Dynamic (Stochastic) General Equilibrium and Growth

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1 Dynamic (Stochastic) General Equilibrium and Growth Martin Ellison Nuffi eld College Michaelmas Term 2018 Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

2 Macroeconomics is Dynamic Decisions are taken over time t 1 t t+1 Expectations Expectations make economics special Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

3 Macroeconomics is General Equilibrium Markets are interconnected Income ( ) Labour maximise utility Households Goods Firms maximise profit Expenditure ( ) Need to analyse all markets together Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

4 Simple General Equilibrium model Utility function U(c i t ) intertemporally separable Household i receives known endowment { y i t } 0 Savings { a i t } 0 remunerated at known interest rate {R t = (1 + r t )} 0 No production, labour market, firms or uncertainty Household chooses savings and consumption Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

5 Household budget constraint Flow budget constraint Iterate forward a i t+s+1 = R t+s a i t+s + y i t+s c i t+s for s 0 T R 1 t=0 t a i T +1 = a i 0 + Present value budget constraint T t=0 ( ) t (y Rs 1 i t c i ) t s=0 a i T +1 R T = a i 0 + T t=0 y i t c i t R t with R t = R 0 R 1 R 2... R t Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

6 Transversality condition No Ponzi condition to rule out explosive borrowing at i +1 lim 0 T R T Present value of terminal saving cannot be negative Condition holds with equality as not optimal to save in limit Present value budget constraint ct i t=0 R t = a i 0 + t=0 Present value of consumption = Present value of resources y i t R t Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

7 Household problem max {ct+s i,ai t+s+1 } s=0 β s U(c i t+s ) s.t. at+s+1 i = R t+s at+s i + yt+s i ct+s i for s 0 at i at i +1 given, lim 0 T R T Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

8 Method #1: Direct substitution Substitute for c i t+s in utility function using flow budget constraint max {at+s+1 i } s=0 β s U(R t+s a i t+s + y i t+s a i t+s+1) First order condition with respect to at+1 i U c i,t + βu c i,t+1r t+1 = 0 Intertemporal Euler equation for consumption βr t+1 U c i,t+1 U c i,t 1 = 0 Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

9 Method #2: Graphical Expand utility function s=0 β s U(c i t+s ) = U(c i t ) + βu(c i t+1) +... = Ū Total differentiation holding Ū and c i t+s constant for s 2 dc i t+1 dc i t = 1 β U c i,t U c i,t+1 = MRS Indifference curve in (c i t, c i t+1 ) space Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

10 Method #2: Graphical Expand budget constraint a i t+2 = R t+1 ( Rt a i t + y i t c i t ) + y i t+1 c i t+1 Total differentiation holding a i t, a i t+2, y i t and y i t+1 constant dc i t+1 dc i t = R t+1 = MRT Budget constraint in (c i t, c i t+1 ) space Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

11 Method #2: Graphical Optimising household sets MRS=MRT βr t+1 U c i,t+1 U c i,t 1 = 0 Optimality in (c i t, c i t+1 ) space Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

12 Method #3: Value function Value function V (at) i [ = max U(Rt a i at+1 i t + yt i at+1) i + βv (at+1) i ] First order condition U c i,t = βv (a i t+1) Differentiate V (a i t) with respect to a i t V (a i t) = U c i,tr t Roll forward one period and substitute βr t+1 U c i,t+1 U c i,t 1 = 0 Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

13 Method #4: Lagrangian Define (present value) Lagrangian L i = s=0 β s U(c i t+s ) + First order conditions s=0 λ i t+s β s ( R t+s at+s i + yt+s i ct+s i at+s+1 i ) ct i : U c i,t = λ i t ct+1 i : U c i,t+1 = λ i t+1 at+1 i : λ i t+1βr t+1 λ i t = 0 Combine βr t+1 U c i,t+1 U c i,t 1 = 0 Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

14 Interpretation of Euler equation for consumption Growth in c i t determined by R t through growth in U c i,t βr t+1 U c i,t+1 U c i,t 1 = 0 βr t+1 = 1 U c i,t+1 = U c i,t c i t+1 = ci t βr t+1 > 1 U c i,t+1 < U c i,t c i t+1 > ci t βr t+1 < 1 U c i,t+1 > U c i,t c i t+1 < ci t Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

15 General equilibrium Aggregate household Euler equations with log utility i ct+1 i = βr t+1 ct i i Market clearing with no aggregate savings i yt i = ct i i for t Market interest rate βr t+1 = i y i t+1 i y i t = ȳt+1 ȳ t Behaves as a representative agent economy! Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

