Macroeconomics I - GRIPSjulen/teaching/macro1_13/todai... · Macroeconomics I University of Tokyo OLG Economy with Production I: The Diamond Model: Markets and Equilibrium Julen Esteban-Pretel

Lecture 3

Macroeconomics IUniversity of Tokyo

OLG Economy with Production I:The Diamond Model:

Markets and EquilibriumJulen Esteban-Pretel

National Graduate Institute for Policy Studies

Lecture 3: OLG with Production- Environment and EquilibriumCore Macro I - Spring 2013

OLG Model with Production§ The endowment OLG model ignored production.

§ Do the pathologies of the OLG model only appear in pure exchange versions of the model?

§ We will see that the possibility of inefficient competitive equilibrium extends to the OLG model with production.

2


Physical Environment§ Two types of agents:

• Consumers.• Firms.

§ Same OLG structure as in the endowment model for:

• Time: Discrete. t = 1,2,..., ∞. Stationary environment.• Population: Consumers live for two periods, except for the initial old.

- N0 initial old, Nt agents born in period t.- Population grows at a rate Nt/Nt-1 = n.

• Utility: u( ) for initial old and U(ct) for the rest.§ New assumptions:

• Endowment is now in terms of labor - (ly,lo), and initial capital (K1).- Endowment profile: (ly,lo) = (1,0) for all generations (initial old have lo=0).- Initial old are endowed, in the aggregate, with K1 units of capital.

• No utility is derived from leisure: all labor endowment will be supplied.

3

c01


Physical Environment (cont.)

§ New assumptions (cont.):

• The good can be consumed or saved (invested and used as productive capital).• Consumers own the capital.

• Capital depreciates at a rate δ.• Firms operate in perfect competition.• Firms have access to a neoclassical production technology F(K,N).

- F(·,·) satisfies:– Twice continuous differentiable.– Strictly increasing in both arguments.– Strictly concave.– Satisfies Inada conditions.

- Assume that F(·,·) is CRS.

4


Detour about the CRS Production Function§ If F(·,·) is CRS (linearly homogenous, homogenous of degree 1)

then F(λK,λN)= λF(K,N) for all λ>0.§ It can be shown that:

(i) F(K,N) = N·F(K/N,1) ⇒ F(K,N)=Nf(k), where k≡K/N and f(k)≡F(k,1),

(ii) FK(K,N) = f ’(k), FN(K,N) = f(k) - k·f ’(k).

(iii) (Euler’s Theorem) F(K,N) = K·FK(K,N) + N·FN(K,N).

(iv) If F(K,N) is homogenous of degree 1, then FK and FN are homogeneous of degree 0.

FK(λK,λN) = FK(K,N), FN(λK,λN) = FN(K,N).

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Feasible and Efficient Allocations§ Def: A feasible allocation given K1 > 0 is a non-negative sequence of

consumption allocations and a non-negative sequence of capital stocks satisfying the resource constraint (RC):

or

for t = 1,2,... where

§ We could also have defined a feasible allocation given K1 > 0 as a non-negative sequence satisfying the RC.

§ As in the endowment economy, Def: An allocation is Pareto optimal it is feasible and there is no other

feasible allocation that Pareto dominates it.

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�c01, {ct}

�t=1

⇥

{Kt+1}�t=1

Ntctt + Nt�1c

t�1t + Kt+1 � (1� �)Kt ⇥ F (Kt,Nt) ,

ctt +ct�1tn

+ nkt+1 � (1� �)kt ⇥ f (kt) .

kt �Kt

Nt.

(3.1)

�c01, {ct}

�t=1 , {Kt+1}

�t=1

⇥


Market Structure§ Households/Consumers

• Buy goods, which they either consume or save.• Supply labor to firms.• Own the capital, which they rent to firms.

§ Firms

• Operate in perfect competition.• Rent capital from households.

- Rental rate: rt.- Price of capital: 1.

• Hire labor.- Wage rate: wt.

§ Three markets: goods, capital and labor.§ We could also consider consumption loans, but it would make no difference.

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Timing§ The timing of events for a given generation t is as follows.

