the Solow model missing according to Lucas (article)? Lectures 5, 6 & 7 37/37 Topics in Macroeconomics