equilibrium in two periods

Equilibrium in Two Periods

Dynamic Macroeconomic Analysis

Universidad Autonoma de Madrid

Fall 2012

Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 1 / 95

1 Outline

2 Households

3 Firms

4 EquilibrioDefinition of EquilibriumAggregate demand and supplyEquilibrium in the labor market

5 Numerical exampleEquilibrium in the credit marketConsumptionLabor supplyAggregate demand and supply

6 FluctuacionesUn Aumento Permanente en la ProductividadAumento Transitorio en la Productividad (aumenta A1)Cambios Anticipados en la Funcion de Produccion (aumenta A2)


Roadmap

So far, we have studied consumption and savings decisions and laborsupply decisions in a two-period setting. However, in the case of laborsupply, we analyzed a partial equilibrium setting with exogenous wages

In this theme we derive the general equilibrium allocations in twoperiods with endogenous labor supply, savings and consumption.

Second, we will analyze the response of the equilibrium to transitoryand permanent shocks to TFP.

Our model is still incomplete (we ignore the role of capital and theeconomy only lasts for two periods), but it neatly illustrates essentialfeatures of RBC theory.


The environment

The economy lasts for two periods.

There are two groups of agents with different discount rates. Allagents value consumption and leisure and need to decide labor supplyin both periods. The labor market is competitive.

At the other side of the labor market there is a representative firmthat hires labor services taking the competitive wage in each periodas given.

At the end of the first period the agents can access the credit marketto borrow or lend resources.


The Household Problem

The households choose (l1, l2, c1, c2) taking prices (w1, w2, R) asgiven. Formally,

maxc1,c2,l1,l2

{ln c1 + ln(1− l1) + β (ln c2 + ln(1− l2))

+ λ[w1l1 +1

1 + Rw2l2 − c1 −

1

1 + Rc2]},

where λ is the Lagrange multiplier.

Notice that we have solved this problem in the previous lecture.


The Household ProblemThe first-order conditions

1

c1− λ = 0 (c1)

β1

c2− λ

1

1 + R= 0 (c2)

− 1

1− l1+ λw1 = 0 (l1)

−β1

1− l2+ λ

w2

1 + R= 0. (l2)


Consumption decisions

In a first step we derive the consumption Euler condition:

c2 = β(1 + R)c1,

Inserting this condition in the budget constraint we obtain:

(1 + β)c1 = x =⇒ c1 =x

1 + β, (1)

c2 = c1β(1 + R) =βx(1 + R)

1 + β. (2)



In the next step we combine the FOCs for c1 and l1:

1

1− l1=

1

c1w1 =⇒ w1l1 = w1 − c1. (3)

Similarly, inserting λ = u′(c1) into the FOC for l2

−β1

1− l2+ λ

w2

1 + R= 0.

we obtain the following condition:

β1

1− l2=

1

c1

w2

1 + R.

Hence,

c1β(1 + R) = w2 − w2l2 =⇒ w2l21 + R

=w2

1 + R− c1β. (4)



The value of the agent’s lifetime resources can now be written as

x = w1l1 +w2l2

1 + R= w1 − c1 +

w2

1 + R− c1β.

Given that c1 = x1+β , we can thus write

c1 =w1 − c1 + w2

1+R − c1β

1 + β,

leading to the following solutions

c∗1 =1

2(1 + β)

[w1 +

w2

(1 + R)

].

c∗2 = β(1 + R)1

2(1 + β)

[w1 +

w2

(1 + R)

]Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 8 / 95

Labor supply decisions

Given that we have demonstrated that w1l1 = w1 − c1, we have

w1l∗1 = w1 − c∗1

= w1 −1

2(1 + β)

[w1 +

w2

(1 + R)

]

w1l1 = w1

[1− 1

2(1 + β)

]− 1

2(1 + β)w2

(1 + R)

l∗1 =1 + 2β

2(1 + β)− 1

2(1 + β)1

1 + R

w2

w1.