§ In period t:• At the beginning of period t production takes place:

- Uses labor of generation t, capital saved by the now old generation t-1.

• At the end of period t:- Generation t gets paid wt.- Decides how much to consume in period t ( ) and how much to save ( ).

– Saving occur in the form of physical capital (only asset in the economy).

§ In period t+1:

• At the beginning of period t+1 production takes place:- Uses labor of generation t+1 and capital saved by the now old generation t.

• At the end of period t+1:- Generation t obtains the return on its savings: rt+1 - δ.

– (rt+1 - δ) is the real interest rate from period t to t+1.

- Consumes its savings and interests. i.e.

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ctt s

tt

ctt+1 = (1+ rt+1 � �)stt.


Decision Problems

§ Households/Consumers:• Initial old (generation 0):

• Generation t:

§ Firms:• Each firm solves the same problem:

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(3.2)

(3.4)

maxKdt ,N

dt�0F(Kdt ,N

d

t )� rtKdt � wtNdt (3.7)

s.t. c01 ⇥ (1+ r1 � �)

K1

N0.

maxc01�0u(c01)

maxctt,ct

t+1,stt�0U(ctt, c

tt+1)

s.t. ctt + s

tt � wt,

ctt+1 ⇥ (1+ rt+1 � �)stt.

(3.3)

(3.5)

(3.6)


Competitive Equilibrium with Production§ Def: A competitive equilibrium with production given K1 is a sequence of

factor prices and associated allocations for households

and allocations for firms such that:

(i) Optimization: Given ,

- solves initial old’s decision problem (3.2)-(3.3).

- solves generation t’s decision problem (3.4)-(3.6) for t = 1,2,...

- solves the firm’s problem (3.7) for t = 1,2,...

(ii) Markets clearing: For t = 1,2,...

- Goods market:

- Asset market:

- Labor market:

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{rt,wt}�t=1�c01, {ct, stt}

�t=1

⇥ �Kd

t ,Nd

t

⇥�t=1�

K1, {rt,wt}�t=1⇥

c01

{ct, stt}�t=1�

Kd

t ,Nd

t

⇥�t=1

Nt = Ndt .

(3.8)

(3.9)

(3.10)

Nt�1st�1t�1 = Kdt , where s

00 �Kd

1

N0

Ntctt + Nt�1c

t�1t + Kt+1 � (1� �)Kt = F

�Kd

t ,Nd

t

⇥,


Steady State§ Def: A steady state (SS) or stationary equilibrium is such

that the equilibrium sequences , and

are defined by

are an equilibrium, for a given k1 = k*.

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(k�, s�, c�y , c�o , r�,w�){rt,wt}�t=1

�c01, {ct, stt}

�t=1

⇥ �Kd

t ,Nd

t

⇥�t=1

ctt = c�y

stt = s

�

rt = r�

wt = w�

Kd

t = k� · NdtNt = Ndt

(3.21)

ct�1t = c⇥o


Computing the Equilibrium - Households§ Initial old:

• Given that u(·) is increasing, (3.3) holds with equality, and hence:

§ Generation t:

• Given that U(·, ·) is increasing in both arguments, (3.5) and (3.6) hold with equality, and we can substitute and in the utility function and solve the unconstrained problem:

• FOC:

• From (3.13) we can solve for the saving function:

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c01 = (1+ r1 � �)

K1

N0.

ctt c

tt+1

(3.11)

(3.12)

(1+ rt+1 � �) =U1 (wt � stt, (1+ rt+1 � �) stt)U2 (wt � stt, (1+ rt+1 � �) stt)

=U1(ctt, ctt+1)U2(ctt, ctt+1)

⇥ MRS(ctt, ctt+1). (3.13)

stt = s(wt, rt+1)

maxstt�0U (wt � stt, (1+ rt+1 � �) stt)


Computing the Equilibrium - Firms§ Firm’s problem restated:

§ FOC (they are the marginal product (MP) conditions):

where

§ Since F(·,·) is strictly concave, the FOC are not only necessary but also sufficient for profit maximization.

