Advanced Macroeconomics The Ramsey Model Marcin Kolasa Warsaw School of Economics Marcin Kolasa (WSE) Ad. Macro - Ramsey model 1 / 30
Introduction Authors: Frank Ramsey (1928), David Cass (1965) and Tjalling Koopmans (1965) Basically the Solow model with endogenous savings - explicit consumer optimization Probably the most important model in contemporaneous macroeconomics, workhorse for many areas, including business cycle theories Marcin Kolasa (WSE) Ad. Macro - Ramsey model 2 / 30
Basic setup Closed economy No government One homogeneous final good Price of the final good normalized to 1 in each period (all variables expressed in real terms) Two types of agents in the economy: Firms Households Firms and households identical: one can focus on a representative firm and a representative household, aggregation straightforward Marcin Kolasa (WSE) Ad. Macro - Ramsey model 3 / 30
Firms Final output produced by competitive firms Neoclassical production function with Harrod neutral technological progress Y t = F (K t, A t L t ) (1) Capital and labour inputs rented from households Technology is available for free and grows at a constant rate g > 0: Maximization problem of firms: A t+1 = (1 + g)a t max{f (K t, A t L t ) W t L t R K,t K t } L t,k t Firms maximize their profits, taking factor prices as given (competitive factor markets) First order conditions: W t = F L t R K,t = F K t = f (k t ) (2) Marcin Kolasa (WSE) Ad. Macro - Ramsey model 4 / 30
Households I Own production factors (capital and labour), so earn income on renting them to firms Labour supplied inelastically, grows at a constant rate n > 0: L t+1 = (1 + n)l t Capital is accumulated from investment I t and subject to depreciation: K t+1 = (1 δ)k t + I t (3) Total income of households can be split between consumption or savings (equal to investment): W t L t + R K,t K t = C t + S t = C t + I t (4) Make optimal consumption-savings decisions Marcin Kolasa (WSE) Ad. Macro - Ramsey model 5 / 30
Households II Households maximize the lifetime utility of their members (present and future): U 0 = β t u( C t )L t (5) t=0 where: C t = Ct L t - consumption per capita β - discount factor (0 < β < 1) u(c t ) - instantaneous utility from consumption: where: θ > 0 If θ = 1 then u( C t) = ln C t u( C t ) = C t 1 θ 1 θ (6) Marcin Kolasa (WSE) Ad. Macro - Ramsey model 6 / 30
Households III Remarks: Literally: household members live forever Justification: intergenerational transfers, people care about utility of their offspring Discounting: households are impatient Remarks on the utility function: u( C t ) is a constant relative risk aversion function (CRRA): C t u ( C t ) u ( C t ) = θ u( C t ) is a constant intertemporal elasticity of substitution function: ( ln C1 ln ) C ( 2 u ( C 1) u ( C 2) ) = 1 θ CRRA form essential for balanced growth (steady state) Marcin Kolasa (WSE) Ad. Macro - Ramsey model 7 / 30
Households IV Households optimization problem: maximize (5) subject to the model s constraints: Capital law of motion (capital is the only asset held by households), incorporating income definition and savings-investment equality (4) Transversality condition: ( ) lim t t K t+1 s=1 1 1 + r s 0 (7) where: r t = R K,t δ = f (k t ) δ is the market rate of return on capital (real interest rate) Interpretation of the transversality condition: Analog of a terminal condition in a finite horizon Non-negativity constraint on the terminal (net present) value of assets held by households (capital) No-Ponzi game condition: proper lifetime budget constraint on households Optimization by households implies (7) holds with equality Marcin Kolasa (WSE) Ad. Macro - Ramsey model 8 / 30
