The Ramsey-Cass-Koopmans Model: Romer (2001), Ch. 2, Part A
The Ramsey-Cass-Koopmans Model: Romer (2001), Ch. 2, Part A
The Ramsey-Cass-Koopmans Model: Romer (2001), Ch. 2, Part A
Now: Two basic models that are based on microeconomic decision-making. These form the basis for the microeconomic approach of modern macroeconomics. They are also referred to as dynamic general equilibrium models. Similar to Solow model, but: Both the Ramsey-Cass-Koopmans (RCK) model and the overlapping-generations (OLG) model endogenise the savings decision. Solution quite dicult; intuitive approach used here (as in Romer, Ch. 2). The supplies of labour and technology are still exogenous. RCK model: no market imperfections benchmark model; key feature: shows decentralised economy to reach a pareto-ecient outcome.
Assumptions
Firms: large number; identical; produce with identical (neoclassical) production function Y = F (K, AL); competitive factor and output markets; take A as given (exog. growth rate g); maximise prots; owned by households who receive all prots Households (HH): large number (H); identical; size of each household grows at rate n; capital rented out to rms (no depreciation); income is used for consumption and saving; HH maximises lifetime utility U=
Z
et u (C(t))
L(t) dt H
(2.1)
t=0
u(t) = instantaneous utility function (of each HH member); C(t) = consumption per person; L(t) = total size of population; L(t) = number of household members; = H discount rate et discounting in continuous time instantaneous utility function: u (C(t)) = C(t)1 1 > 0, n (1 ) > 0 (2.2)
constant relative risk aversion (CRRA) (i.e. Cu00 (C)/u0 (C) is constant) constant intertemporal elasticity of substitution (CIES) (i.e. 1/ is constant) (see IS2) C 1 is increasing in C if < 1, decreasing if > 1 and equals ln C for = 1; dividing by (1 ) ensures that MU (marginal utility) of consumption is always positive; the assumption n (1 ) > 0 ensures that lifetime utility converges (see below)
2
2.1
2.2
Households
intertemporal budget constraint Z Z K(0) L(t) L(t) R(t) dt + dt e C(t) eR(t) W (t) H H H t=0 t=0
(2.6)
Rt R(t) = =0 r( )d , so that eR(t) shows the eect of continuously compounding interest and eR(t) the eect of continuously discounting up to time t. the limiting behaviour Z K(0) L(t) + dt 0 (2.7) eR(t) [W (t) C(t)] | {z } H H t=0
saving at t
K(0) lim + s H
Household capital holdings at time s are Z s K(s) L(t) R(s) K(0) = e dt + eR(s)R(t) [W (t) C(t)] H H t=0 | {z H }
contribution of initial wealth
R(t)
t=0
(2.8)
(2.9)
This is equation (2.8) times eR(s) , so that (2.8) can be equivalently written as lim eR(s) K(s) 0 H (2.10)
(no-Ponzi-game condition, i.e. continuous rollover of debt not possible) Writing everything in per eective worker units: instantaneous utility function (NB: consumption per eective worker is c(t) = C(t)/A(t), and thus consumption per worker equals C(t) = A(t)c(t)) c(t)1 [A(t)c(t)]1 [A(0)egt ] c(t)1 C(t)1 = = = A(0)1 e(1)gt 1 1 1 1 lifetime utility becomes U=
Z 1
(2.11)
t=0
e U = A(0) {z H } t=0 |
=B
1 L(0)
dt
= B
t=0
et
c(t)1 dt 1
(2.12)
K(t) = total capital stock in the economy; K(t) = capital stock per household; H K(t) per eective labour units: k(t) = A(t)L(t) ; and thus K(t) = k(t)A(t)L(t); so the intertemporal budget constraint becomes Z Z A(0)L(0) A(t)L(t) A(t)L(t) R(t) dt k(0) + dt (2.13) e c(t) eR(t) w(t) H H H t=0 t=0 Note that A(t)L(t) = A(0)L(0)e(n+g)t Z Z R(t) (n+g)t e c(t)e dt k(0) +
t=0
eR(t) w(t)e(n+g)t dt
(2.14)
t=0
(2.15)
Household maximisation: choose the path of c(t) to maximise lifetime utility, subject to its budget constraint Lagrangian (from 2.12 and 2.13): L = B c(t)1 dt + 1 t=0 Z Z R(t) (n+g)t k(0) + e w(t)e dt et
t=0 Z
(2.16)
R(t)
c(t)e
(n+g)t
dt
t=0
HH chooses innitely many c(t)s; dynamic optimisation (optimal control problem; method not covered in this course); rst-order condition for an individual c(t) is Bet c(t) = eR(t) e(n+g)t in logs ln B t ln c(t) = ln R(t) + (n + g) t Z t r( )d + (n + g) t = ln
