Download - 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)
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 2 / 95
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.
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 2 / 95
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.
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 3 / 95
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.
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 4 / 95
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)
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 5 / 95
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)
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 6 / 95
Consumption decisions
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)
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 7 / 95
Consumption decisions
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.
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 9 / 95
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.
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 10 / 95
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
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 11 / 95
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 .
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 12 / 95
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.
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 13 / 95
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)
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 14 / 95
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
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 15 / 95
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)
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 16 / 95
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.
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 17 / 95
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
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 18 / 95
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..
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 19 / 95
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
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 20 / 95
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.
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 21 / 95
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.
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 22 / 95
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!
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 23 / 95
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.
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 24 / 95
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
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 25 / 95
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.
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 26 / 95
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
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 27 / 95
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%).
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 28 / 95
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
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 29 / 95
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
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 30 / 95
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
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 31 / 95
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.
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 32 / 95
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.
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 33 / 95
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
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 34 / 95
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
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 35 / 95
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.
Dynamic Macroeconomic Analysis (UAM) Equilibrium in Two Periods Fall 2012 36 / 95