General equilibrium Market clearing conditions: Output produced by firms must be equal to households total spending (on consumption and investment): Y t = C t + I t (8) Labour supplied by households must be equal to labour input demanded by firms Capital supplied by households must be equal to capital input demanded by firms Definition of competitive equilibrium A sequence of {K t, Y t, C t, I t, W t, R K,t } t=0 for a given sequence of {L t, A t } t=0 and an initial capital stock K 0, such that (i) the representative household maximizes its utility taking the time path of factor prices {W t, R K,t } t=0 as given; (ii) firms maximize profits taking the time path of factor prices as given; (iii) factor prices are such that all markets clear. Marcin Kolasa (WSE) Ad. Macro - Ramsey model 9 / 30
Households optimization problem Lifetime utility rewritten: U 0 = t=0 = L 0 β t C t 1 θ 1 θ L t = t=0 β t C t 1 θ 1 θ (9) where: β = β(1 + n) β(1 + g) 1 θ (1 + n) < 1 the last inequality is assumed and ensures that utility is bounded Marcin Kolasa (WSE) Ad. Macro - Ramsey model 10 / 30
Households optimization problem II Capital accumulation rewritten in per capita terms (using (4)): K t+1 = 1 δ 1 + n K t + 1 1 + n (W t + R K,t K t C t ) (10) Capital accumulation rewritten in intensive form (using (4) and defining w t = Wt A t ): k t+1 = 1 δ (1 + g)(1 + n) k 1 t + (1 + g)(1 + n) (w t + R K,t k t c t ) (11) Transversality condition rewritten in intensive form: ( ) t (1 + n)(1 + g) lim 0 (12) t 1 + r s k t+1 s=1 Marcin Kolasa (WSE) Ad. Macro - Ramsey model 11 / 30
Lagrange function and FOCs Lagrange function (normalizing L 0 ): LL = t=0 ( C β t t 1 θ [ (1 δ) 1 θ + λ K t + W t + R K,t K t C t t (1 + n) K t+1 First order conditions (FOCs): ] ) LL C t = 0 = θ C t = λ t (13) LL K t+1 = 0 = βλ t+1 (1 δ + R K,t+1 )) = (1 + n)λ t (14) Marcin Kolasa (WSE) Ad. Macro - Ramsey model 12 / 30
Euler equation I Equations (13) and (14) imply: ( ) θ C t+1 = β R K,t+1 + 1 δ C t (1 + n) (15) Using the definition of β and rewriting in intensive form: ( ct+1 c t ) θ = β R K,t+1 + 1 δ (1 + g) θ (16) For consumption per capita, using also the definition of the interest rate r t : ( ) θ ( ) C t+1 ct+1 A θ t+1 = = β (1 + r t+1) (17) C t c t A t Marcin Kolasa (WSE) Ad. Macro - Ramsey model 13 / 30
Euler equation II Interpretation of the Euler equation (17): For θ > 0: Ct+1 > C t 1 + r t+1 > β 1 Interpretation: For (per capita) consumption to grow the (market) interest rate must exceed households rate of time preference Interpretation: It is optimal for households to postpone consumption (i.e. save in the current period and consume more in the next period) iff the related utility loss is more than offset by the rate of return on savings Role of θ: The higher θ the less responsive consumption to changes in the interest rate In other words: The higher θ the stronger the consumption smoothing motive (the lower intertemporal substitution) Marcin Kolasa (WSE) Ad. Macro - Ramsey model 14 / 30
Equilibrium dynamics in intensive form - summary The equilibrium dynamics of the model at any time t can be characterized by 3 equations: (16), (11) and (12). They are (after some rewriting and using (4) and (8) in intensive form): k t+1 k t = ( ct+1 c t ) θ = β f (k t+1 ) + 1 δ (1 + g) θ (18) 1 δ (1 + g)(1 + n) + 1 f (k t ) c t (19) (1 + g)(1 + n) k t ( ) t (1 + n)(1 + g) k t+1 f = 0 (20) (k t+1 ) + 1 δ lim t s=1 At t = 0 capital is fixed. For given initial k 0 and c 0, equations (18) and (19) describe the future evolution of these variables: k t and c t. The transversality condition (20) pins down the initial level of consumption c 0. Marcin Kolasa (WSE) Ad. Macro - Ramsey model 15 / 30
