Growth Theory: Review

Transcription

1 Growth Theory: Review Lecture 1.1, Exogenous Growth Topics in Growth, Part 2 June 11, 2007 Lecture 1.1, Exogenous Growth 1/76 Topics in Growth, Part 2

2 Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Growth Accounting: Objective 2 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 1.1, Exogenous Growth 2/76 Topics in Growth, Part 2

3 Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Technical framework 3 Ignore the demand side for now Carefully specify the supply side Inputs: capital, K, and labour, L Output, Y State of technology, A Lecture 1.1, Exogenous Growth 3/76 Topics in Growth, Part 2

4 Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Technology, F 4 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 1.1, Exogenous Growth 4/76 Topics in Growth, Part 2

5 Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Marginal products 5 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 1.1, Exogenous Growth 5/76 Topics in Growth, Part 2

6 Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Constant returns to scale (CRS) 6 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 1.1, Exogenous Growth 6/76 Topics in Growth, Part 2

7 Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Cobb-Douglas production function 7 F(K, AL) = K α (AL) 1 α 1 > α > 0 CRS Positive and diminishing MP Lecture 1.1, Exogenous Growth 7/76 Topics in Growth, Part 2

8 Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Steps for growth accounting 8 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 1.1, Exogenous Growth 8/76 Topics in Growth, Part 2

9 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 9 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 1.1, Exogenous Growth 9/76 Topics in Growth, Part 2

10 Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results CRS production function with physical and human capital 10 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 Lecture 1.1, Exogenous Growth 10/76 Topics in Growth, Part 2

11 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 11 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 1.1, Exogenous Growth 11/76 Topics in Growth, Part 2

12 Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Compare A t for the 2 production functions 12 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 Lecture 1.1, Exogenous Growth 12/76 Topics in Growth, Part 2

13 Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Decomposing GDP per worker growth: K only 13 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 1.1, Exogenous Growth 13/76 Topics in Growth, Part 2

14 Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Decomposing GDP per worker growth: K and H 14 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 1.1, Exogenous Growth 14/76 Topics in Growth, Part 2

15 Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Decomposing GDP per worker growth: K and H 15 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 1.1, Exogenous Growth 15/76 Topics in Growth, Part 2

16 Growth Accounting: Objective and Technical Framework TFP residual Decomposing growth in GDP per worker Results Summary of Results 16 On my webpage, you can find an example of growth accounting comparing the UK, India and South Korea. The main result is that GDP growth accounting: increase in human capital (average years of education) accounts for a major part of growth in India When accounting for physical capital only, a lot of the growth is assigned to the residual. Keep in mind that this residual may have a different meaning than just technological progress. If you are interested in research in economic growth, this kind of accounting exercise (including variables suspected to contribute to growth) is a useful start. Lecture 1.1, Exogenous Growth 16/76 Topics in Growth, Part 2

17 Model Setup Balanced Growth (with popul. and prod. growth) From Growth Accounting to the Solow Model 17 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 1.1, Exogenous Growth 17/76 Topics in Growth, Part 2

18 Model Setup Balanced Growth (with popul. and prod. growth) From Growth Accounting to the Solow Model 18 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 1.1, Exogenous Growth 18/76 Topics in Growth, Part 2

19 Model Setup Balanced Growth (with popul. and prod. growth) Kaldor facts: Stylized facts of economic growth 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 1.1, Exogenous Growth 19/76 Topics in Growth, Part 2

20 Model Setup Balanced Growth (with popul. and prod. growth) Understanding growth differences across time and across countries 20 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 1.1, Exogenous Growth 20/76 Topics in Growth, Part 2

21 Model Setup Balanced Growth (with popul. and prod. growth) Assumptions of the Solow model 21 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 1.1, Exogenous Growth 21/76 Topics in Growth, Part 2

22 Model Setup Balanced Growth (with popul. and prod. growth) Assumptions of the Solow model 22 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 1.1, Exogenous Growth 22/76 Topics in Growth, Part 2

