Advanced marcoeconomics test
Will.Luke
AdvancedUMacroSlides1TheStochasticGrowthModel.pdf
ECON 401
Advanced Macroeconomics
Topic 1
The Stochastic Growth Model
Fabio Ghironi
University of Washington
Introduction
• This course explores modern theories of macroeconomic fluctuations.
• The goal will be to take you as close as possible to understanding how many macroecono- mists at academic and policy institutions think about business cycles and policy questions,
including the crisis created by COVID-19, by studying a sequence of models.
• We will start from the stochastic growth model (also known as real business cycle—RBC— model), in which fluctuations are the result of random shocks to technology and economic
outcomes are efficient.
1
Introduction, Continued
• In this course, the word efficiency has a very precise meaning:
– The market economy is efficient when the outcome it generates is the same as the
outcome that would be chosen by a benevolent social planner in charge of allocating
resources.
• The market economy is efficient in the RBC model: A benevolent planner who acts to maximize social welfare would not do better than the market.
• This implies that this is a model in which there is no role for policy to improve on market outcomes.
2
Introduction, Continued
• We will study the RBC model not because we believe that it is an accurate, realistic theory of how the macroeconomy works, but because it is a useful starting point to become familiar
with concepts, tools, and techniques that we will use many times throughout the quarter.
• We will then introduce a number of more realistic features into our framework: monopoly power, nominal rigidity, financial market frictions, labor market imperfections, producer entry
dynamics, heterogeneity across agents, and more.
• These features will imply that the economy we model is no longer efficient: Policy can improve outcomes relative to the market.
• We will conclude the course with an example of how the tools we study can be used to analyze the COVID-19 crisis.
3
Introduction, Continued
• The tools used in the RBC model became the foundation of the mainstream framework for studying macroeconomic fluctuations in the 1980s, starting with seminal work by Finn
Kydland of U.C. Santa Barbara and Edward Prescott of Arizona State University published
in Econometrica in 1982.
• The model studies fluctuations of the economy around its growth trend (business cycles) triggered by unexpected, random shocks, assuming that agents in the economy act to
optimize intertemporal objective functions under rational expectations about the future.
• In its standard versions, the analysis assumes that shocks generate departures from trend that disappear over time: For instance, an unexpected improvement in technology causes
the economy’s GDP to rise above trend for some time, but eventually the effect of the shock
disappears, and the economy is back chugging along its trend growth path.
• The figure in the next slide shows the behavior of U.S. GDP since 1947. It gives you an illustration of situations in which the standard approach can work (much of the time) but also
situations in which it will do very poorly (the aftermath of the Great Recession that followed
the Global Financial Crisis of 2007-08).
– U.S. GDP appears to have returned to its post-Great Recession trend in the aftermath of the COVID-19 recession, but the standard solution technique for many models that we will learn (log-linearization) may have problems because of the size of this shock.
4
B ill io n s o f C h ai n ed 2 01 2 D o lla rs
1950 1960 1970 1980 1990 2000 2010 2020 2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
Real Gross Domestic ProductReal Gross Domestic Product
Shaded areas indicate U.S. recessions. Source: U.S. Bureau of Economic Analysis fred.stlouisfed.org
Introduction, Continued
• As we were saying, market outcomes are efficient in the basic RBC model.
• This happens because there are no distortions (or frictions) that would cause a benevolent social planner to choose outcomes that differ from the market ones.
• But the usefulness of the model does not depend the assumption that business cycles are triggered by technology shocks or that of an efficient model-economy.
• Many scholars have studied the consequences of other types of shocks and of departures from an efficient environment by modifying the model and the approach to it in the directions
we will talk about.
5
Introduction, Continued
• We will introduce distortions, obstacles to the smooth functioning of markets, realistic features that will allow us to tackle questions that the basic RBC setup cannot address—
including issues that have taken center stage in discussions on macroeconomics since the
Great Recession.
• Studying the RBC model will prepare us to study those more realistic models, and it will help us understand exactly when and why market outcomes in those models are not efficient, and
therefore when and why there is a role for policy in addressing those suboptimal outcomes.
– Put differently, understanding the outcomes in an efficient model-economy helps us
understand the mechanisms through which distortions lead to inefficient outcomes when
we introduce realistic features in our models, and when and how policy action can be
optimal.
6
Solving Models
• In studying the RBC model, we will pay special attention to the procedure for solving it.
• The difficulty in solving the model is a fundamental non-linearity that arises from the interaction between multiplicative elements, such as Cobb-Douglas production, and additive
elements, such as capital accumulation and depreciation.
• This non-linearity makes it impossible, in general, to solve the model without resorting to some kind of approximation.
– The only case in which this problem does not arise is when capital depreciates fully in
one period and agents utility from consumption takes the logarithmic form.
– This is a very special, unrealistic combination of assumptions.
7
Solving Models, Continued
• The solution method that we will study for the more general scenario is called log- linearization.
• It starts from the model’s optimality conditions and budget constraints and transforms them into a system of log-linear expectational difference equations in which all endogenous
variables are function of the capital stock and of the exogenous shocks that cause
fluctuations.
• Variables in the log-linearized model measure percentage deviations of original variables from their trend (or steady-state) levels.
– We will use the words trend and steady state interchangeably, with the understanding
that underlying variables are constant in steady state only if long-run growth is zero,
otherwise they are moving at their trend-growth rate.