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maxKdt ,N

dt�0F(Kdt ,N

d

t )� rtKdt � wtNdt (3.7)

(3.14)

(3.15)

rt = FK(Kdt ,Ndt ) = f '

�Kdt

Ndt

⇥= f '(kt)

kt �Kd

t

Nd

t

wt = FN(Kdt ,Ndt ) = f

�Kdt

Ndt

⇥� K

dt

Ndtf '

�Kdt

Ndt

⇥= f(kt)� ktf '(kt)


Computing the Equilibrium - Firms (cont.)

§ Consequences of CRS:

(i) The MP conditions only pin down the capital-labor ratio .- Since FK(·,·) and FN(·,·) are homogeneous of degree 0, if (Kd,Nd) satisfies

the MP conditions, so does (λKd,λNd).

(ii) We can show that many competitive firms can be treated as one big firm.

- If K1/N1 = K2/N2, then F(K1 + K2, N1 + N2) = F(K1,N1) + F(K2,N2).

(iii) Equilibrium profits are equal to zero:

- That is why we did not need to worry about who owned the firms.

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Kd

t /Nd

t

(3.16)

(3.17)

= 0 (by Euler's Theorem)

F�Kd

t ,Nd

t

⇥� rtKdt + wtNdt

= F�Kd

t ,Nd

t

⇥� FK�Kd

t ,Nd

t

⇥Kd

t + FN�Kd

t ,Nd

t

⇥Nd

t


Computing the Equilibrium - Saving Function

§ Optimization by generation t implies

§ Factor markets clearing conditions (3.9) and (3.10) imply

§ Combining (3.18) and (3.19) with the MP conditions we obtain

- This is a first order diff. equation with an initial condition k1=K1/N1.

§ Equilibrium factor prices for t = 1,2,... can be found using the MP conditions.

§ We can prove that the goods market clearing condition is satisfied.

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(3.18)

(3.19)

stt = s(wt, rt+1), t = 1, 2, . . . .

(3.20)

(3.8)

stt = s(wt, rt+1) = nkt+1, t = 1, 2, . . . .

Ntctt + Nt�1c

t�1t + Kt+1 � (1� �)Kt ⇥ F

�Kd

t ,Nd

t

⇥,

kt+1 =s (f(kt)� ktf '(kt)), f '(kt+1))

n, t = 1, 2, . . .


Example§ Assume U(cy,co) = ln(cy) + βln(co), F(K,N) = AKαN(1-α), where β > 0, A > 0, and

0 < α < 1.

§ The young agent solves

which after solving delivers the following saving function

§ Given the Cobb-Douglass production function: f(k)=Akα and f ’(k)=Aαkα-1. Using the MP conditions, we obtain

§ Using (3.20), (3.23) and (3.24) we get

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stt = s(wt, rt+1) =

�

1+ �wt (3.23)

(3.22)

rt = A�k��1t and wt = A(1� �)k�t (3.24)

kt+1 =1

n

⇥

1+ ⇥A(1� �)k�t (3.25)

maxstt

[ln(wt � stt) + �ln((1+ rt+1 � ⇥)stt)],


Example (cont.)

§ (3.25) determines a unique sequence given k1 which converges to k*, the unique steady state capital-labor ratio.

• We can find k* from (3.25) by setting kt+1 = kt = k*

Figure 3.1

§ Given (3.26) we can solve for the factor prices using the MP conditions

§ Now we can solve for the steady state consumption allocations

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(3.27)

{kt}�t=1

k� =�⇥A(1� �)n(1+ ⇥)

⇥ 1

1��

. (3.26)

r� =�n(1+ ⇥)⇥(1� �) and w

� = A(1� �)�⇥A(1� �)n(1+ ⇥)

⇥ �1��

.

(3.28)cy =w�

1+ �and co =

�

1+ �w�(1+ r� � ⇥).


Figure 3.1

18

0

Figure 3.1: SS capital-labor ratio and convergence to the SS.

kt

kt+1

45°

k1 k*

kt+1 =1

n

⇥

1+ ⇥A(1� �)k�t

Documents

Macroeconomics I - GRIPSjulen/teaching/macro1_13/todai... · Macroeconomics I University of Tokyo OLG Economy with Production I: The Diamond Model: Markets and Equilibrium Julen Esteban-Pretel