Growth Theory: Review

Growth Theory: Review Lecture 1, Endogenous Growth Economic Policy in Development 2, Part 2 March 2009 Lecture 1, Exogenous Growth 1/104 Economic Policy in Development 2, Part 2

Outline Growth Accounting Growth Accounting Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model Main Ingredients of the Model Definition of Equilibrium Characterizing Equilibrium Quantities Steady state and Dynamics Lecture 1, Exogenous Growth 2/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Growth Accounting: Objective 3 Many factors play role to determine output in a country Certainly, size of the labour force and capital stock do But also, education, government regulation, weather,... Any theory of economic growth chooses which of these factors to emphasize as sources of GDP growth within countries explanation for differences in levels/growth rates across countries Growth accounting: tool to evaluate relative importance of such factors Theory & Policy Implications Lecture 1, Exogenous Growth 3/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Technical framework 4 Ignore the demand side for now Carefully specify the supply side Inputs: capital, K, and labour, L Output, Y State of technology, A Lecture 1, Exogenous Growth 4/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Technology, F 5 Y = F(K, AL) where Y = output K = capital (input / factor) L = labour (input / factor) A = state of technology H = AL = effective labour Assumptions Marginal products positive and diminishing Constant returns to scale Lecture 1, Exogenous Growth 5/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Marginal products 6 Marginal product of labour F L = F L > 0 positive 2 F L 2 = F LL < 0 and diminishing Marginal product of capital F K = F K > 0 positive 2 F = F K 2 KK < 0 and diminishing Lecture 1, Exogenous Growth 6/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Constant returns to scale (CRS) 7 F(λK, AλL) = λf(k, AL) for λ > 0 Implications of CRS Size (of firms) does not matter representative firm Euler s theorem: Factor payments exhaust the output See example Lecture 1, Exogenous Growth 7/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Cobb-Douglas production function 8 F(K, AL) = K α (AL) 1 α 1 > α > 0 CRS? F(λK, AλL) = (λk) α (AλL) 1 α = λ α K α λ 1 α (AL) 1 α = λ α λ 1 α K α (AL) 1 α = λk α (AL) 1 α = λf(k, AL) Yes, Cobb-Douglas production function is CRS. Lecture 1, Exogenous Growth 8/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Cobb-Douglas production function 9 F(K, AL) = K α (AL) 1 α 1 > α > 0 CRS? Yes. Positive and diminishing MP? F K (K, AL) = αk α 1 (AL) 1 α > 0 F L (K, AL) = (1 α)k α A 1 α L α > 0 F KK (K, AL) = α(α 1)K α 2 (AL) 1 α < 0 F LL (K, AL) = (1 α)( α)k α A 1 α L α 1 < 0 Yes, Cobb-Douglas production function has positive and diminishing MPs. Lecture 1, Exogenous Growth 9/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Profit maximizing firm(s) 10 Π(K, AL) = F(K, AL) rk wl Firms take prices as given and choose inputs K and L First order conditions Π K = F K(K, AL) r = 0 Π L = F L(K, AL) w = 0 Firm picks K and L such that FK (K, AL) = r FL (K, AL) = w Lecture 1, Exogenous Growth 10/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Profit maximizing firms with CD production function 11 With CD production function, FOCs become F K (K, AL) = αk α 1 (AL) 1 α = r F L (K, AL) = (1 α)k α A 1 α L α = w Or, rearranging ( AL ) 1 α F K (K, AL) = α = r K ( K ) α F L (K, AL) = (1 α)a = w AL Lecture 1, Exogenous Growth 11/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Euler s Theorem with CD production function 12 rk + wl = F(K, AL) Factor payments exhaust production. We have ( AL ) 1 α r = F K (K, AL) = α K ( K α w = F L (K, AL) = (1 α)a AL) Therefore, rk + wl = F K (K, AL)K + F L (K, AL)L Lecture 1, Exogenous Growth 12/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Euler s Theorem with CD production function 13 We had rk + wl = F K (K, AL)K + F L (K, AL)L Therefore, substituting in functional forms for F K (K, AL) and F L (K, AL) from the previous slide, we get: [ ( AL ) 1 α ] [ ( K α ] rk + wl = α K + (1 α)a L K AL) = αk α (AL) 1 α + (1 α)k α (AL) 1 α = K α (AL) 1 α = F(K, AL) Lecture 1, Exogenous Growth 13/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results One large firm or many small firms 14 Since firms take prices as given, and assuming A and α are the same for all firms from FOCs, we get r = F K (K i, AL i ) = αa 1 α( L i K i ) 1 α w = F L (K i, AL i ) = (1 α)a 1 α( K i L i ) α Capital-labor ratio, k = L i K i, chosen is the same for all firms (indexed by i). Using the CRS assumption, total output by many firms can be represented by output of one firm Lecture 1, Exogenous Growth 14/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Steps for growth accounting 15 TFP residual, A t, for K only production function TFP residual, A t, across countries: K only TFP residual, A t, including human capital TFP residual, A t, across countries: K and H Decomposing growth in GDP per worker: K only Decomposing growth in GDP per worker: K and H Summary of results Critique Lecture 1, Exogenous Growth 15/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results TFP, A t, as residual for K only production function 16 From the production function in year t Y t = A t K α t L 1 α t Denoting per worker variables in lower case letters, i.e., output per w. y t = Y t L t and capital per w. k t = K t L t After dividing by L t, we rewrote production function Y t L t = A t ( K t L t ) α ( L t L t ) 1 α as y t = A t k α t Hence, by rearranging we got A t = y t k α t Lecture 1, Exogenous Growth 16/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results TFP residual, A t, across countries: K only 17 UK, South Korea and