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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)
Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, 2010 2 / 23
Problem 1 (Utility maximization and budget constraints)
Contents
1 Problem 1 (Utility maximization and budget constraints)
2 Problem 2 (Consumption smoothing, canceled)
3 Problem 3 (Linearization)
Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, 2010 3 / 23
Problem 1 (Utility maximization and budget constraints)
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.
Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, 2010 4 / 23
Problem 1 (Utility maximization and budget constraints)
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)
Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, 2010 5 / 23
Problem 1 (Utility maximization and budget constraints)
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(·).
Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, 2010 6 / 23
Problem 1 (Utility maximization and budget constraints)
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)
Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, 2010 7 / 23
Problem 1 (Utility maximization and budget constraints)
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)
Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, 2010 8 / 23
Problem 1 (Utility maximization and budget constraints)
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)
Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, 2010 9 / 23
Problem 1 (Utility maximization and budget constraints)
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.
Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, 2010 10 / 23
Problem 2 (Consumption smoothing, canceled)
Contents
1 Problem 1 (Utility maximization and budget constraints)
2 Problem 2 (Consumption smoothing, canceled)
3 Problem 3 (Linearization)
Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, 2010 11 / 23
Problem 3 (Linearization)
Contents
1 Problem 1 (Utility maximization and budget constraints)
2 Problem 2 (Consumption smoothing, canceled)
3 Problem 3 (Linearization)
Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, 2010 12 / 23
Problem 3 (Linearization)
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.
Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, 2010 13 / 23
Problem 3 (Linearization)
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.
Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, 2010 14 / 23
Problem 3 (Linearization)
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.
Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, 2010 15 / 23
Problem 3 (Linearization)
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.
Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, 2010 16 / 23
Problem 3 (Linearization)
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.
Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, 2010 17 / 23
Problem 3 (Linearization)
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.
Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, 2010 18 / 23
Problem 3 (Linearization)
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.
Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, 2010 19 / 23
Problem 3 (Linearization)
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.
Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, 2010 20 / 23
Problem 3 (Linearization)
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.
Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, 2010 21 / 23
Problem 3 (Linearization)
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.
Markus Roth (Advanced Macroeconomics) Problem set 4 January 14, 2010 22 / 23