5 The Solow Growth Model 5.1 Models and Assumptions ² What is a model? A mathematical description of the economy. ² Why do we need a model? The world is too complex to describe it in every detail. ² What makes a model successful? When it is simple but e®ective in de- scribing and predicting how the world works. ² A model relies on simplifying assumptions. These assumptions drive the conclusions of the model. When analyzing a model it is crucial to spell out the assumptions underlying the model. ² Realism may not a the property of a good assumption. 67 5.2 Basic Assumptions of the Solow Model 1. Continuous time. 2. Single good produced with a constant technology. 3. No government or international trade. 4. All factors of production are fully employed. L_ 5. Labor force grows at constant rate n = L. 6. Initial values for capital, K0 and labor, L0 given. 68 Production Function ² Neoclassical (Cobb-Douglas) aggregate production function: Y (t) = F [K(t);L(t)] = K(t)® L(t)1¡® ² To save on notation write: Y = AK® L1¡® ² Constant returns to scale: F (¸K; ¸L) = ¸ F (K; L) = ¸A K® L1¡® ² Inputs are essential: F (0; 0) = F (K; 0) = F (0;L) = 0 69 ² Marginal productivities are positive: @F = ®AK®¡1L1¡® > 0 @K @F = (1 ¡ ®) AK®L¡® > 0 @L ² Marginal productivities are decreasing, @2F = (® ¡ 1) ®A K®¡2L1¡® < 0 @K2 @2F = ¡® (1 ¡ ®) AK®L¡®¡1 < 0 @L2 70 Per Worker Terms X ² De¯ne x = L as a per worker variable. Then µ ¶ µ ¶ Y AK®L1¡® K a L 1¡® y = = = A = A k® L L L L ² Per worker production function has decreasing returns to scale. 71 Capital Accumulation ² Capital accumulation equation: K_ = sY ¡ ±K ² Important additional assumptions: 1. Constant saving rate (very speci¯c preferences: no r) 2. Constant depreciation rate 72 K_ Y ² Dividing by K in the capital accumu equation: K = sK ¡ ±. Y K_ Y L y ² Some Algebra: K = sK ¡ ± = s K ¡ ± = sk ¡ ± L ² Now remember that: k_ K_ L_ K_ K_ k_ = ¡ = ¡ n ) = + n k K L K K k ² We get k_ y + n = s ¡ ± ) k_ = sy ¡ (± + n) k k k ² Fundamental Di®erential Equation of Solow Model: k_ = s A k® ¡ (± + n) k 73 Graphical Analysis ² Change in k; k_ is given by di®erence of s A k® and (± + n)k ² If s A k® > (± + n)k; then k increases. ² If s A k® < (± + n)k; then k decreases. ² Steady state: a capital stock k¤ where, when reached, k_ = 0 ² Unique positive steady state in Solow model. ² Positive steady state (locally) stable. 74 Steady State Analysis ² Steady State: k_ = 0 ² Solve for steady state µ ¶ 1 s A 1¡® 0 = s A (k¤)® ¡ (n + ±)k¤ ) k¤ = n + ± ³ ´ ® ¤ s A 1¡® ² Steady state output per worker y = n+± ² Steady state output per worker depends positively on the saving (invest- ment) rate and negatively on the population growth rate and depreciation rate. 75 Comparative Statics ² Suppose that of all a sudden saving rate s increases to s0 > s: Suppose that at period 0 the economy was at its old steady state with saving rate s: ² (n + ±)k curve does not change. ² s A k® = sy shifts up to s0y: ² New steady state has higher capital per worker and output per worker. ² Monotonic transition path from old to new steady state. 76 Evaluating the Basic Solow Model ² Why are some countries rich (have high per worker GDP) and others are poor (have low per worker GDP)? ² Solow model: if all countries are in their steady states, then: 1. Rich countries have higher saving (investment) rates than poor coun- tries 2. Rich countries have lower population growth rates than poor countries ² Data seem to support this prediction of the Solow model 77 The Solow Model and Growth ² No growth in the steady state ² Positive or negative growth along the transition path: k_ = s A k® ¡ (n + ±)k k_ g ´ = s A k®¡1 ¡ (n + ±) k k 78 Introducing Technological Progress ² Aggregate production function becomes Y = K® (AL)1¡® ² A : Level of technology in period t. ² Key assumption: constant positive rate of technological progress: A_ = g > 0 A ² Growth is exogenous. 