problem set 4 - welcome to the chair in macroeconomics! · problem set 4 the decentralized economy...

Problem set 4The decentralized economy and log-linearization

Markus Roth

Chair for MacroeconomicsJohannes Gutenberg Universität Mainz

January 14, 2010

Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, 2010 1 / 23

Contents

1 Problem 1 (Utility maximization and budget constraints)

2 Problem 2 (Consumption smoothing, canceled)

3 Problem 3 (Linearization)


Problem 1 (Utility maximization and budget constraints)

Contents






The problem

• The problem is to maximize

Vt =∞

∑s=0

βsU(ct+s). (1)

subject to three different budget constraints.

• We know from our previous analysis that the three budgetconstraints are equivalent.

• Equivalent means that we can transform the period budgetconstraint into the lifetime budget constraint and vice versa.

• Hence, we expect that the solution (the consumption Eulerequation) should be identical.



Budget constraint 1

• The first budget constraint we consider is

at+1 = (1+ r)(at + xt − ct). (2)

• Current assets and income that are not consumed is invested.

• The Lagrangian is

L =∞

∑s=0

βsU(ct+s) + λt+s [(1+ r)(at+s + xt+s − ct+s)− at+s+1] .

• The first order conditions are

∂L∂ct+s

= βsU′(ct+s)− λt+s(1+ r)!= 0 (I)

∂L∂at+s+1

= λt+s+1(1+ r)− λt+s!= 0 (II)



Solution 1

• The solution is then

U′(ct+s) = (1+ r)βU′(ct+s+1), (3)

or equivalentlyU′(ct+s)

βU′(ct+s+1)= 1+ r,

where the marginal rate of substitution between consumption inperiod t and consumption t+ 1 equals the marginal rate oftransformation of consumption between both periods.

• This is the standard Euler equation for a general period utilityfunction U(·).



Budget constraint 2

• The second budget constraint we consider is

∆at + ct = xt + rat−1 or at + ct = xt + (1+ r)at−1 (4)

• where the dating convention is that at denotes the end of periodstock of assets and ct and xt are consumption and income duringperiod t.

• The Lagrangian is

L =∞

∑s=0

βsU(ct+s) + λt+s [xt+s + (1+ r)at+s−1 − ct+s − at+s] .

• The first order conditions are

∂L∂ct+s

= βsU′(ct+s)− λt+s!= 0 (I)

∂L∂at+s

= λt+s+1(1+ r)− λt+s!= 0 (II)



Solution 2

• The solution to this problem is again the Euler equation for theoptimal intertemporal consumption decision

U′(ct+s) = (1+ r)βU′(ct+s+1). (3)



Budget constraint 3

• The last budget constraint we consider is

∞

∑s=0

(

1

1+ r

)s

ct+s =∞

∑s=0

(

1

1+ r

)s

xt+s + (1+ r)at, (5)

• The Lagrangian to this problem is

L =∞

∑s=0

βsU(ct+s) + λ

{

∞

∑s=0

[(

1

1+ r

)s

(xt+s − ct+s)

]

+ (1+ r)at

}

.

• Note that in this case we have only one constraint.• Thus, there is only one Lagrangian multiplier λ.• The first order conditions are

∂L∂ct+s

= βsU′(ct+s)− λ

(

1

1+ r

)s!= 0 (I)

∂L∂ct+s+1

= βs+1U′(ct+s+1)− λ

(

1

1+ r

)s+1!= 0 (II)



Solution 3

• As already expected, this leads again to the same Euler equation

U′(ct+s) = (1+ r)βU′(ct+s+1),

which we usually write for period t

U′(ct) = (1+ r)βU′(ct+1).

• We have shown that expressing the household‘s problem in any ofthese three different alternative ways produces the same Eulerequation.


Problem 2 (Consumption smoothing, canceled)

Contents





Problem 3 (Linearization)

Contents






The model

• Consider a nonlinear difference equation of the form

xt+1 = f (xt). (6)

• The linear approximation of this equation around the steady statex = xt = xt+1 is given by

xt+1 = f (xt) ≃ f (x) + f ′(x)(xt − x)

or equivalently using x = f (x)

xt+1 = f (xt) ≃ x+ f ′(xt − x).

• This approximation is simply the tangent of this function at thesteady-state.



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

xt

x t+1

xt+1

Approx.

Figure: Approximation xt+1 =√xt.



The approximation error

• Of course, the approximation error depends on the specificfunction you look at.

• In general you will have a good approximation when xt is close toits steady-state value x.

• In our economic models we usually assume that his is the case.

• Note that the approximation technique we apply here is based ona first order Taylor series expansion.

• In general it is also possible to compute higher orderapproximations but we focus on the simplest case here.



Rewrite the function

• In order to log-linearize the difference equation we have torewrite it in terms of log deviations around the steady-state.

• Therefore we define a new variable

xt ≡ ln(xtx

)

.

• Note that we can express every variable as

xt = xext .

• In the steady-state this newly defined variable has to be zero

x = ln(x

x

)

= ln(1) = 0.



Rewriting the equation

• Using the new variable we can write the function as

xext+1 = f(

xext)

.

• Now we can linearize this expression as we did above.

• However, we now use xt instead of xt and linearize the left handside (LHS) and the right hand side (RHS) of the equationseparately

LHS ≃ xex + xex(xt+1 − x)

= x+ xxt+1

RHS ≃ f (xex) + f ′(xex)xex(xt − x)

= f (x) + f ′(x)xxt.



Equating both sides

• Equating the left hand side approximation and the right hand sideapproximation yields

x+ xxt+1 = f (x) + f ′(x)xxt

xxt+1 = f ′(x)xt

xt+1 = f ′(x)xt.

• Next, consider a Cobb-Douglas production function

Yt = F(Kt, Lt) = Kαt L

1−αt . (7)

• We write it asYeyt = KαLαeαkte(1−α)lt.



The production function

• The left hand side is approximated by

LHS ≃ Y+ Yyt.

• The right hand side is approximated by

RHS ≃ KαL1−α + KαL1−α[αkt + (1− α)lt] (8)

• Note that some intermediate steps are omitted.

• Equating both sides yields

Y+ Yyt = KαL1−α + KαL1−α[αkt + (1− α)lt]

Yyt = KαL1−α[αkt + (1− α)lt]

yt = αkt + (1− α)lt.



The production function (general)

• In order to interpret the coefficients consider a general productionfunction

Yt = F(Kt, Lt).

• It is rewritten toYeyt = F

(

Kekt , Lelt)

(9)

• The left hand side is approximated by

LHS ≃ Y+ Yyt.

• The right hand side is approximated by

RHS ≃ F(K, L) + FK(K, L)Kkt + FL(K, L)Llt.



The production function (general) 2

• Equating both approximations yields

Y+ Yyt = F(K, L) + FK(K, L)Kkt + FL(K, L)Llt

Yyt = FK(K, L)Kkt + FL(K, L)Llt

yt =FKK

Ykt +

FLAL

Ylt.

• From this representation it becomes clear that the coefficient oncapital equals the elasticity of output with respect to capital.

• Maybe you are more familiar with

FKK

Y=

∂Y

∂K

K

Y= εYK.



Log-linearization and elasticities

• In general, when you log-linearize a function say

y = f (x, z).

• The log-linearized function is given by

yt = εyxxt + εyzzt.

where εyx and εyz are the respective elasticities.


References

References

Wickens, M. (2008).Macroeconomic Theory: A Dynamic General Equilibrium Approach.Princeton University Press.


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