India 1400.00 TFP (Accounting for Physical Capital Only) 1200.00 1000.00 800.00 600.00 U.K. Korea India 400.00 200.00 0.00 1960 1965 1970 1975 1980 1985 1990 1995 Lecture 1, Exogenous Growth 17/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results CRS production function with physical and human capital 18 Now include human capital (e.g., years of education) into production function (Make sure production function is CRS and positive and diminishing MP) We will use Y t = A t F(K t, H t, L t ) = A t K α with α,β [0, 1] (parameters) t H β t L1 α β t, Note: A t F(λK t,λh t,λl t ) = λa t F(K t, H t, L t ) CRS Homework: Show that this production function is CRS Check that MP of physical capital, K, human capital, H t and labor L t are positive and diminishing Lecture 1, Exogenous Growth 18/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results TFP as residual for production function with K and H 19 From the production function in year t Y t = A t K α t H β t L1 α β t Denoting per worker variables in lower case letters, i.e., output per w. y t = Y t L t, phys. capital per w. k t = K t L t human capital per w. h t = h t L t and After dividing by L t, we can rewrite the production function Y t L t = A t ( K t L t ) α ( H t L t ) β ( L t L t ) 1 α β as y t = A t kt α h β t Hence, by rearranging we get A t = y t kt α h β t Lecture 1, Exogenous Growth 19/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Compare A t for the 2 production functions 20 Residual for production function with physical capital only A t = y t kt α Residual for production function with human capital A t = y t kt α h β t Difficult to measure β In data computations below, I used α = β = 0.3 Lecture 1, Exogenous Growth 20/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results TFP residual, A t, across countries: K and H 21 UK, South Korea and India 1400.00 TFP (Accounting for Physical and Human Capital) 1200.00 1000.00 800.00 600.00 U.K. Korea India 400.00 200.00 0.00 1960 1965 1970 1975 1980 1985 1990 1995 Lecture 1, Exogenous Growth 21/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results TFP residual, A t, across countries: K only 22 UK, South Korea and India 1400.00 TFP (Accounting for Physical Capital Only) 1200.00 1000.00 800.00 600.00 U.K. Korea India 400.00 200.00 0.00 1960 1965 1970 1975 1980 1985 1990 1995 Lecture 1, Exogenous Growth 22/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results TFP ratio: Korea and India relative to U.K. (K only) 23 80% 70% 60% TFP ratios (Accounting for Physical Capital Only) TFP Ratio Korea/UK (K only) TFP Ratio India/UK (K only) 50% 40% 30% 20% 10% 0% 1960 1965 1970 1975 1980 1985 1990 1995 Lecture 1, Exogenous Growth 23/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results TFP ratio: Korea and India relative to U.K. (K and H) 24 80% 70% 60% 50% TFP ratios (Accounting for Physical and HumanCapital) TFP Ratio Korea/UK (K only) TFP Ratio India/UK (K only) TFP Ratio Korea/UK (K and H) TFP Ratio India/UK (K and H) 40% 30% 20% 10% 0% 1960 1965 1970 1975 1980 1985 1990 1995 Lecture 1, Exogenous Growth 24/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Decomposing GDP per worker growth: K only 25 Now that we have a series for A t, we want to decompose growth in GDP per worker into growth in the capital stock versus growth in productivity. Last time, we derived ( ) log y t+1 log y t = log A t+1 log A t + α log k t+1 log k t log y t+1 log y t log y t+1 log y t = log A t+1 log A t log y t+1 log y t + α log k t+1 log k t log y t+1 log y t 1 = 100% = log A t+1 log A t log y t+1 log y t + α log k t+1 log k t log y t+1 log y t Lecture 1, Exogenous Growth 25/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results UK, Korea and India 26 Growth Accounting with physical capital only Lecture 1, Exogenous Growth 26/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Decomposing GDP per worker growth: K and H 27 Now that we have a series for A t, we want to decompose growth in GDP per worker into growth in the capital stock versus growth in human capital versus growth in productivity (TFP). Growth in output per worker is y t+1 y t = A t+1(k t+1 ) α (h t+1 ) β A t (k t ) α (h t+1 ) β y t+1 y t = A t+1 A t ( k t+1 k t ) α ( h t+1 h t ) β Next, as before we go to logs so we have a sum. Lecture 1, Exogenous Growth 27/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Decomposing GDP per worker growth: K and H 28 log y t+1 log y t = ( ) ( ) log A t+1 log A t + α log k t+1 log k t + β log h t+1 log h t log y t+1 log y t log y t+1 log y t = log A t+1 log A t log y t+1 log y t + α log k t+1 log k t log y t+1 log y t + β log h t+1 log h t log y t+1 log y t 1 = 100% = log A t+1 log A t log y t+1 log y t + α log k t+1 log k t log y t+1 log y t + β log h t+1 log h t log y t+1 log y t Lecture 1, Exogenous Growth 28/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results UK, Korea and India 29 Growth Accounting with physical and human capital Lecture 1, Exogenous Growth 29/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Summary of Results 30 TFP in the UK > TFP Korea AND India For given inputs output in the UK is higher than in Korea and India When accounting for higher educational attainment, differences in TFP are smaller Adding H as input shows that India is not that much less productive than the UK; educational attainment (on average) is lower Lecture 1, Exogenous Growth 30/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Summary of Results 31 GDP growth accounting: increase in human capital (average years of education) accounts for a major part of growth in India Hence, omitting human capital in growth accounting can lead to erroneous conclusions Lecture 1, Exogenous Growth 31/104 Economic Policy in Development 2, Part 2

Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Critique 32 Interpretation of TFP? Technological change? Deregulation? Regulation?? Why did trend change? Other factors Human capital? Done Capital-skill complementarities? Quality of capital? Lecture 1, Exogenous Growth 32/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) From Growth Accounting to the Solow Model 33 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) Lecture 1, Exogenous Growth 33/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) From Growth Accounting to the Solow Model 34 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 Lecture 1, Exogenous Growth 34/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Kaldor facts: Stylized facts of economic growth 35 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. Lecture 1, Exogenous Growth 35/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Understanding growth differences across time and across countries 36 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 Lecture 1, Exogenous Growth 36/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Assumptions of the Solow model 37 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 Lecture 1, Exogenous Growth 37/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Assumptions of the Solow model 38 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 Lecture 1, Exogenous Growth 38/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Aggregating consumers 39 Savings per person are: sy t = r t k t + w t Multiplying by the number of people in period t 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 ) Lecture 1, Exogenous Growth 39/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Firm s problem (see above) 40 ] 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 Lecture 1, Exogenous Growth 40/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Law of motion of aggregate capital stock 41 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 Lecture 1, Exogenous Growth 41/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Law of motion: simple case n = 0 and g = 0 42 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 Lecture 1, Exogenous Growth 42/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Solow s law of motion 43 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 Lecture 1, Exogenous Growth 43/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Solow s law of motion 44 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 Lecture 1, Exogenous Growth 44/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Steady state 45 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 Lecture 1, Exogenous Growth 45/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Comparative statics 46 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* Lecture 1, Exogenous Growth 46/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Comparative dynamics 47 Suppose the level of the capital stock in some economy (country) at time 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...until the capital stock reaches the new steady state...k Lecture 1, Exogenous Growth 47/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Homework 48 Derive the same reasoning for * If δ decreases or increases If A decreases or increases If N decreases or increases* Lecture 1, Exogenous Growth 48/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Balanced growth: n 0 and g 0 49 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 Lecture 1, Exogenous Growth 49/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Balanced Growth: Steady state in units of effective labour 50 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 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 Lecture 1, Exogenous Growth 50/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Law of Motion 51 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 Lecture 1, Exogenous Growth 51/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Law of Motion 52 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 Lecture 1, Exogenous Growth 52/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Steady state in detrended variables (i.e. per unit of effective labour) 53 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 Show that ˆk t+1 is an increasing and concave function of ˆk t if α,δ [0, 1], g, n [ 1, 1] Lecture 1, Exogenous Growth 53/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Solow s law of motion (capital per u. of eff. labour) 54 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 Lecture 1, Exogenous Growth 54/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Balanced Growth: Per capita/worker var s 55 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 Higher population growth implies lower level of capital stock per unit of effective labor in the long run, but growth rate of per capita variables unaffected Lecture 1, Exogenous Growth 55/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Capital per worker and GDP per capita 56 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 ) α ( At+1 ) 1 α = (1 + g) α (1 + g) 1 α k t A t = (1 + g) Lecture 1, Exogenous Growth 56/104 Economic Policy in Development 2, Part 2