Labor supply decisions

Similarly, our previous results imply that w2l∗2 = w2 − c∗1 β(1 + R),

w2l2 = w2 − c1β(1 + R)

= w2 −β(1 + R)2(1 + β)

[w1 +

w2

(1 + R)

]= w2[1−

β(1 + R)2(1 + β)

1

1 + R]− β(1 + R)

2(1 + β)w1.

l∗2 =2 + β

2(1 + β)− β(1 + R)

2(1 + β)w1

w2.


Summary household decisions

In equilibrium, the household decisions concerning (c1, c2, l1, l2) satisfy

c∗1 =1

2(1 + β)

[w1 +

w2

(1 + R)

]

c∗2 = β(1 + R)1

2(1 + β)

[w1 +

w2

(1 + R)

]

l∗1 =1 + 2β

2(1 + β)− 1

2(1 + β)1

1 + R

w2

w1

l∗2 =2 + β

2(1 + β)− β(1 + R)

2(1 + β)w1

w2


The representative firm

Like before we assume that there is a representative firm that acts asa price taker in the labor and the product market.

For simplicity, we assume that production function is linear in labor:

Y fj = AjL

fj ,

where Aj denotes the value of TFP in period j , Y fj is output in period

j and Lfj is the firm’s labor demand in j .


The problem of the representative firm

In each period the representative firm solves the following staticproblem:

πj = maxlf

[AjLfj − wjL

fj ],

where wj denotes the competitive wage in period j .

The FOC associated with the firm’s problem implies that

Aj = wj .

In other words, in each period wages are uniquely determined by thevalue of TFP, i.e. w1 = A1 y w2 = A2.


EquilibriumConsumption

To generate a credit market, we will once again assume that there arevarious groups of agents with different discount rates.

Let’s define βi as the discount factor of agent i ,where i = 1, ...., N.

Accordingly,

c∗1,i =1

2(1 + βi )

[w1 +

w2

(1 + R)

]. (5)

c∗2,i = βi (1 + R)1

2(1 + βi )

[w1 +

w2

(1 + R)

](6)


EquilibriumNet-borrowing or lending

Recall thatb1,i = w1l1,i − c1,i

Furthermore, we know that w1l1,i = w1,i − c1,i . Hence,

b∗1,i = w1l1,i − c (7)

= w1 − 2c1,i

= w1 − 21

2(1 + βi )

[w1 +

w2

(1 + R)

]= w1 −

1

1 + βi

[w1 +

w2

(1 + R)

]=

w1βi − w21+R

1 + βi


Labor supply

Finally, recall that labor supply in both periods satisfies:

l∗1,i =1 + 2βi

2(1 + βi )− 1

2(1 + βi )1

1 + R

w2

w1. (8)

y en el segundo perıodo

l∗2,i =2 + βi

2(1 + βi )− βi (1 + R)

2(1 + βi )w1

w2. (9)


Definition of equilibrium

Following standard conventions, the competitive equilibrium in thistwo-period economy can be defined as follows

Given A1, A2 and βi for i = 1, ..., N, the equilibrium of this economyconsists of a set of prices w1, w2, and R and allocations c∗1,i ,

c∗2,i , b∗1,i , l

∗1,i , and l∗2,i for the households and Lf

1 and Lf2 for the firm.

Given w1, w2, and R, the allocations of the household solve thehousehold problem, i.e. c∗1,i , c∗2,i , b

∗1,i , l

∗1,i , and l∗2,i estan dadas por las

ecuaciones (5), (6), (7), (8) y (9).

The firms maximize profits given A1 and A2, i.e.

w1 = A1 and w2 = A2.

All markets clear.


Market clearing conditions

Credit market:N

∑i

b∗1,i = 0,

Goods market:

N

∑i

c∗1,i = A1Lf1 and

N

∑i

c∗2,i = A2Lf2

Labor market:N

∑i

l∗1,i = Lf1 and

N

∑i

l∗2,i = Lf2


Walras’ Law

Remember that the final good is perishable. Hence, in every period weexpect that the agents consume the entire value of aggregate output.

