The Ramsey Model - Economics

Main IngredientsThe Model

Equilibrium and PlannerSteady State and Dynamics

The Ramsey ModelLecture 11 & 12

Topics in Macroeconomics

November 12 & 13, 2006

Lecture 11 & 12 1/28 Topics in Macroeconomics



The Ramsey Model 2

Main Ingredients

◮ Neoclassical model of the firm

(Topics 1 & 2)

◮ Consumption-savings choice for consumers

(Topic 3, Certainty)

◮ “Solow model + incentives to save”

(recall example with taxes)




Markets and ownershipRepresentative FirmRepresentative Household

Model: Markets and ownership 3

Agents◮ Firms produce goods, hire labor and rent capital◮ Households own labor and assets (capital),

receive wages and rental payments, consume and save◮ There are Nt households and many firms

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





Model: Firms / Representative Firm* 4

(Recall equivalence!)Seeks to maximize profits

Profitt = F (Kt , Lt) − RtKt − wtLt

The FOCs for this problem deliver

∂F (t)∂Kt

= Rt∂F (t)∂Lt

= wt

In per unit of labor terms, let f (kt) ≡ F (kt , 1)

f ′(kt ) = Rt f (kt ) − kt f ′(kt ) = wt

Recall Euler’s Theorem: factor payments exhaust outputLecture 11 & 12 4/28 Topics in Macroeconomics




Model: Households / Representative household 5

Preferences

U0 =

∞∑

t=0

βtu(ct)

Budget constraint

ct + at+1 = wt lt + (1 + rt)at ,

for all t = 0, 1, 2, ...

a0 given

Note: labor supplied inelastically, lt = 1, i.e. Lt = Nt





Model: Households / Representative household 6

Intertemporal version of budget constraint

∞∑

t=0

t∏

s=0

(

11 + rs

)

ct = a0 +∞∑

t=0

t∏

s=0

(

11 + rs

)

wt

We rule out that debt explodes (no Ponzi games)

at+1 ≥ −B for some B big, but finite

More compactly, PDV (c) = a(0) + PDV (w)





Model: Household’s problem 7

max(at+1,ct )

∞

t=0

∞∑

t=0

βtu(ct)

s.t.

ct + at+1 = wt + (1 + rt)at , for all t = 0, 1, 2, ...

at+1 ≥ −B for some B big, but finite

a0 given






Euler equationIn general,

u′(ct) = β(1 + rt+1)u′(ct+1)

From here on, CES utility, u(c) = c1−σ

1−σ , Euler eqn. becomes,

(

ct+1

ct

)σ

= β(1 + rt+1)






Budget constraint

ct + at+1 = wt + (1 + rt)at , for all t = 0, 1, 2, ...






Transversality conditionHH do not want to “end up” with positive values of assets

limt→∞

βtu′(ct)at ≤ 0

HH cannot think they can borrow at the “end of their life”

limt→∞

βtu′(ct)at ≥ 0

Hence,

limt→∞

βtu′(ct)at = 0




Definition of EquilibriumCharacterizing Equilibrium QuantitiesBenevolent Planner’s Problem

Definition of Equilibrium* 11

A competitive equilibrium is defined by sequences of quantitiesof consumption, {ct}, capital, {kt}, and output, {yt}, andsequences of prices, {wt} and {rt}, such that

◮ Firms maximize profits

◮ Households maximize U0 subject to their constraints

◮ Goods, labour and asset markets clear◮ Choices are consistent with the aggregate law of motion

for capitalKt+1 = (1 − δ)Kt + It





Characterizing Equilibrium Quantities* 12

From the equilibrium conditions derived before, we find:

◮ There cannot be arbitrage opportunities in equilibrium

Rt − δ = rt

In equilibrium it does not pay to invest in capital directly.The riskless asset and capital have the same payoff.





Characterizing Equilibrium Quantities* 13

From the equilibrium conditions derived before, we find:

◮ Substituting out all the prices leads to the following set ofnecessary and sufficient conditions for an equilibrium interms of quantities only.

kt+1 + ct = f (kt) + (1 − δ)kt

ct+1

ct= [β(1 + f ′(kt+1) − δ)]1/σ

limt→∞

βtu′(ct )kt = 0

k0 > 0

Prices can be determined from the firm’s problems FOCs.Lecture 11 & 12 13/28 Topics in Macroeconomics




Benevolent planner’s problem* 14

What is the allocation of resources that an economy shouldfeature in order to attain the highest feasible level of utility?

Central Planner’s optimal choice problem

max(kt+1,ct)

∞

t=0

∞∑

t=0

βtu(ct)

s.t.

ct + kt+1 = f (kt) + (1 − δ)kt , for all t = 0, 1, 2, ...

k0 > 0 given





Benevolent planner’s problem 15

Welfare

Socially optimal allocation coincides with the equilibriumallocation.

The competitive equilibrium leads to the social optimum.

Not surprising: no distortions or externalities→ Welfare Theorems hold




Steady StateDynamics

Notes: simplifying features* 16

◮ We are considering an economy without population growth.

◮ There is no exogenous technological change, either.