Model Setup Steady state (simple case) Balanced Growth (with popul. and prod. growth) Balanced Growth: Wage and rental rate of capital 57 Growth rate of wages w t+1 = F K(t + 1) w t F K (t) (ˆk = = ˆk ) α ( At+1 A t ) = (1 + g) (1 α)k α t+1 (A t+1l t+1 ) α A t+1 (1 α)k α t (A t L t ) α A t Show that the rental rate on capital, r t, is constant along the BGP Lecture 1, Exogenous Growth 57/104 Economic Policy in Development 2, Part 2

Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model Solow model and Savings Behaviour 58 Recall that in the Solow model the savings rate was an exogenous constant (parameter) therefore aggregate investment was a constant fraction of output/aggregate income Suppose you know that whatever you save, the government will tax at 100% next year. How much would you save versus consume this year? About nothing unless you can hide it really well... Hence questions such as: What is the effect of capital gains taxes? cannot seriously be addressed in the Solow model. However, the Solow model DID teach us that the savings rate is important for growth. Lecture 1, Exogenous Growth 58/104 Economic Policy in Development 2, Part 2

Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model The Model 59 Income Consider a household that receives an exogenous flow of income in each period of time We restrict the number of periods to be 2: t and t + 1 Denote income in each period by y t and y t+1 Assume there are perfect financial markets where the household can freely borrow and lend by holding assets or debt, a t+1, at an interest rate r Lecture 1, Exogenous Growth 59/104 Economic Policy in Development 2, Part 2

Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model The Model 60 Preferences Preferences of the household are defined over sequences of consumption {c t, c t+1 } We assume that instantaneous utility can be represented by a standard utility function: u(c) [i.e. u(.) is increasing, twice differentiable, concave and satisfies Inada conditions*] Life-time utility is the discounted sum of instantaneous utilities The agent has a subjective rate of time preference ρ so that the discount factor is 1/(1 + ρ) < 1 high ρ means impatient low ρ means patient Lecture 1, Exogenous Growth 60/104 Economic Policy in Development 2, Part 2

Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model The Model 61 Preferences Life-time utility is V(c t, c t+1 ) = u(c t ) + 1 1 + ρ u(c t+1) Sometimes we will define β 1 1+ρ and write utility as β is the discount factor V(c t, c t+1 ) = u(ct) + βu(ct + 1) Lecture 1, Exogenous Growth 61/104 Economic Policy in Development 2, Part 2

Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model The Model 62 Budget constraint The agent faces two period-by-period constraints c t + a t+1 = y t c t+1 = y t+1 + (1 + r)a t+1 The assumption of perfect financial markets means that consumption is not restricted to equal income Agent can allocate consumption in many different ways In fact, he faces a single constraint: the intertemporal budget constraint It follows from aggregating over time as follows: c t + 1 1+r c t+1 = y t + 1 1+r y t+1 Lecture 1, Exogenous Growth 62/104 Economic Policy in Development 2, Part 2

Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model The Model 63 Budget constraint It follows from aggregating over time that: c t + 1 1+r c t+1 = y t + 1 1+r y t+1 In other words, the present value of consumption cannot exceed the present value of income (or wealth) This can be represented graphically* Only at the point corresponding to the endowment, saving (borrowing (-) or lending (+)) is zero Lecture 1, Exogenous Growth 63/104 Economic Policy in Development 2, Part 2

Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model Household s optimization problem 64 Given y t, y t+1 and r or, equivalently, max u(c t ) + βu(c t+1 ) c t,c t+1 s.t. c t + 1 1 + r c t+1 = y t + 1 1 + r y t+1 max u(c t ) + βu(c t+1 ) c t,c t+1 s.t. c t + a t+1 = y t c t+1 = (1 + r)a t+1 + y t+1 Lecture 1, Exogenous Growth 64/104 Economic Policy in Development 2, Part 2

Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model Solution to the optimization problem 65 Graphical solution* Using the first formulation, the problem and the solution can be represented in the typical indifference-curve diagram on the (c t, c t+1 ) - space The optimal choice is characterized by the allocation where the intertemporal budget constraint (with slope (1 + r)) is tangent to an indifference curve Lecture 1, Exogenous Growth 65/104 Economic Policy in Development 2, Part 2

Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model Solution to the optimization problem 66 Analytical solution This solution is characterized by a FOC and the budget constraint (2 equations for 2 unknowns (c t, c t+1 )) The FOC reads, u (c t ) = 1 + r 1 + ρ u (c t+1 ) = β(1 + r)u (c t+1 ) This is called the Euler equation Lecture 1, Exogenous Growth 66/104 Economic Policy in Development 2, Part 2

Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model Consumption smoothing 67 The FOC implies that the change in consumption over time depends entirely on the form of the utility function, u(.), ρ and r The time-profile of income does not matter for the time-profile of consumption (holding present value of life-time income fixed) The present value of income is only important in determining the level consumption in the two periods, but not the steepness of the consumption path Lecture 1, Exogenous Growth 67/104 Economic Policy in Development 2, Part 2

Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model Consumption smoothing 68 Consider in particular the situation where interest rate equals the rate of time preference: r = ρ In this case, consumption is the same in the two periods even if income is not This captures the implication of concave utility functions for consumption: agents tend to prefer smooth consumption paths They can do that because they can borrow and lend To see more specifically how the interest rate can alter the optimal path of consumption, it proves convenient to use a specific yet fairly general form for the utility function... Lecture 1, Exogenous Growth 68/104 Economic Policy in Development 2, Part 2

Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model Constant-elasticity-of-substitution utility 69 We use the following, u(c) = c1 σ 1 σ if σ 1 = log c if σ = 1 It turns out that σ determines the household s willingness to shift consumption across periods: the smaller is σ, the more slowly marginal utility as consumption the more willing is the household to allow its consumption to vary over time (if r differs from ρ) Lecture 1, Exogenous Growth 69/104 Economic Policy in Development 2, Part 2

Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model Constant-elasticity-of-substitution utility 70 This can be seen from the FOC, u (c t ) = 1 + r 1 + ρ u (c t+1 ) Using the CES utility function*, c ( t+1 1 + r ) 1 σ = c t 1 + ρ If σ is close to zero, then utility is close to linear and the household is willing to accept large swings in consumption to take advantage of small differences between ρ and r Lecture 1, Exogenous Growth 70/104 Economic Policy in Development 2, Part 2

Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model Constant-elasticity-of-substitution utility 71 In fact the intertemporal elasticity of substitution, IES, is closely related to σ The IES is defined as θ(c) = u (c) u (c)c This is essentially a measure of the curvature of the utility functions and, therefore, of the willingness to accept swings in consumption over time With the CES utility function, the IES becomes* θ(c) = u (c) u (c)c = 1/σ Lecture 1, Exogenous Growth 71/104 Economic Policy in Development 2, Part 2

Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model Constant-elasticity-of-substitution utility 72 With the CES utility function, the IES becomes* θ(c) = u (c) u (c)c = 1/σ That is σ is the inverse of θ: θ = 1/σ Since σ is constant, θ is constant and u(.) is said to be of CES type Note that with uncertainty σ characterizes the degree of risk-aversion and this type of utility functions are also known as constant-relative-risk-aversion (CRRA) utility functions (see that later) Clearly, θ = 1/σ determines the responsiveness of the slope of the consumption path to changes in the interest rate Lecture 1, Exogenous Growth 72/104 Economic Policy in Development 2, Part 2

Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model Intertemporal-substitution and wealth effects 73 Intertemporal substitution and r θ = 1/σ determines the responsiveness of the slope of the consumption path to changes in the interest rate Higher r implies that optimal consumption grows faster over time This does not depend on the time path of income This is the intertemporal-substitution effect of a change in the interest rate (1 + r) is just the relative price of c t in terms of c t+1 Lecture 1, Exogenous Growth 73/104 Economic Policy in Development 2, Part 2

Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model Intertemporal-substitution and wealth effects 74 Intertemporal substitution and r Thus intertemporal substitution is the standard substitution effect when the relative price of two commodities changes This effect of an increase in r tends to increase saving a = y t c t Lecture 1, Exogenous Growth 74/104 Economic Policy in Development 2, Part 2

Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model Intertemporal-substitution and wealth effects 75 Wealth effect and r But, as usual, there is also a wealth effect Here, it is useful to draw the indifference-curve diagram If initially saving is zero, then the wealth effect is nil and the substitution effect dictates an increase in saving If initially the household is borrowing, both the wealth and substitution effects go in the direction of increasing saving (or reducing borrowing) If the household is initially saving, then the wealth effect tends to reduce saving and the net effect is ambiguous Follow graphical analysis and discussion in D.Romer (1996,p325-327)]. Lecture 1, Exogenous Growth 75/104 Economic Policy in Development 2, Part 2

Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model Intertemporal-substitution and wealth effects 76 Savings is a = y t c t Suppose r increases Substitution effect a Income effect a? If initially a = 0, no wealth effect a If initially a > 0, positive wealth effect a? If initially a < 0, negative wealth effect a Lecture 1, Exogenous Growth 76/104 Economic Policy in Development 2, Part 2

Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model Capital gains taxes in the two-period model 77 Budget constraint The agent faces two period-by-period constraints c t + a t+1 = y t c t+1 = y t+1 + (1 + r(1 τ))a t+1 The intertemporal budget constraint c t + 1 1+r(1 τ) c t+1 = y t + 1 1+r(1 τ) y t+1 Note: preferences unchanged optimal choice may change because the budget set and relative price of consumption in t versus t + 1 changes. Lecture 1, Exogenous Growth 77/104 Economic Policy in Development 2, Part 2

Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model Household s optimization problem with taxes 78 Given y t, y t+1, r and τ max u(c t ) + βu(c t+1 ) c t,c t+1 s.t. c t + 1 1 + (1 τ)r c 1 t+1 = y t + 1 + (1 τ)r y t+1 Lecture 1, Exogenous Growth 78/104 Economic Policy in Development 2, Part 2

Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Taxes in the two-period model Euler equation with taxes 79 Euler equation with capital-gains tax c t+1 c t = ( 1 + r(1 τ) 1 + ρ ) 1 σ Let ˆr = (1 τ)r denote the after tax interest rate (effective interest rate) Higher tax rate, τ, implies lower effective interest rate, ˆr Increasing taxes affects consumers in the same way as a decrease in the interest rate (substitution of consumption from t + 1 to t & wealth effect) Side question: In light of the Solow model for example, should we increase or decrease taxes? Lecture 1, Exogenous Growth 79/104 Economic Policy in Development 2, Part 2

Main Ingredients of the Model Definition of Equilibrium Characterizing Equilibrium Quantities Steady state and Dynamics 80 Main Ingredients Neoclassical model of the firm Consumption-savings choice for consumers Solow model + incentives to save (recall example with taxes) Lecture 1, Exogenous Growth 80/104 Economic Policy in Development 2, Part 2

Main Ingredients of the Model Definition of Equilibrium Characterizing Equilibrium Quantities Steady state and Dynamics The Model 81 Markets and ownership Agents Firms produce goods, hire labor and rent capital Households own labor and assets (capital), receive wages and rental payments, consume and save Markets Inputs: competitive wage rates, w, and rental rate, R Assets: free borrowing and lending at interest rate, r Output: competitive market for consumption good Lecture 1, Exogenous Growth 81/104 Economic Policy in Development 2, Part 2

Main Ingredients of the Model Definition of Equilibrium Characterizing Equilibrium Quantities Steady state and Dynamics The Model 82 Firms / Representative Firm Seeks to maximize profits Profit = F(K, L) RK wl The FOCs for this problem deliver F K = R F L = w In per unit of labor terms, let f(k) F(k, 1) f (k) = R f(k) kf (k) = w Recall Euler s Theorem: factor payments exhaust output Lecture 1, Exogenous Growth 82/104 Economic Policy in Development 2, Part 2

Main Ingredients of the Model Definition of Equilibrium Characterizing Equilibrium Quantities Steady state and Dynamics The Model 83 Households / Representative household Preferences U 0 = β t u(c t ) t=0 Budget constraint c t + a t+1 = w t + (1 + r)a t, for all t = 0, 1, 2,... a 0 given Note: labor supplied inelastically, l t = 1 Lecture 1, Exogenous Growth 83/104 Economic Policy in Development 2, Part 2