Notice also that the equilibrium in the labor market is satisfiedtrivially: the firm is willing to hire any amount of labor at theequilibrium wages w ∗j = Aj .

Given the equilibrium in the labor market, we can show that theequilibrium in the goods market implies equilibrium in the creditmarket, and vice versa..


Proof

For each agent i the first-period budget constraint can be written as

b1,i = w1l1,i − c1.

Summing over all agents we get

N

∑i

b1,i =N

∑i

w1l1,i −N

∑i

c1,i = w1

N

∑i

l1,i −N

∑i

c1,i .

Finally, if the labor market and the goods market are in equilibrium,then aggregate savings satisfies

N

∑i

b∗1,i = A1

N

∑i

l∗1,i −N

∑i

c∗1,i

= A1Lf1 −

N

∑i

c∗1,i = 0,

Hence, the credit market is also in equilibrium


Aggregate demand

Let’s start with period 1.

We have shown that

c∗1,i =1

2(1 + βi )

[A1 +

A2

(1 + R)

].

Hence, aggregate demand for the final good is equal to

Cd1 =

N

∑i

c∗1,i =[A1 +

A2

(1 + R)

] N

∑i

1

2(1 + βi ). (10)

According to the above equation, aggregate demand in period 1 is adecreasing function of the interest rate.


Aggregate supply

Now let’s derive the solution for aggregate supply:

Y s1 = A1L

f1 = A1

N

∑i

l∗1,i = A1l∗1,i (11)

= A1

N

∑i

[1 + 2βi

2(1 + βi )− 1

2(1 + βi )1

1 + R

A2

A1

]In this case we find a positive relationship between the level ofaggregate supply and the interest rate. The reason is that agentschoose to supply more labor in period 1 if the interest rate rises forsome reason.


Equilibrium in the product marketThe intersection of the aggregate demand and supply schedules uniquelydetermines the equilibrium interest rate. Notice that 1 + R reflects therelative price of future consumption in terms of current consumption!


Labor supply

Remember that individual’s labor supply in period 1 is given by:

l∗1,i =1 + 2βi

2(1 + βi )− 1

2(1 + βi )1

1 + R

w2

w1.

Summing over all agents we obtain the expression for aggregate laborsupply

Ls1 =

N

∑i

l∗1,i =N

∑i

[1 + 2βi

2(1 + βi )− 1

2(1 + βi )1

1 + R

w2

w1

]Not surprisingly aggregate labor supply in period 1 is an increasingfunction of the wage rate w1.


Equilibrium in the labor market

Given that the maximization problem of the firm uniquely determines thewage rate (w1 = A1,), labor demand is perfectly elastic.

w1 Ls

A1

Ls


A numerical example

Consider an economy with 100 agents, so that N = 100.

Suppose that 20 of these agents have a discount factor β1 = 0.8while the rest of the agents have a discount factor equal to β2 = 0.9.Notice that the type-2 agents are more patient than the type-1agents.

Suppose for simplicity that A1 = A2 = 1.

Hence in equilibrium w1 = 1 and w2 = 1 and so w2/w1 = 1.


Equilibrium in the credit market

In any equilibrium aggregate saving is equal to zero and so:

N1

β1A1 − A21+R

1 + β1+ N2

β2A1 − A21+R

1 + β2= 0,

which implies that

200.8− 1

1+R

1 + 0.8+ 80

0.9− 11+R

1 + 0.9= 0,

and so

200.8− 1

1+R

1.8= 80

11+R − 0.9

1.9


Equilibrium in the credit market

200.8− 1

1+R

1.8= 80

11+R − 0.9

1.9

From the above equation it follows that

(0.8)20(1.9)− 1.9(20)1 + R

=80(1.8)1 + R

− (80)0.9(1.8),

or equivalently

30.4− 38

1 + R=

144

1 + R− 129.6.