Steady state equilibrium I In the steady state equilibrium k t and c t must be constant. Using (18) and (19), the long-run solution to the Ramsey model: f (k ) = (1 + g)θ β 1 + δ (21) c = f (k ) (n + g + δ + ng)k (22) Marcin Kolasa (WSE) Ad. Macro - Ramsey model 16 / 30
Steady state equilibrium II Plotting (21) and (22) in the (k, c) space: c t c t+1 =c t c* k t+1 =k t k* k G k t Marcin Kolasa (WSE) Ad. Macro - Ramsey model 17 / 30
Modified golden rule I How do we know that k < k G? From (22): f (k G ) = n + g + δ + ng The transversality condition (20) written in the steady-state: This implies: lim t k Since f (k) < 0 for any k > 0 ( ) t (1 + n)(1 + g) f (k = 0 ) + 1 δ f (k ) > n + g + δ + ng f (k ) > f (k G ) = k < k G (23) Marcin Kolasa (WSE) Ad. Macro - Ramsey model 18 / 30
Modified golden rule II Equivalently, we can show that the steady-state savings rate s falls short of the savings rate consistent with the golden rule: From (22), the steady-state savings rate is: Using (23): s = 1 c f (k ) = (n + g + δ + ng) k f (k ) k s < f (k ) f (k ) = α(k ) Intuitive explanation: households are impatient (β < 1) and smooth consumption (θ > 0, relevant if g > 0). Marcin Kolasa (WSE) Ad. Macro - Ramsey model 19 / 30
The role of the discount factor Higher β implies more patient consumers From (21): if β goes up, f (k ) goes down, which means that k goes up The c t+1 = c t locus on the (k, c) chart shifts right Steady-state consumption goes up Intuition: if households are more patient, they save more, which brings them closer to the standard golden rule Marcin Kolasa (WSE) Ad. Macro - Ramsey model 20 / 30
Phase diagram From (18): k k = c 0 From (19): c f (k) (n + g + δ + ng)k = k 0 c t c t+1 =c t c* k t+1 =k t k* k t Marcin Kolasa (WSE) Ad. Macro - Ramsey model 21 / 30
Saddle path (stable arm) Transversality condition (20) pins down the inital level of c 0 for any initial k 0, so that the system converges to the steady-state: c t c t+1 =c t c* k t+1 =k t c 0 k 0 k* k t Marcin Kolasa (WSE) Ad. Macro - Ramsey model 22 / 30
Uniqueness of equilibrium How do we know that the saddle path is a unique equilibrium? If the initial level of consumption were below c 0 : capital would eventually reach its maximal level k > k G this implies: f ( k) < f (k G ) = n + g + δ + ng which violates the transversality condition (20) since: ( ) t (1 + n)(1 + g) lim k = t f ( k) + 1 δ informally (but more intuitively): at the end of their planning horizon, households would hold very valuable assets, which cannot be optimal If the initial level of consumption were above c 0 : capital would eventually reach 0 but consumption would stay positive, which is clearly not feasible Marcin Kolasa (WSE) Ad. Macro - Ramsey model 23 / 30
Speed of convergence Compared to the Solow model, the speed of convergence in the Ramsey model depends additionally on the behaviour of the savings rate along the transition path For very small time intervals, the following implications hold (see Barro and Sala-i-Martin, 2004, ch. 2.6.4): 1 = s t s depends positively on k t k θ < s 1 θ = s 1 θ > s = s t = s = s t s depends negatively on k t k Intuition (suppose the economy starts from k 0 < k, so c 0 < c ): if households care much about consumption smoothing (θ is high), they wll try to shift consumption from the future to the present if households care little about consumption smoothing (θ is low), they will try to postpone consumption to reach steady-state sooner For standard parameter values 1 θ > s, so the Ramsey model predicts relatively fast pace of convergence Marcin Kolasa (WSE) Ad. Macro - Ramsey model 24 / 30