79 Balanced Growth Path ² Situation in which output per worker, capital per worker and consumption per worker grow at constant (but potentially di®erent) rates ² Steady state is just a balanced growth path with zero growth rate ² For Solow model, in balanced growth path gy = gk = gc 80 Proof ² Capital Accumulation Equation K_ = sY ¡ ±K K_ Y ² Dividing both sides by K yields gK ´ K = sK ¡ ± k_ K_ ² Remember that gk ´ k = K ¡ n ² Hence k_ Y g ´ = s ¡ (n + ±) k k K Y ² In BGP gk constant. Hence K constant. It follows that gY = gK: Therefore gy = gk 81 What is the Growth Rate? ² Output per worker Y K® (AL)1¡® K® (AL)1¡® y = = = = k®A1¡® L L L® L1¡® ² Take logs and di®erentiate gy = ®gk + (1 ¡ ®)gA ² We proved gk = gy and we use gA = g to get gk = ®gk + (1 ¡ ®)g = g = gy ² BGP growth rate equals rate of technological progress. No TP, no growth in the economy. 82 Analysis of Extended Model ² in BGP variables grow at rate g: Want to work with variables that are constant in long run. De¯ne: y Y y~ = = A AL k K k~ = = A AL ² Repeat the Solow model analysis with new variables: y~ = k~® k~_ = sy~ ¡ (n + g + ±)k~ k~_ = sk~® ¡ (n + g + ±)k~ 83 Closed-Form Solution ² Repeating all the steps than in the basic model we get: ³ ³ ´ ´ 1 ~ s ~1¡® s ¡¸t 1¡® k(t) = ±+n+g + k0 ¡ ±+n+g e ³ ³ ´ ´ ® s ~1¡® s ¡¸t 1¡® y~(t) = ±+n+g + k0 ¡ ±+n+g e ² Interpretation. 84 Balanced Growth Path Analysis ² Solve for k~¤ analytically ® 0 = sk~¤ ¡ (n + g + ±)k~¤ à ! 1 s 1¡® k~¤ = n + g + ± ² Therefore à ! ® s 1¡® y~¤ = n + g + ± 85 à ! 1 s 1¡® k(t) = A(t) n + g + ± à ! ® s 1¡® y(t) = A(t) n + g + ± à ! 1 s 1¡® K(t) = L(t)A(t) n + g + ± à ! ® s 1¡® Y (t) = L(t)A(t) n + g + ± 86 Evaluation of the Model: Growth Facts 1. Output and capital per worker grow at the same constant, positive rate in BGP of model. In long run model reaches BGP. K 2. Capital-output ratio Y constant along BGP 3. Interest rate constant in balanced growth path 4. Capital share equals ®; labor share equals 1 ¡ ® in the model (always, not only along BGP) 5. Success of Solow model along these dimensions, but source of growth, technological progress, is left unexplained. 87 Evaluation of the Model: Development Facts 1. Di®erences in income levels across countries explained in the model by di®erences in s; n and ±. 2. Variation in growth rates: in the model permanent di®erences can only be due to di®erences in rate of technological progress g: Temporary dif- ferences are due to transition dynamics. 3. That growth rates are not constant over time for a given country can be explained by transition dynamics and/or shocks to n; s and ±. 4. Changes in relative position: in the model countries whose s moves up, relative to other countries, move up in income distribution. Reverse with n: 88 Interest Rates and the Capital Share ² Output produced by price-taking ¯rms ² Hire workers L for wage w and rent capital Kfrom households for r ² Normalization of price of output to 1: ² Real interest rate equals r ¡ ± 89 Pro¯t Maximization of Firms max K® (AL)1¡® ¡ wL ¡ rK K;L ² First order condition with respect to capital K ®K®¡1 (AL)1¡® ¡ r = 0 µ ¶ K ®¡1 ® = r AL ®k~®¡1 = r ² In balanced growth path k~ = k~¤; constant over time. Hence in BGP rconstant over time, hence r ¡ ± (real interest rate) constant over time. 90 Capital Share ² Total income = Y; total capital income = rK ² Capital share rK capital share = Y ®K®¡1 (AL)1¡® K = K® (AL)1¡® = ® ² Labor share = 1 ¡ ®. 91 Wages ² First order condition with respect to labor L (1 ¡ ®)K®(LA)¡®A = w (1 ¡ ®)k~®A = w ² Along BGP k~ = k~¤; constant over time. Since A is growing at rate g; the wage is growing at rate g along a BGP. 92.