# 程序代写代做代考 Lecture 2: Ramsey-Cass-Koopmans Model

Lecture 2: Ramsey-Cass-Koopmans Model
Patrick Macnamara
ECON60111: Macroeconomic Analysis University of Manchester
Fall 2020

Introduction: Ramsey-Cass-Koopmans Model
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Introduction
Two major shortcomings of the Solow model:
(1) The saving rate is exogenous, not determined by the decisions of economic agents.
• The Ramsey-Cass-Koopmans model address this issue. They endogenize the saving rate. Saving and investment are determined in general equilibrium.
(2) Technological progress is exogenous, not explained. • Endogenous growth models address this issue.
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Main Ingredients of Ramsey-Cass-Koopmans
(1) Two types of optimizing agents: • Firms
• Households
• (Closed economy, still no public sector)
(2) Several competitive markets:
• Goods market: all-purpose output goods
• Labor market
• Rental market for capital goods • Loan/bond market
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General Equilibrium
Firms
• employ labor at the market wage W(t),
• rent capital goods at the rental rate r(t),
• produce homogeneous output good; price normalized to 1.
Households
• supply labor at the market wage W(t), • consume the output good,
• save in bonds and capital.
No arbitrage between capital and bond markets
• For simplicity, we assume δ = 0
• Return on bonds, return on capital both equal to r(t)
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The Model
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Technology
• Firms’ production function is:
Y =F(K,AL)
where Y is output, K is capital and AL is effective labor.
• Same assumptions imposed on F as last week:
• (1)F exhibitsCRS,(2)f(0)=0,f′ >0,f′′ <0where f(k)≡F􏰀K ,1􏰁,(3)Inadaconditions AL • Initial levels of K(t), L(t) and A(t) are taken as given e.g., fix K(0) > 0, L(0) > 0 and A(0) > 0
• A(t) = exogenous technology level, grows at constant rate g: A ̇/A=g ⇒ A(t)=A(0)egt
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Firms
• Large number of firms, who: • are owned by households,
• take prices as given,
• hire labor and rent capital to maximize profits.
• Firm’s problem:
Π(t)= max {F(K(t),A(t)L(t))−r(t)K(t)−W(t)L(t)}
K(t),L(t)
r(t) is also the rental rate of capital, since δ = 0
Y (t) real int. rate real wage • First order conditions (FOC):
r(t) = ∂Y(t)/∂K(t) W (t) = ∂Y (t)/∂L(t)
• CRS assumption and perfect competition ⇒ Π(t) = 0.
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Firms’ FOC in Intensive Form
• Using Y = ALf(k), k = K/(AL) and f(k) ≡ F (k,1): 􏰂K􏰃 􏰂K􏰃
Y=ALf AL Y=ALf AL
∂Y ∂K
􏰂K􏰃 ∂Y AL ∂L
􏰂K􏰃 AL
􏰂K􏰃 K AL AL2
− ALf ′ ∂Y =f′(k) ∂Y =A[f(k)−kf′(k)]
= f ′
∂K ∂L
= Af
• Then, capital and bond returns are:
r(t) = f ′(k(t))
• Wage per unit of effective labor, w(t) ≡ W(t)/A:
w(t) = f (k(t)) − k(t)f ′(k(t))
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Households
• H identical households
• Total population is L(t), which grows at constant rate n:
L ̇/L = n ⇒ L(t) = L(0)ent
• Each household has L(t)/H members
• Each member of household supplies 1 unit of labor at every point in time
• Household, taking prices as given, chooses consumption and saving to maximize lifetime utility.
• This is the key difference with the Solow growth model
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Households: Utility
• Household’s utility function:
􏰗∞ −ρt L(t) U= e u(C(t)) dt
t=0 H household’s total
• ρ = discount rate
• C(t) = consumption of each household member • u(·) = instantaneous utility function
• Consumption per unit of effective labor is: c(t) = C(t)
A(t )
Be careful with the notation: C(t) is consumption per household member, NOT total consumption.