Collecting terms we get:

30.4 + 129.6 =144

1 + R+

38

1 + R

160 =182

1 + R=⇒ 1 + R = 1.1375 =⇒ R = 0.1375, (13.75%).


Consumption

Given our solutions for w1, ww and R, we can now derive the valuesof consumption:

c∗1,1 =1

2(1 + β1)

[w1 +

w2

(1 + R)

]=

1

2(1 + 0.8)

(1 +

1

1.1375

)= 0.522.

Also,

c∗2,1 = β1(1 + R)1

2(1 + β1)

[w1 +

w2

(1 + R)

]= 0.8(1.1375)

1

2(1 + 0.8)

(1 +

1

1.1375

)= 0.475


Consumption

For the type-2 agents, consumption in period 1 is given by

c∗1,2 =1

2(1 + β2)

[w1 +

w2

(1 + R)

]=

1

2(1 + 0.9)

(1 +

1

1.1375

)= 0.494.

Similarly, in period 2 these agents consume a quantity:

c∗2,2 = β2(1 + R)1

2(1 + β2)

[w1 +

w2

(1 + R)

]= 0.9(1.1375)

1

2(1 + 0.9)

(1 +

1

1.1375

)= 0.506


Consumption

Note that the relatively impatient agents of type 1 consume more inthe first period than the patient agents of type 2.

Summarizing, aggregate demand in the first period is equal to:

Cd1 = 20c∗1,1 + 80c∗1,2

= 20(0.52198) + 80(0.49451)= 50


Labor supply

Again let’s start with the agents of type 1:

l∗1,1 =1 + 2β1

2(1 + β1)− 1

2(1 + β1)1

1 + R

A2

A1

=1 + 2(0.8)2(1 + 0.8)

− 1

2(1 + 0.8)1

1.1375= 0.47802

and

l∗2,1 =2 + β1

2(1 + β1)− β1(1 + R)

2(1 + β1)A1

A2

=2 + 0.8

2(1 + 0.8)− 0.8(1.1375)

2(1 + 0.8)= 0.525.


Labor supply

Similarly, for the agents of type 2 we find the following values:

l∗1,2 =1 + 2β2

2(1 + β2)− 1

2(1 + β2)1

1 + R

A2

A1

=1 + 2(0.9)2(1 + 0.9)

− 1

2(1 + 0.9)1

1.1375

= 0.50549

and

l∗2,1 =2 + β2

2(1 + β2)− β2(1 + R)

2(1 + β1)A1

A2

=2 + 0.9

2(1 + 0.9)− 0.9(1.1375)

2(1 + 0.9)= 0.4935.


The labor market

Hence, labor market equilibrium in both periods requires that:

Lf1 = Ls

1

=(20l∗1,1 + 80l∗1.2

)= 20(0.47802) + 80(0.50549)= 50

and

Lf2 = Ls

2

=(20l∗2,1 + 80l∗2.2

)= 20(0.525) + 80(0.4935)= 50


Aggregate demand and supply

en el segundo perıodo. Al agente le importa menos la desutilidad detrabajar que sufre en el futuro.The aggregate supply of goods is givenby

Y s1 = A1

(20l∗1,1 + 80l∗1.2

)= 20(0.47802) + 80(0.50549)= 50.

which is equal to aggregate consumption (demand)

Cd1 = 20c∗1,1 + 80c∗1,2

= 20(0.52198) + 80(0.49451)= 50


Aggregate demand and supply

Similarly, in the second period aggregate demand and supply aregiven by

Cd2 = 20c∗2,1 + 80c∗2,2

= 20(0.475) + 80(0.50625)= 50

and

Y s2 = A2

(20l∗2,1 + 80l∗2.2

)= 20(0.525) + 80(0.4935)= 50.

Hence, the goods market is in equilibrium in both periods.


equilibrium in two periods

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