16 Neoclassical growth theory Endowment economy as production economy where only labour enters production function Accumulation of savings at household but not aggregate level Workhorse model in macroeconomics is Walrasian economy with aggregate capital accumulation Maintain exogenous labour supply decisions for now Overlapping generations (OLG) vs representative agent (Ramsey) Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

17 Two-generation OLG Work when young and retire when old C Y t and C O t+1 are consumption when young and old Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

18 Household in OLG Household supplies labour and capital to firm Work when young for labour income w t and retire when old Kt+1 O is saving of young which pays return r t+1 when old Capital depreciates at 100% after production Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

19 Household problem in OLG Household maximises utility max C Y t,c O t+1,k O t+1 ( ) log Ct Y + β log Ct+1 O s.t. C Y t + K O t+1 = w t C O t+1 = r t+1 K O t+1 First order condition is Euler equation for consumption C O t+1 βc Y t = r t+1 Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

20 Firm problem in OLG Firm maximises profits ( max K α t L 1 α ) t w t L t r t K t K t,l t First order conditions define factor prices r t = αkt α 1 L 1 α t w t = (1 α) Kt α Lt α Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

21 Equilibrium in OLG Equilibrium is a sequence { r t, w t, C Y t, C O t, K O t+1, K t, L t } t=0 s.t. 1 { C Y t, Ct O, Kt+1 O } solves household problem given {rt, w t } 2 {K t, L t } solves firm problem given {r t, w t } 3 Markets clear K t = Kt O, L t = 1 and Ct Y + Ct O + K t+1 = Kt α L 1 α t 4 K t given 5 Transversality Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

22 Calculating equilibrium in OLG Euler equation for consumption and household budget constraint imply r t+1 Kt+1 O β ( ) w t Kt+1 O = r t+1 Substitute in for wage and market clearing conditions K t+1 = β 1 + β (1 α) K α t Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

23 Equilibrium in OLG Monotonic convergent dynamics to steady state In long run all variables constant Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

24 Check Kaldor (1957) facts 1 Output per worker grows at a roughly constant rate NO 2 Capital per worker grows over time NO 3 Capital/output ratio is roughly constant YES 4 Rate of return to capital is constant YES 5 Shares of capital and labour in net income are nearly constant YES 6 Real wage grows over time NO 7 Ratios of consumption and investment to GDP are constant YES Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

25 Technological change in OLG Add (labour-augmenting) technological progress in production Y t = Kt α (θ t L t ) 1 α with θ t+1 = (1 + g)θ t Euler equation for consumption unchanged C O t+1 βc Y t = r t+1 Firm profit maximising conditions r t = αkt α 1 (θ t L t ) 1 α w t = (1 α) Kt α θ t (θ t L t ) α Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

26 Equilibrium in OLG with technological change Euler equation for consumption and household budget constraint as before r t+1 Kt+1 O β ( ) w t Kt+1 O = r t+1 Substitute in for wage and market clearing conditions Normalised equilibrium K t+1 = K t+1 = β 1 + β (1 α) K t α θ 1 α t β (1 + β) (1 + g) (1 α) K α t with K t = K t θ t Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

27 Equilibrium in OLG with technological change Monotonic convergent dynamics to steady state in K t In long run 1 θ t, K t, Y t, C Y t, C O t grow at rate g 2 K t /Y t constant 3 r t = αk α 1 t (θ t L t ) 1 α = α K α 1 t constant 4 w t = (1 α) K α t θ t (θ t L t ) α = (1 α) K α t θ t grows at rate g Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

28 Check Kaldor (1957) facts 1 Output per worker grows at a roughly constant rate YES 2 Capital per worker grows over time YES 3 Capital/output ratio is roughly constant YES 4 Rate of return to capital is constant YES 5 Shares of capital and labour in net income are nearly constant YES 6 Real wage grows over time YES 7 Ratios of consumption and investment to GDP are constant YES Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

29 Ramsey model Retain existing assumptions 1 Log utility 2 Labour supply exogenous 3 Constant returns to scale in production function 4 100% depreciation 5 Technology grows at rate g Replace OLG structure with infinitely-lived representative agent Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

30 Household problem in Ramsey Representative household maximises lifetime utility max {C t+s,k t+s+1 } s=0 s.t. β s log C t+s K t+s+1 = r t+s K t+s + w t+s C t+s for s 0 K T +1 K t given, lim 0 T R T Euler equation for consumption as before C t+1 βc t = r t+1 Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