Main Ingredients of the Model Definition of Equilibrium Characterizing Equilibrium Quantities Steady state and Dynamics The Model 84 Households / Representative household Intertemporal version of budget constraint t=0 s=0 t ( 1 1 + r s ) c t = a 0 + t=0 s=0 t ( 1 1 + r s We rule out that debt explodes (no Ponzi games) a t+1 B for some B big, but finite ) w t More compactly, PDV(c) = a(0) + PDV(w) Lecture 1, Exogenous Growth 84/104 Economic Policy in Development 2, Part 2

Main Ingredients of the Model Definition of Equilibrium Characterizing Equilibrium Quantities Steady state and Dynamics The Model 85 Household s problem max (a t+1,c t ) t=0 s.t. β t u(c t ) t=0 c t + a t+1 = w t + (1 + r)a t, for all t = 0, 1, 2,... a t+1 = B for some B big, but finite a 0 given Lecture 1, Exogenous Growth 85/104 Economic Policy in Development 2, Part 2

Main Ingredients of the Model Definition of Equilibrium Characterizing Equilibrium Quantities Steady state and Dynamics The Model 86 Euler equation In general, u (c t ) = β(1 + r t+1 )u (c t+1 ) From here on, CES utility, u(c) = c1 σ 1 σ, Euler eqn. becomes, ( ) σ ct+1 = β(1 + r t+1) c t Lecture 1, Exogenous Growth 86/104 Economic Policy in Development 2, Part 2

Main Ingredients of the Model Definition of Equilibrium Characterizing Equilibrium Quantities Steady state and Dynamics The Model 87 Transversality condition HH do not want to end up with positive values of assets lim t βt u (c t )a t 0 HH cannot think they can borrow at the end of their life lim t βt u (c t )a t 0 Hence, lim t βt u (c t )a t = 0 Lecture 1, Exogenous Growth 87/104 Economic Policy in Development 2, Part 2

Main Ingredients of the Model Definition of Equilibrium Characterizing Equilibrium Quantities Steady state and Dynamics Definition of Equilibrium 88 A competitive equilibrium is defined by sequences of quantities of consumption, {c t }, capital, {k t }, and output, {y t }, and sequences of prices, {w t } and {r t }, such that Firms maximize profits Households maximize U 0 subject to their constraints Goods, labour and asset markets clear Lecture 1, Exogenous Growth 88/104 Economic Policy in Development 2, Part 2

Main Ingredients of the Model Definition of Equilibrium Characterizing Equilibrium Quantities Steady state and Dynamics Characterizing Equilibrium Quantities* 89 k t+1 + c t = f(k t ) + (1 δ)k t c t+1 c t = [β(1 + f (k t+1 ) δ)] 1/σ lim t βt u (c t )k t = 0 k 0 > 0 Lecture 1, Exogenous Growth 89/104 Economic Policy in Development 2, Part 2

Main Ingredients of the Model Definition of Equilibrium Characterizing Equilibrium Quantities Steady state and Dynamics Characterizing Equilibrium Quantities* 90 From the equilibrium conditions derived before, we find: There cannot be arbitrage opportunities in equilibrium R t δ = r t In equilibrium it does not pay to invest in capital directly. The riskless asset and capital have the same payoff. Lecture 1, Exogenous Growth 90/104 Economic Policy in Development 2, Part 2

Main Ingredients of the Model Definition of Equilibrium Characterizing Equilibrium Quantities Steady state and Dynamics Characterizing Equilibrium Quantities* 91 From the equilibrium conditions derived before, we find: Substituting out all the prices leads to the following set of necessary and sufficient conditions for an equilibrium in terms of quantities only. k t+1 + c t = f(k t ) + (1 δ)k t c t+1 c t = [β(1 + f (k t+1 ) δ)] 1/σ lim t βt u (c t )k t = 0 k 0 > 0 Prices can be determined from the firm s problems FOCs. Lecture 1, Exogenous Growth 91/104 Economic Policy in Development 2, Part 2

Main Ingredients of the Model Definition of Equilibrium Characterizing Equilibrium Quantities Steady state and Dynamics Benevolent planner s problem* 92 What is the allocation of resources that an economy should feature in order to attain the highest feasible level of utility? Central Planner s optimal choice problem max (k t+1,c t ) t=0 s.t. β t u(c t ) t=0 c t + k t+1 = f(k t ) + (1 δ)k t, for all t = 0, 1, 2,... k 0 > 0 given Lecture 1, Exogenous Growth 92/104 Economic Policy in Development 2, Part 2