Steady state* 17

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 isconstant (i.e. the growth rate is zero)

Recall from the Solow model:

aggregate capital stock for (n = 0, g = 0)capital per unit of labor for (n > 0, g = 0)capital per unit of effective labor for (n > 0, g > 0)





Steady state 18

From the Euler equation,

ct+1

ct= [β(1 + f ′(kt+1) − δ)]1/σ , for all t

If consumption grows at a constant rate (BGP), say γ

1 + γ = [β(1 + f ′(kt+1) − δ)]1/σ , for all t

Thus RHS must be constant→ kt+1 = kt = k∗ must be constant along the BGP





Steady state 19

But then, from the resource constraint with kt = kt+1 = k∗:

ct + kt+1 = f (kt) + (1 − δ)kt , for all t

i.e.,ct = f (k∗) − δk∗

ct+1 = f (k∗) − δk∗

We find that consumption must be constant along the BGP,→ ct+1 = ct = c∗ or γ = 0

Hence we have a steady state in per capita variables.





Steady state* 20

Hence from the Euler equation

1 + γ = 1 = [β(1 + f ′(k∗) − δ)]1/σ

or, simplified

f ′(k∗) =1β− (1 − δ) = ρ + δ

we can solve for k∗and from the (simplified) resource constraint

c∗ = f (k∗) − δk∗

we can solve for c∗





Modified golden rule* 21

The capital stock that maximizes utility in steady state is calledthe modified golden rule level of capital

f ′(k∗) = ρ + δ

Using f (k) = kα, we get

k∗ = kMGR =

[

α

ρ + δ

]1

1−α

Compare to golden rule level of capital(max conso in st. st.)

kGR =[α

δ

]1

1−α

(see Problem set 2, Q 2.2, assume A = 1 and set s = α)Lecture 11 & 12 21/28 Topics in Macroeconomics




Modified golden rule 22

Since ρ > 0 and α ∈ (0, 1),

kMGR =

[

α

ρ + δ

]1

1−α

<[α

δ

]1

1−α

= kGR

This result reflects the impatience of agents.

As long as ρ > 0, they’d always prefer to consume earlier ratherthan later, thereby reducing investments for next period andhence the steady state level of capital (and consumption)!

One of Ramsey’s points was that this is the steady state that weshould aim at because it makes people the happiest - not theone that maximizes consumption per se.





Off steady state dynamics* 23

Off the steady state, consumption and capital adjust to reachthe steady state eventually.

To analyze these dynamics, consider the movements of c and kseparately.

Let ∆c = ct+1 − ct and ∆k = kt+1 − kt . See graphical analysis.






We use 2 equilibrium conditions:

◮ Euler equation (EE)

ct+1

ct= [β(1 + f ′(kt+1) − δ)]1/σ

◮ Resource constraint (RC)

ct + kt+1 = f (kt) + (1 − δ)kt






◮ Let ∆c = ct+1 − ct and ∆k = kt+1 − kt

◮ Use EE to determine points where ∆c = 0

◮ Use RC to determine points where ∆k = 0

◮ Look at dynamics left and right of ∆c = 0

◮ Look at dynamics above and below ∆k = 0

◮ Steady state is where ∆c = 0 and ∆k = 0






◮ Consider the set of points such that ∆c = 0, then from theEuler eqn, the optimal k satisfies f ′(k) = ρ + δ

→ draw vertical line at k∗(< kGR)

To the left: kt < k∗ ⇒ f ′(kt) > f ′(k∗) ⇒ ∆c > 0 ⇒ c ↑To the right: kt > k∗ ⇒ f ′(kt) < f ′(k∗) ⇒ ∆c < 0 ⇒ c ↓

◮ Consider the set of points such that ∆k = 0, then from theResource cstrt, the optimal c satisfies c = f (k) − δk

→ draw hump-shaped line from origin, maximized at kGR

cross 0 again for k such that f (k) = δk

Above: ct > f (kt) − δkt ⇒ ∆k = f (kt ) − δkt − ct < 0 ⇒ k ↓Below: ct < f (kt) − δkt ⇒ ∆k = f (kt) − δkt − ct > 0 ⇒ k ↑





Equilibrium path toward steady state 27

◮ Suppose k0 < k∗

Then, what consumption level should the household pick?

◮ above ∆k = 0 – curve?

This has c rising but would eventually lead to k = 0 andfrom RC jump of c to c = 0 → violates EE

→ cannot be an equilibrium decision◮ below ∆k = 0 – curve?

Yes, for some c0 all equilibrium conditions will be satisfied

Intuition:K-stock too low, marginal product high → invest a lot

◮ if too low

HH oversaving → leads to c = 0 and ∆k = 0 → violates TC





Equilibrium path toward steady state 28

◮ Suppose kt > k∗

Then, what consumption level should the household pick?

◮ below ∆k = 0 – curve?

This would lead to k such that f (k) = δk and c = 0(u′(0) = ∞, >< transversality)

→ cannot be an equilibrium decision◮ above ∆k = 0 – curve?

Yes, for some ct all equilibrium conditions will be satisfied

Intuition:K-stock too high, marginal product low → consume a lot

◮ If too high, get to k = 0 and jump to c = 0 again.


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The Ramsey Model - Economics