Definition of terms:
instantaneous utility
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Households: Utility
• Assume utility is constant-relative-risk-aversion (CRRA): C 1−θ
u(C)=1−θ, θ>0 θ = coefficient of relative risk aversion
1/θ = intertemporal elasticity of substitution
• Assumption required to ensure finite lifetime utility:
ρ−n−(1−θ)g >0
(1)
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Dynamics of Capital and Effective Capital
• Law of motion for capital (assuming δ = 0):
K ̇ (t ) = Y (t ) − C (t )L(t ) (2)
• Use k = K/(AL) and apply quotient rule (and product rule): k ̇(t)= K ̇(t)A(t)L(t)−K(t)􏰉A(t)L ̇(t)+A ̇(t)L(t)􏰊
[A(t )L(t )]2
k ̇(t)= Y(t) −C(t)− K(t) L ̇(t)− K(t) A ̇(t)
A(t)L(t) A(t) A(t)L(t) L(t) A(t)L(t) A(t) 􏰏 􏰎􏰍 􏰐 􏰏􏰎􏰍􏰐 􏰏 􏰎􏰍 􏰐􏰏􏰎􏰍􏰐 􏰏 􏰎􏰍 􏰐􏰏􏰎􏰍􏰐
• We get:
y(t) c(t) k(t) n k(t) g
k ̇(t) = f (k(t)) − c(t) − (n + g)k(t)
(3)
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Households: Budget Constraint
􏰗 ∞ −R(t) L(t) K(0) 􏰗 ∞ −R(t) L(t)
e C(t) dt≤ + e W(t) dt
t=0 HHt=0 H
􏰏 􏰎􏰍 􏰐 􏰏 􏰎􏰍 􏰐
PDV of household consumption PDV of household income
where R(t) = 􏰖 t r(τ)dτ is the continuously compounded 0
interest over the period [0, t].1
s→∞
1This specification accommodates the fact that r(t) may change over time. If r(t) is constant at ̄r, then R(t) = ̄rt.
􏰄K(0) 􏰗s −R(t) lim + e
L(t) 􏰅
[W (t ) − C (t )]
This version of the BC will be useful on the next slide.
H t=0
H
dt ≥ 0 (4)
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Households: Budget Constraint (alternative)
• Household’s wealth at time s, K(s)/H:
􏰗 s HH0H
K(s) R(s) K(0) = e
+
e
R(s)−R(t) L(t) [W (t ) − C (t )] dt
−R(s)K(s) K(0) 􏰗 s −R(t) L(t)
e = + e [W(t)−C(t)] dt
HH0 H
• Notice the similarity of the RHS to (4) on the previous slide.
Household’s budget constraint can be written as: lim e−R(s)K(s) ≥0
s→∞ H
This is the no-Ponzi-game condition. This says it is not
possible for households to issue debt and roll it over forever.
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FOC for Household’s Maximization Problem
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Household Maximization Problem
• Representative household chooses path of consumption to maximize lifetime utility subject to its budget constraint.
• It is easier to work with variables normalized by the quantity of effective labor.
• To formally solve the household’s maximization problem, first re-write utility and the budget constraint in terms of …
• consumption per effective labor c(t), rather than C(t), and • wage per effective labor w(t), rather than W(t)
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Re-Write Household Utility
• Make substitutions to original utility: C(t) =A(t)c(t) =A(0)egtc(t)
􏰗 ∞ −ρt C(t)1−θ L(t) U=e dt
L(t) = L(0)ent
t=0 1−θ H
􏰗 ∞ −ρt 1−θ = e A(0) e
t=0
gt(1−θ) c(t)1−θ L(0)ent 1−θ H
dt gt(1−θ) nt c(t)1−θ
Define as B
• We get:
1−θ L(0) 􏰗 ∞ −ρt
=A(0) eee dt
H t=0 1−θ
􏰗 ∞ −βt c(t)1−θ U=Be dt
t=0 1−θ
where β ≡ ρ − n − (1 − θ) g is the effective rate of time preference.
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Re-Write Budget Constraint
• Original budget constraint:
􏰗 ∞ H t=0
L(t) H
K(0)
+
e
−R(t)
[W(t)−C(t)]
dt ≥0
• Substitute W(t) = w(t)A(t) and C(t) = c(t)A(t):
􏰗 ∞ H t=0
A(t)L(t) H
K(0)
+
e
−R(t)
[w(t) − c(t)]
dt ≥ 0
• Substitute A(t)L(t) = A(0)L(0)e(n+g)t and divide both sides by A(0)L(0)/H:
k(0)+
􏰗 ∞ −R(t) (n+g)t
e e [w(t)−c(t)]dt ≥0
t=0
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Households’ Problem: Lagrangian and FOC
• Lagrangian: L=B
􏰒
􏰗 ∞ c(t)1−θ −βt
0 1−θ
0
e dt
+λ k(0)+
where λ is the Lagrange multiplier.
􏰗∞ −R(t) (n+g)t e e
[w(t)−c(t)]dt
􏰓
• First order condition (FOC):
∂L = Bc(t)−θe−βt − λe−R(t)e(n+g)t = 0
∂c(t)
• At every point in time, household behavior is characterized by (1) this FOC and (2) the budget constraint.
