The Neoclassical Growth Model

1 Setup

• Three goods:

– Final output

– Capital

– Labour

• One household, with preferences∞∑t=0

βtu (ct)

(Later we will introduce preferences with respect to labour/leisure)

• Endowment

– Initial stock of capital k0

– One unit of labour in each period

• Technology

– Capital and labour can be combined to produce final output according to productionfunction

Y = f (K,L)

– Final output good can be consumed or invested. Investing converts output into capitalone-for-one:

Yt = It + Ct

(we may or may not impose It ≥ 0, depending on whether we want to allow capital tobe converted back into final output; typically it doesn’t matter whether we do)

– Capital depreciated at rate δ, so

Kt+1 = (1− δ)Kt + It

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Econ 210, Fall 2013 Pablo Kurlat

2 Competitive Equilibrium

• Markets:

– For output at time t. Price pt. Output at time zero is numeraire (p0 ≡ 0)

– For labour at time t. Price is wtpt. This means that the price of labour in terms oftime-t output is wt.

– For capital services at time t. Price is rKt pt. This means that the price of capital servicesin terms of time-t output is rKt

• What does it mean to hire capital services?

– I own one unit of capital (a machine)

– I let you use it for one period

– You return it to me (depreciated) at the end of the period

• Production is carried out by a representative firm.

• The representative firm is owned by the household

• The firm takes prices wt, rKt , pt as given and has access to the productive technology, but itdoesn’t own anything.

• It hires capital and labour and combines them to produce output seeking to maximize profits,which it then pays out to the household that owns it

• The household gets income from selling its labour, from renting the capital that it owns andfrom the firm’s profits. It chooses how much to consume each period and how much capitalto build.

• Firm’s problem:

π = max{yt,kt,Lt}∞t=0

∞∑t=0

ptyt − ptwtLt − ptrKt kt

s.t.

yt = f (kt, Lt)

(1)

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Econ 210, Fall 2013 Pablo Kurlat

• Household problem:

max{ct,it,kt+1}∞t=0

∞∑t=0

βtu (ct)

s.t.

kt+1 = (1− δ) kt + it∞∑t=0

pt (ct + it) ≤∞∑t=0

[ptwt + ptr

Kt kt]

+ π

k0 given

(2)

• Note that implicit in the way we formulate the household problem is that there exist marketsfor trading goods of different periods at prices pt.

– Since there is a single household, in equilibrium they will not be trading these claims,but the prices will have to be such that this is an optimal choice.

Definition 1. A competitive equilibrium is given by an allocation {yt, ct, kt, Lt}∞t=0 and prices{pt, wt, r

Kt

}such that:

1. The firm solves problem (1), taking prices as given

2. The household solve problem (2), taking prices as given

3. Markets clear:

(a) yt = ct + it

(b) Lt = 1

(c) kt chosen by the firm equals kt chosen by the household

3 Characterizing the Equilibrium

3.1 Firm’s problem

• Static problem: maximize profits period by period

• Simplifies to:maxk,L

f (k, L)− rKk − wL

• FOC:

fK (k, L)− rK = 0

fL (k, L)− w = 0

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Econ 210, Fall 2013 Pablo Kurlat

• Assumption: f (·) is homogenous degree 1 and differentiable.

Definition 2. f (k, L) is h.d.1 if f (λk, λL) = λf (k, L) for any λ > 0

• h.d.1 is just another way of saying constant returns to scale

Proposition 1. (Euler’s Theorem) If f (k, L) is h.d.1 and differentiable, then

fKk + fLL = f (k, L)

Proof. We know thatf (λk, λL) = λf (k, L)

Differentiate both sides w.r.t. λ:

fK (λk, λL) k + fL (λk, λL)L = f (k, L)

Evaluating at λ = 1 gives the result.

Proposition 2. In equilibrium, firms make zero profits

Proof. Profits are given by:

π =∞∑t=0

pt[f (kt, Lt)− rKt kt − wtLt

]=

∞∑t=0

pt [f (kt, Lt)− fK (kt, Lt) kt − fL (kt, Lt)Lt]

= 0

where the first step follows from the firm’s FOC and the second from Proposition 1.

3.2 Household Problem

• Household FOC:βtu′ (ct)− λpt = 0

• This leads to a difference equation:

βtu′ (ct)

βt+1u′ (ct+1)=

λptλpt+1

u′ (ct) = βptpt+1

u′ (ct+1)

u′ (ct) = βRt+1u′ (ct+1) (3)

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Econ 210, Fall 2013 Pablo Kurlat

whereRt+1 ≡

ptpt+1

is the gross interest rate between periods t and t+ 1. Why?

– Give up one unit at time t ⇒ obtain pt

– Use this to buy t+ 1 goods ⇒ obtain ptpt+1

• Equation (3) is an Euler equation - we’ve encountered it before

• No arbitrage requiresRt+1 = rKt+1 + 1− δ

– Two ways of transfering resources from one period to the next (or viceversa)

– Trading goods of differet dates in the market

– Converting goods into capital and renting it out

3.3 Equilibrium

• From household:u′ (ct) = β

[rKt+1 + 1− δ

]u′ (ct+1)

• From firm:fK (k, 1) = rK

• Combining:u′ (ct) = β [fK (kt+1, 1) + 1− δ]u′ (ct+1) (4)

• Plus the resource constraint

kt+1 = f (kt, 1)− ct + (1− δ) kt (5)

• (4) and (5) define a system of two difference equations.

• We need two initial / terminal conditions to have the solution

– k0 given is one of them

– What is the other?

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Econ 210, Fall 2013 Pablo Kurlat

3.4 The steady state and a phase diagram

• A steady state is defined as {css, kss} such that if ct = css and kt = kss, then according to(4) and (5), ct+1 = css and kt+1 = kss

• From (4):β [fK (kss, 1) + 1− δ] (6)

• From (5):

kss = f (kss, 1)− css + (1− δ) kss⇒ css = f (kss, 1)− δkss (7)

• Interpretation

• Phase diagram:

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Econ 210, Fall 2013 Pablo Kurlat

• Does the system converge to steady state?

4 The Social Planner’s Problem

• Single household, so no need to specify weights in a welfare function

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Econ 210, Fall 2013 Pablo Kurlat

• Planner solves

max{ct,it,kt+1}∞t=0

∞∑t=0

βtu (ct)

s.t.

kt+1 = (1− δ) kt + it

ct + ii ≤ f (kt)

k0 given

• This is a special case of a general problem of the form (SLP notation)

sup{xt}

∞∑t=0

βtF (xt, xt+1)

s.t.

xt+1 ∈ Γ (xt) ∀t

x0 given

• Mapping:

– xt → kt

– F (xt, xt+1)→ u (f (kt) + (1− δ) kt − kt+1)

– Γ (xt) = [0, f (kt) + (1− δ) kt]

• We’ll spend the next few classes looking at this mathematical problem

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