The budget constraint of the government In Ramsey model with lump sum taxes Ricardian equivalence holds: Government plans sustainable - initial debt plus net present value of expenditures equals net present value of revenues Households live forever No financial frictions - one interest rate Hence: financing expenditures with lump-sum taxes equivalent to financing expenditures with debt With distortionary taxation Ricardian equivalence breaks Still, for simplicity let us assume that the government runs a balanced budget each period: G t = V t + τ w W tl t + τ k (R K,t δ) K t + τ cc t + τ i I t + τ f (Y t W tl t δk t) (24) where: G t - government purchases (exogenous) τ w, τ k, τ c, τ i, τ f - proportional tax rates on wage income, capital income, consumption, investment and firms taxable profits, respectively (all exogenous) V t - lump-sum taxes (net of lump-sum transfers) from households, adjusted so that the balanced budget constraint (24) holds Marcin Kolasa (WSE) Ad. Macro - Ramsey model 25 / 30
Modified firms problem Maximization problem of firms: max L t,k t {F (K t, A t L t ) W t L t R K,t K t τ f (F (K t, A t L t ) W t L t δk t )} First order conditions (using definition r t = R K,t δ): W t = F L t r t 1 τ f + δ = F K t = f (k t ) Firms are competitive so earn zero profits: Y t = W t L t + R K,t K t + τ f (Y t W t L t δk t ) (25) Market clearing on the product market: Y t = C t + I t + G t (26) Marcin Kolasa (WSE) Ad. Macro - Ramsey model 26 / 30
Modified households problem We assume that government actions do not affect utility directly, so households lifetime utility is still given by (5) Households budget constraint (4) becomes: W t L t +R K,t K t τ w W t L t τ k (R K,t δ) K t V t = (1+τ c )C t +(1+τ i )I t (27) Note that, by (25), households factor income is no longer equal to output Modified transversality condition: ( t lim t K t+1 s=1 1 1 + (1 τ k )r s ) 0 (28) Marcin Kolasa (WSE) Ad. Macro - Ramsey model 27 / 30
Modified households optimization problem Lifetime utility (identical to (9)): U 0 = L 0 t=0 1 θ t C t β 1 θ (29) Capital accumulation (substituting for investment from (27)): K t+1 = 1 δ 1 + g K 1 t + (1 + g)(1 + τ i ) [ (1 τw )W t + (1 τ k )R K,t K t + τ k δ K t (1 + τ c ) C t Ṽ t ] (30) Transversality condition: ( lim t t K t+1 s=1 1 + n 1 + (1 τ k )r s ) 0 (31) Marcin Kolasa (WSE) Ad. Macro - Ramsey model 28 / 30
Equilibrium dynamics The equilibrium dynamics of the model at any time t is given by the following equations: Euler equation (maximizing (29), subject to (30) and using firms FOC): ( ct+1 c t ) θ = β 1 + τ i(1 δ) + (1 τ k )(1 τ f )(f (k t+1 ) δ) (1 + g) θ (1 + τ i ) (32) Capital accumulation equation (30) (merged with government budget constraint (24) and firms zero profit condition (25), with g t = Gt A tl t ): k t+1 1 δ = k t (1 + g)(1 + n) + 1 f (k t ) c t g t (33) (1 + g)(1 + n) k t Transversality condition (31): ( ) t (1 + n)(1 + g) lim t f = 0 (34) (k t+1 ) + 1 δ k t+1 s=1 Marcin Kolasa (WSE) Ad. Macro - Ramsey model 29 / 30
Main implications of the Ramsey model As in the Solow model, long-run growth (of output per capita) possible only with technological progress (exogenous in both models) We should observe conditional, but not necessarily unconditional, convergence (in line with the data) Compared to the Solow model: Explicit optimality criterion - households utility If there is no distortionary taxation (i.e. if there is no government or all taxes are lump-sum), allocations are Pareto optimal: decentralized equilibrium coincides with allocations dictated by a benevolent social planer (markets are competitive and complete, so the first welfare theorem applies) Savings rate endogenous and in the long-run always lower than implied by the golden rule Speed of convergence higher than in the Solow model (for standard parameter values) Marcin Kolasa (WSE) Ad. Macro - Ramsey model 30 / 30