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Households’ Problem: the Euler Equation
• From the FOC:
• Take logs and differentiate w.r.t. t (use R(t) = 􏰖 t r(τ)dτ):
Bc(t)−θe−βt = λe−R(t)e(n+g)t
0
lnB−θlnc(t)−βt =lnλ−R(t)+(n+g)t − θ c ̇ ( t ) − β = − r ( t ) + n + g
c(t)
• Rearrangeanduseβ=ρ−n−(1−θ)g:
c ̇(t) = r(t)−ρ−θg c(t) θ
This is the Euler equation.
• It describes how c must behave over time, given c(0). • c(0) determined by requirement that BC is satisfied
(5)
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Equilibrium and Dynamics
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Equilibrium
The general equilibrium is represented by factor prices (r(t),w(t)) and allocations (c(t), k(t)) such that
(1) firms optimize, given the path of prices
(2) households optimize, given the path of prices
(3) Markets clear (i.e., labor supply equals labor demand and capital supply equals capital demand)
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Dynamics
• Given k(0) and c(0), we use two equations to determine the evolution of c and k over time:
(1) the Euler Equation (see Equation (5)) (2) law of motion for k (see Equation (3))
• The initial k(0) is pinned down by assumption, but c(0) must be determined.
• How can we determine c(0)?
• Analyze the path of (c,k) for different values of c(0) • We can rule out all paths except one
• If c(0) too low, c converges to zero and household does not exhaust lifetime wealth
• If c(0) too high, c explodes and k becomes negative
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The Dynamics of c
• Equilibrium in the capital rental market requires
r(t) = f ′(k(t))
• Substitute for r(t) in the Euler equation (Equation (5)):
c ̇(t) = f′(k(t))−ρ−θg c(t) θ
• Therefore, c ̇(t) = 0 whenever
f ′(k(t)) = ρ + θg
• Steady-state capital per effective worker k∗ determined by: f ′(k∗) = ρ + θg
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Phase Diagram: Consumption
c increasing to left of k∗
c decreasing to right of k∗ 26/38

The Dynamics of k
• The dynamics of k similar to the Solow model, except that here investment is determined endogenously. (An additional difference is that here we assume zero depreciation.)
k ̇(t)=f (k(t))−c(t)− (n+g)k(t) 􏰏 􏰎􏰍 􏰐 􏰏 􏰎􏰍 􏰐
actual investment
See Equation (3) for a derivation. • Therefore, k ̇(t) = 0 whenever
break-even investment
c(t) = f (k(t)) − (n + g)k(t)
• Golden-rule level of capital determined by: f ′(kGR ) = n + g
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Phase Diagram: Capital
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Phase Diagram: Both
Note that k∗ < kGR 29/38 Phase Diagram 30/38 Phase Diagram: Low Initial Consumption A D Households can afford to increase consumption 31/38 Phase Diagram: High Initial Consumption D C Consumption explodes and capital becomes negative 32/38 Phase Diagram: Saddle Path B D For points above or below B, economy doesn’t converge to D 33/38 Phase Diagram: Summary C B A D Only B satisfies budget constraint 34/38 Phase Diagram: Saddle Path A B D 35/38 Balanced Growth • The behavior of the economy once it has converged to the steady state is identical to that of the Solow model. • Capital, output, and consumption per unit of effective labor are constant. • The total capital stock, output, and consumption grow at the constant rate n + g. • Per capita/worker (unit of labor) quantities grow at the constant rate g. • Growth in effectiveness of labor remains the only source of long-run growth in output per worker. • As a result, the saving rate is constant at the steady state ∗ y∗ −c∗ s= y∗ . 36/38 Welfare • Given that (1) markets are competitive and complete, and (2) there are no externalities, the first Welfare theorem applies in this economy. • That is, any competitive (or, Walrasian) equilibrium, leads to a Pareto-efficient allocation. 37/38 The Modified Golden Rule • Saving rate is derived from the optimizing behavior of households, whose utility depends on consumption. • Thus, saving rates higher than the golden-rule level are not possible. • In fact, k∗ is less than the golden-rule level (kGR). ⇐⇒ kGR>k∗ ⇐⇒ρ+θg>n+g
kGR>k∗ ⇐⇒ρ−n−(1−θ)g=β>0
We assumed β > 0.
• Since, k∗ maximizes households’ lifetime utility. We call it the modified-golden-rule level of capital.
f ′(k∗) = ρ + θg
f ′(kGR ) = n + g
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