31 Firm problem in Ramsey Firm maximises profits ( ) max Kt α (θ t L t ) 1 α w t L t r t K t K t,l t First order conditions define factor prices as before r t = αkt α 1 (θ t L t ) 1 α w t = (1 α) Kt α θ t (θ t L t ) α Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

32 Equilibrium in Ramsey Equilibrium is a sequence {r t, w t, C t, K t, L t } t=0 s.t. 1 {C t, K t+1 } solves household problem given {r t, w t } 2 {K t, L t } solves firm problem given {r t, w t } 3 Markets clear L t = 1 and C t + K t+1 = Y t = Kt α (θ t L t ) 1 α 4 K t given 5 Transversality Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

33 Equilibrium in Ramsey Euler equation for consumption and return to capital imply 1 = αβ 1 Kt+1 α 1 C t C (θ t+1l t+1 ) 1 α t+1 Substitute in for market clearing conditions and rearrange Y t+1 = 1 C t+1 αβ + 1 Y t αβ C t Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

34 Transversality Only stable solution is Y t C t = 1 1 αβ for t 0 Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

35 Equilibrium in Ramsey with technological change 1 Y t /C t, Y t /K t+1 constant 2 θ t, C t, K t, Y t grow at rate g 3 r t = αk α 1 t (θ t L t ) 1 α = α K α 1 t constant 4 w t = (1 α) K α t θ t (θ t L t ) α = (1 α) θ t K α t grows at rate g Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

36 Check Kaldor (1957) facts 1 Output per worker grows at a roughly constant rate YES 2 Capital per worker grows over time YES 3 Capital/output ratio is roughly constant YES 4 Rate of return to capital is constant YES 5 Shares of capital and labour in net income are nearly constant YES 6 Real wage grows over time YES 7 Ratios of consumption and investment to GDP are constant YES Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

37 Endogenous growth models Growth is exogenous in neoclassical OLG and Ramsey models Endogenous growth models seek to explain growth Extended accumulation models overcome diminishing returns to capital by adding externalities in capital accumulation (learning-by-doing and AK model) or additional factors of production (human capital) Innovation models explain technological progress as a function of endogenous variables (investment in research and development) Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

38 Learning-by-doing, Arrow (1962) Production function of representative firm Y t = K α t (θ t L t ) 1 α with θ t given Knowledge advances as aggregate capital stock increases θ t = ψk t 1 Aggregate production linear in K t when K t K t 1 Y t = ψ 1 α K t L 1 α t Returns are decreasing at firm level but constant in aggregate Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

39 Equilibrium in AK model Factor prices r t = αkt α 1 (θ t L t ) 1 α αψ 1 α w t = (1 α) Kt α θ t (θ t L t ) α (1 α) ψ 1 α K t Euler equation for consumption C t+1 C t = βr t+1 = αβψ 1 α Transversality as before θ t, C t, K t, Y t grow at constant rate αβψ 1 α Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

40 Growth accounting Take logs and differentiate Y t = K α t (θ t L t ) 1 α 1 dy t Y t dt = α 1 dk t K t dt + (1 α) 1 dθ t θ t dt + (1 α) 1 dl t L t dt α 1/3 as it is capital share of income in equilibrium Growth in Y t = 1 3 Growth in K t (Growth in θ t + Growth in L t ) Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

41 Growth accounting in the 20th century, Crafts (2000) Output growth Contribu TFP Contribution of capital Contribution of labour Japan 2.2% 0.7% 1.2% 0.3% UK 1.3% 0.4% 0.8% 0.1% US 2.8% 1.3% 0.9% 0.6% Germany 1.3% 0.3% 0.6% 0.4% Output growth Contribu TFP Contribution of capital Contribution of labour Japan 9.2% 3.6% 3.1% 2.5% UK 3.0% 1.2% 1.6% 0.2% US 3.9% 1.6% 1.0% 1.3% Germany 6.0% 3.3% 2.2% 0.5% Output growth Contribu TFP Contribution of capital Contribution of labour Japan 3.8% 1.0% 2.0% 0.8% UK 1.6% 0.7% 0.9% 0.0% US 2.4% 0.2% 0.9% 1.3% Germany 2.3% 1.5% 0.9% 0.1% Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

42 Growth accounting in emerging markets Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43

43 Growth accounting for the UK , ONS (2015) Average UK growth of 2.1% decomposes into contributions of 0.4% from technology, 1.2% from capital and 0.5% from labour Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term / 43