23 Model Setup Balanced Growth (with popul. and prod. growth) Aggregating consumers 23 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 1.1, Exogenous Growth 23/76 Topics in Growth, Part 2

24 Model Setup Balanced Growth (with popul. and prod. growth) Profit maximizing firm(s) 24 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 1.1, Exogenous Growth 24/76 Topics in Growth, Part 2

25 Model Setup Balanced Growth (with popul. and prod. growth) Profit maximizing firms with CD production function 25 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 1.1, Exogenous Growth 25/76 Topics in Growth, Part 2

26 Model Setup Balanced Growth (with popul. and prod. growth) Euler s Theorem with CD production function 26 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 1.1, Exogenous Growth 26/76 Topics in Growth, Part 2

27 Model Setup Balanced Growth (with popul. and prod. growth) Euler s Theorem with CD production function 27 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 1.1, Exogenous Growth 27/76 Topics in Growth, Part 2

28 Model Setup Balanced Growth (with popul. and prod. growth) One large firm or many small firms 28 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 1.1, Exogenous Growth 28/76 Topics in Growth, Part 2

29 Model Setup Balanced Growth (with popul. and prod. growth) Law of motion of aggregate capital stock 29 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 1.1, Exogenous Growth 29/76 Topics in Growth, Part 2

30 Model Setup Balanced Growth (with popul. and prod. growth) Balanced growth: n 0 and g 0 30 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 1.1, Exogenous Growth 30/76 Topics in Growth, Part 2

31 Model Setup Balanced Growth (with popul. and prod. growth) Balanced Growth: Steady state in units of effective labour 31 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 1.1, Exogenous Growth 31/76 Topics in Growth, Part 2

32 Model Setup Balanced Growth (with popul. and prod. growth) Law of Motion 32 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 1.1, Exogenous Growth 32/76 Topics in Growth, Part 2

33 Model Setup Balanced Growth (with popul. and prod. growth) Law of Motion 33 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 1.1, Exogenous Growth 33/76 Topics in Growth, Part 2

34 Model Setup Balanced Growth (with popul. and prod. growth) Steady state in detrended variables (i.e. per unit of effective labour) 34 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 1.1, Exogenous Growth 34/76 Topics in Growth, Part 2

35 Model Setup Balanced Growth (with popul. and prod. growth) Solow s law of motion (capital per u. of eff. labour) K_t Kt+1 = Kt (45 degree line) Kt+1 = (1-delta) Kt + s F(Kt,AL) K_t Lecture 1.1, Exogenous Growth 35/76 Topics in Growth, Part 2

36 Model Setup Balanced Growth (with popul. and prod. growth) Solow s law of motion (capital per u. of eff. labour) Kt+1 = Kt (45 degree line) Kt+1 = (1-delta) Kt + s F(Kt,AL) K_t K_t Lecture 1.1, Exogenous Growth 36/76 Topics in Growth, Part 2

37 Model Setup Balanced Growth (with popul. and prod. growth) Balanced Growth: Per capita/worker var s 37 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 1.1, Exogenous Growth 37/76 Topics in Growth, Part 2

38 Model Setup Balanced Growth (with popul. and prod. growth) Capital per worker and GDP per capita 38 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 1.1, Exogenous Growth 38/76 Topics in Growth, Part 2

39 Model Setup Balanced Growth (with popul. and prod. growth) Balanced Growth: Wage and rental rate of capital 39 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 1.1, Exogenous Growth 39/76 Topics in Growth, Part 2

40 Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Solow model and Savings Behaviour 40 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 1.1, Exogenous Growth 40/76 Topics in Growth, Part 2

41 Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects A two period model of saving 41 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 1.1, Exogenous Growth 41/76 Topics in Growth, Part 2

42 Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects A two period model of saving 42 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 1.1, Exogenous Growth 42/76 Topics in Growth, Part 2