Main Ingredients of the Model Definition of Equilibrium Characterizing Equilibrium Quantities Steady state and Dynamics Benevolent planner s problem 93 Welfare Socially optimal allocation coincides with the equilibrium allocation. The competitive equilibrium leads to the social optimum. Not surprising: no distortions or externalities Welfare Theorems hold Lecture 1, Exogenous Growth 93/104 Economic Policy in Development 2, Part 2

Main Ingredients of the Model Definition of Equilibrium Characterizing Equilibrium Quantities Steady state and Dynamics Notes: simplifying features* 94 We are considering an economy without population growth. There is no exogenous technological change, either. We include these two at the end of these notes. Lecture 1, Exogenous Growth 94/104 Economic Policy in Development 2, Part 2

Main Ingredients of the Model Definition of Equilibrium Characterizing Equilibrium Quantities Steady state and Dynamics Steady state* 95 Definition A balanced growth path (BGP) is a situation in which output, capital and consumption grow at a constant rate. If this constant rate is zero, it is called a steady state. We can usually redefine the state variable so that the latter is constant (i.e. the growth rate is zero) Recall from the Solow model: capital per unit of labor for (n > 0, g = 0) capital per unit of effective labor for (n > 0, g > 0) Lecture 1, Exogenous Growth 95/104 Economic Policy in Development 2, Part 2

Main Ingredients of the Model Definition of Equilibrium Characterizing Equilibrium Quantities Steady state and Dynamics Steady state 96 From the Euler equation, c t+1 c t = [β(1 + f (k t+1 ) δ)] 1/σ, for all t If consumption grows at a constant rate (BGP), say γ 1 + γ = [β(1 + f (k t+1 ) δ)] 1/σ, for all t Thus RHS must be constant k t+1 = k t = k must be constant along the BGP Lecture 1, Exogenous Growth 96/104 Economic Policy in Development 2, Part 2

Main Ingredients of the Model Definition of Equilibrium Characterizing Equilibrium Quantities Steady state and Dynamics Steady state 97 But then, from the resource constraint with k t = k t+1 = k : c t + k t+1 = f(k t ) + (1 δ)k t, for all t i.e., c t = f(k ) δk c t+1 = f(k ) δk We find that consumption must be constant along the BGP, c t+1 = c t = c or γ = 0 Hence we have a steady state in per capita variables. Lecture 1, Exogenous Growth 97/104 Economic Policy in Development 2, Part 2

Main Ingredients of the Model Definition of Equilibrium Characterizing Equilibrium Quantities Steady state and Dynamics Steady state* 98 Hence from the Euler equation 1 + γ = 1 = [β(1 + f (k ) δ)] 1/σ or, simplified f (k ) = 1 β (1 δ) = ρ + δ we can solve for k and from the (simplified) resource constraint we can solve for c c = f(k ) δk Lecture 1, Exogenous Growth 98/104 Economic Policy in Development 2, Part 2

Main Ingredients of the Model Definition of Equilibrium Characterizing Equilibrium Quantities Steady state and Dynamics Modified golden rule* 99 The capital stock that maximizes utility in steady state is called the modified golden rule level of capital Using f(k) = k α, we get f (k ) = ρ + δ k = k MGR = [ ] 1 α 1 α ρ + δ Compare to golden rule level of capital(max cons o in st. st.) [ α ] 1 k GR 1 α = δ (see Problem set 1, Q 3.3, assume A = 1 and set s = α (from Q 3.4)) Lecture 1, Exogenous Growth 99/104 Economic Policy in Development 2, Part 2

Main Ingredients of the Model Definition of Equilibrium Characterizing Equilibrium Quantities Steady state and Dynamics Modified golden rule 100 Since ρ > 0 and α (0, 1), k MGR = [ ] 1 α 1 α ρ + δ < [ α δ ] 1 1 α = k GR This result reflects the impatience of agents. As long as ρ > 0, they d always prefer to consume earlier rather than later, thereby reducing investments for next period and hence the steady state level of capital (and consumption)! One of Ramsey s points was that this is the steady state that we should aim at because it makes people the happiest - not the one that maximizes consumption per se. Lecture 1, Exogenous Growth 100/104 Economic Policy in Development 2, Part 2