=0
(2.17)
(2.18)
(2.20)
growth rate of consumption per worker C(t) C(t) c(t) r(t) = +g = C(t) c(t) positive if r(t) > ; negative if r(t) < the smaller , the higher the intertemporal elasticity of substitution and the larger are the changes in consumption over time Intuitive derivation of the Euler equation consider the trade-o of a HH thinking about consuming a little less in period t and consuming a little more in period t + t loss in utility in period t u c = Bet c(t) c c gain in utility in period t + t u c = Be(t+t) [c(t + t)] c(t + t) c [r(t)ng]t e c = Be(t+t) c(t)e[c(t)/c(t)]t (2.21)
[r(t)ng]t e c Bet c(t) c = Be(t+t) c(t)e[c(t)/c(t)]t [r(t)ng]t e 1 = et e[c(t)/c(t)]t c(t) t + [r(t) n g] t 0 = t c(t) c(t) 0 = [( n) (1 ) g] + [r(t) n g] c(t) c(t) = r(t) g c(t)
The Euler equation describes the optimal behaviour of c(t) given some initial value c(0). The latter is determined by the HHs budget constraint, i.e. given (2.20) there is only one value of c(0) that just exhausts the HHs lifetime wealth.
when k > k f 0 (k) < + g c < 0 The dynamics of k evolution of k identical to the Solow model k(t) = f (k(t)) c(t) (n + g)k(t) c = f (k) (n + g)k Figure 2.2 points above the curve: c > f (k) (n + )k k < 0 points below the curve: c < f (k) (n + )k k > 0 c c k > 0 for small k and k < 0 when k gets large; it is zero at the golden rule level 0 of k, i.e. where f (k) = n + g The phase diagram Figure 2.3 E: joint equilibrium where both c = 0 and k = 0 k must be smaller than the golden rule level of k, because f 0 (k ) = + g > f 0 (kgoldenrule ) = n + g (since we assumed earlier that n (1 ) g > 0). The initial value of c Figure 2.4 many divergent paths that satisfy the laws of motion (2.23) and (2.24) but: do not satisfy the intertemporal budget constraint ruled out only the path beginning at F is possible (2.24)
The saddle path Figure 2.5 for any initial k, the economy must be on this saddle path
Welfare
Is the equilibrium a desirable outcome? First welfare theorem decentralised outcome is pareto-ecient (competitive markets, no externalities, large number of agents, ...) a benevolent social planner would choose the same outcome
only dierence: capital stock above the golden rule level is not possible (because choosing a sub-optimal level of consumption every period would be irrational) the chosen k will always be below the golden-rule k, because subjective discount rate > 0, i.e. HHs discount future consumption modied golden-rule capital stock 5
6.1
Qualitative eects:
c(t) f 0 (k(t)) g = c(t) f 0 (k) = + g (2.23)
a lower implies a higher k (because of diminishing MPK), so curve shifts to the right immediate decline in c to point A and then gradual movement to E similar behaviour as in Solow model (one dierence: saving rate is not constant during adjustment)
6.2
not required for the exam brief intuition: can derive the speed of adjustment via linearisation adjustment speed higher than in the Solow model reason: saving rate initially higher (lower) than in the steady state if we start o with k < k (k > k ), i.e. savings rate is not constant
7
7.1
(2.39)
7.2
Unanticipated temporary increase a little more complicated smaller reduction in consumption Figure 2.9 initial downward jump depends on how long the higher government purchases last; compare panels a) and c) no new saddle path instead: dynamics of k take the economy back onto the old saddle path important: there cannot be a discontinuous jump because HHs know that government spending will later return to old level note also behaviour of the real interest rate in panel b) of Figure 2.9 this reects the behaviour of the capital stock during the adjustment path when the rise in government spending is permanent: no interest rate eect
7.3
Empirical application
Barro (1987): wars and real interest rates war: good example of temporarily high government spending, so expect real interest rates to be higher during wars evidence: mixed