43 Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects A two period model of saving 43 Preferences Life-time utility is V(c t, c t+1 ) = u(c t ) 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 1.1, Exogenous Growth 43/76 Topics in Growth, Part 2

44 Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects A two period model of saving 44 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 r c t+1 = y t r y t+1 Lecture 1.1, Exogenous Growth 44/76 Topics in Growth, Part 2

45 Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects A two period model of saving 45 Budget constraint It follows from aggregating over time that: c t r c t+1 = y t 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 1.1, Exogenous Growth 45/76 Topics in Growth, Part 2

46 Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Household s optimization problem 46 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 r c t+1 = y t 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 1.1, Exogenous Growth 46/76 Topics in Growth, Part 2

47 Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Solution to the optimization problem 47 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 1.1, Exogenous Growth 47/76 Topics in Growth, Part 2

48 Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Solution to the optimization problem 48 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 1.1, Exogenous Growth 48/76 Topics in Growth, Part 2

49 Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Consumption smoothing 49 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 1.1, Exogenous Growth 49/76 Topics in Growth, Part 2

50 Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Consumption smoothing 50 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 1.1, Exogenous Growth 50/76 Topics in Growth, Part 2

51 Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Constant-elasticity-of-substitution utility 51 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 1.1, Exogenous Growth 51/76 Topics in Growth, Part 2

52 Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Constant-elasticity-of-substitution utility 52 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 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 1.1, Exogenous Growth 52/76 Topics in Growth, Part 2

53 Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Constant-elasticity-of-substitution utility 53 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 1.1, Exogenous Growth 53/76 Topics in Growth, Part 2

54 Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Intertemporal-substitution and wealth effects 54 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 1.1, Exogenous Growth 54/76 Topics in Growth, Part 2

55 Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Intertemporal-substitution and wealth effects 55 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 1.1, Exogenous Growth 55/76 Topics in Growth, Part 2

56 Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Intertemporal-substitution and wealth effects 56 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,p )]. Lecture 1.1, Exogenous Growth 56/76 Topics in Growth, Part 2

57 Preferences, Budget constraint, Optimal choice Consumption smoothing Intertemporal-substitution and wealth effects Intertemporal-substitution and wealth effects 57 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 1.1, Exogenous Growth 57/76 Topics in Growth, Part 2

58 Main Ingredients of the Model Definition of Equilibrium Steady state and Dynamics 58 Main Ingredients Neoclassical model of the firm Consumption-savings choice for consumers Solow model + incentives to save (recall example with taxes) Lecture 1.1, Exogenous Growth 58/76 Topics in Growth, Part 2

59 Main Ingredients of the Model Definition of Equilibrium Steady state and Dynamics The 1SGM 59 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 1.1, Exogenous Growth 59/76 Topics in Growth, Part 2

60 Main Ingredients of the Model Definition of Equilibrium Steady state and Dynamics The 1SGM 60 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 1.1, Exogenous Growth 60/76 Topics in Growth, Part 2

61 Main Ingredients of the Model Definition of Equilibrium Steady state and Dynamics The 1SGM 61 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 1.1, Exogenous Growth 61/76 Topics in Growth, Part 2

62 Main Ingredients of the Model Definition of Equilibrium Steady state and Dynamics The 1SGM 62 Households / Representative household Intertemporal version of budget constraint t=0 s=0 t ( r s ) c t = a 0 + t=0 s=0 t ( 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 1.1, Exogenous Growth 62/76 Topics in Growth, Part 2

63 Main Ingredients of the Model Definition of Equilibrium Steady state and Dynamics The 1SGM 63 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 1.1, Exogenous Growth 63/76 Topics in Growth, Part 2

64 Main Ingredients of the Model Definition of Equilibrium Steady state and Dynamics The 1SGM 64 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 1.1, Exogenous Growth 64/76 Topics in Growth, Part 2

65 Main Ingredients of the Model Definition of Equilibrium Steady state and Dynamics The 1SGM 65 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 1.1, Exogenous Growth 65/76 Topics in Growth, Part 2

66 Main Ingredients of the Model Definition of Equilibrium Steady state and Dynamics Definition of Equilibrium 66 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 1.1, Exogenous Growth 66/76 Topics in Growth, Part 2

67 Main Ingredients of the Model Definition of Equilibrium Steady state and Dynamics Benevolent planner s problem* 67 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 1.1, Exogenous Growth 67/76 Topics in Growth, Part 2

68 Main Ingredients of the Model Definition of Equilibrium Steady state and Dynamics Benevolent planner s problem 68 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 1.1, Exogenous Growth 68/76 Topics in Growth, Part 2

69 Main Ingredients of the Model Definition of Equilibrium Steady state and Dynamics Steady state* 69 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 capita for n = 0, g = 0 capital per unit of effective labor for n > 0, g > 0 Lecture 1.1, Exogenous Growth 69/76 Topics in Growth, Part 2

70 Main Ingredients of the Model Definition of Equilibrium Steady state and Dynamics Steady state 70 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 1.1, Exogenous Growth 70/76 Topics in Growth, Part 2

71 Main Ingredients of the Model Definition of Equilibrium Steady state and Dynamics Steady state 71 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 1.1, Exogenous Growth 71/76 Topics in Growth, Part 2

72 Main Ingredients of the Model Definition of Equilibrium Steady state and Dynamics Steady state* 72 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 1.1, Exogenous Growth 72/76 Topics in Growth, Part 2

73 Main Ingredients of the Model Definition of Equilibrium Steady state and Dynamics Off steady state dynamics* 73 Off the steady state, consumption and capital adjust to reach the steady state eventually. To analyze these dynamics, consider the movements of c and k separately. Let c = c t+1 c t and k = k t+1 k t. See graphical analysis. Lecture 1.1, Exogenous Growth 73/76 Topics in Growth, Part 2

74 Main Ingredients of the Model Definition of Equilibrium Steady state and Dynamics Off steady state dynamics* 74 Consider the set of points such that c = 0, then from the Euler eqn, the optimal k satisfies f (k) = ρ + δ draw vertical line at k (< k GR ) To the left: k t < k f (k t ) > f (k ) c > 0 c To the right: k t > k f (k t ) < f (k ) c < 0 c Consider the set of points such that k = 0, then from the Resource cstrt, the optimal c satisfies c = f(k) δk draw hump-shaped line from origin, maximized at k GR cross 0 again for k such that f(k) = δk Above: c t > f(k t ) δk t k = f(k t ) δk t c t < 0 k Below: c t < f(k t ) δk t k = f(k t ) δk t c t > 0 k Lecture 1.1, Exogenous Growth 74/76 Topics in Growth, Part 2

75 Main Ingredients of the Model Definition of Equilibrium Steady state and Dynamics Adding population and productivity growth 75 Evolution of technology: A t+1 = (1 + g)a t, Evolution of population (labour force*): L t+1 = (1 + n)l t Resource constraint in units of effective labour ĉ t + (1 + n)(1 + g)ˆk t+1 = (1 δ)ˆk t + f(ˆk t ) Euler equation in units of effective labour becomes ĉ t+1 = [β(1+f (ˆk σ t+1 ) δ)] 1 ĉ t (1+n)(1+g) Lecture 1.1, Exogenous Growth 75/76 Topics in Growth, Part 2

76 Main Ingredients of the Model Definition of Equilibrium Steady state and Dynamics Balanced Growth: St. state in units of effective labour 76 In steady state: ĉ t+1 ĉ t = 1 and ˆk t+1 ˆk t = 1 Solving the Euler Equation for ˆk gives [ ] ˆk βα = ((1 + n)(1 + g)) σ (1 δ)β decreasing in population growth rate. But long run growth rates of per capita variables are independent of n Thus, in terms of population growth rates, One Sector Growth Model gives same conclusions as Solow model. Lecture 1.1, Exogenous Growth 76/76 Topics in Growth, Part 2