• The approximated model can then be solved using a method known as the method of undetermined coefficients.
• An advantage of this solution method over alternatives is that it can be applied also to models in which the market outcome is not efficient.
8
Solving Models, Continued
• There are plenty of situations in which you would not want to log-linearize your model (and therefore assume that your variables always display a tendency to return to the steady state
around which you approximated the original, non-linear model).
– There are cases in which log-linearization is used also in non-stationary environments
(scenarios in which the economy does not eventually return to its steady state, or trend,
after shocks), but this is appropriate only for specific types of exercises.
• Log-linearization limits the range of questions you can study, or it can yield very misleading conclusions.
• For example, log-linearization cannot handle phenomena like the Great Recession and the years that followed and situations in which accounting for nonlinearity (like the zero—or
effective—lower bound on central bank policy interest rates) is necessary to understand
what is happening.
• But log-linearization is still used to work on many other interesting, important questions and understanding how it works also helps us understand its limitations and why alternative,
more complicated techniques become necessary in other scenarios.
9
Households in the Basic RBC Model
• Consider an economy populated by a large number of identical, infinitely lived households, all subject to the same uncertainty.
• At time t, the representative household maximizes its expected intertemporal utility from t to the infinite future, discounting utility in future periods according to a discount factor β:
Et
[ ∞∑ s=t
βs−tu(Cs)
] = Et
( ∞∑ s=t
βs−t C1−γs 1 −γ
) , 0 < β < 1, (1)
where Et denotes the expectation based on the information available at time t, ∑∞
s=t is the
summation operator for time that goes from the current period (t) all the way to infinite, and
Cs is consumption in period s (s = t, ...,∞).
• We assume that this expectation is rational, i.e., the household uses optimally all the information that is available to it.
– Much macroeconomic literature studies the consequences of departures from rationality.
It is one of the many topics that, unfortunately, we do not have time to study. Michael
Woodford of Columbia University has been doing very interesting work in this area
recently.
10
The Intertemporal Utility Function
• The expression
Et
( ∞∑ s=t
βs−t C1−γs 1 −γ
) is a compact way of writing
Et
( C 1−γ t
1 −γ + β
C 1−γ t+1
1 −γ + β2
C 1−γ t+2
1 −γ + ... + β∞
C 1−γ t+∞
1 −γ
) .
• Discounting by β captures the idea that households care about utility from current consumption more than they care about utility from future consumption.
– Very interesting literature has explored the consequences of different forms of discount-
ing. See, for instance, work by David Laibson of Harvard University.
11
The Intertemporal Utility Function, Continued
• γ > 0 is the coefficient of relative risk aversion:
– It measures the attitude of our representative household toward risk (uncertainty).
– If γ were equal to zero—linear utility—the household would not care about risk. It would
be perfectly indifferent between a certain level of consumption and an uncertain one.
– If you are not familiar with the concept of risk aversion, you find more information in
Appendix A.
12
The Intertemporal Utility Function, Continued
• Let us define the parameter σ ≡ 1 γ .
• This is known as the elasticity of intertemporal substitution.
– As we shall see, it measures the responsiveness of consumption to interest rate
changes: the willingness of agents to postpone or anticipate consumption across periods
in response to movement in interest rates.
• Tight connection between attitude toward uncertainty (γ) and toward intertemporal substitution (σ) is an undesirable feature of the model when studying important questions
(for instance, related to asset pricing).
• Larry Epstein of Boston University and Stanley Zin of NYU developed a framework that unties risk aversion from intertemporal substitution. Their work, published in the Journal of
Political Economy in 1989, became widely used to address important questions.
• We will stick to the basic framework, keeping in mind that it has significant limitations (see Appendix A for an example).
13
Capital Accumulation and Labor Supply
• Households can accumulate a single asset, homogeneous physical capital, Kt and
Kt+1 = (1 − δ)Kt + It, 0 < δ ≤ 1 (2)
where Kt is the capital stock with which the household begins period t, It is investment in
period t, δ is the rate of capital depreciation, and Kt+1 is the stock of capital with which the
household will begin period t + 1.
• Each household supplies a fixed amount of labor (Nt = 1) in each period in a perfectly competitive labor market.
14
The Budget Constraint
• The representative household’s consumption is constrained by:
Ct + It = r̃tKt + wt, (3)
where r̃t is the rental rate the household receives from the firms that rent its capital in a
perfectly competitive rental market, and wt is the wage (Nt = 1 implies wtNt = wt = labor
income).
– This is the household’s period budget constraint: A constraint like this applies to every
period (to every s, for s = 0, ...,∞).
• (2) and (3) imply that the household’s budget constraint can be rewritten as:
Ct + Kt+1 = (1 + r̃t − δ)Kt + wt. (4)
• The problem of the household is to maximize (1) subject to (4).
• How do we solve such intertemporal optimization problem?
15
Solving the Household’s Problem: Intuitive Approach
• Let us start with in intuitive approach.
• Suppose we have a household that must decide what to do with 1 dollar in the current period (t): it can use it to buy consumption in t (let us assume that 1 unit of consumption costs 1
dollar) or it can invest it in an asset that will generate the uncertain gross return Rt+1 at time
t + 1.
• Consider what the two possible choices do for the household:
• If it uses the dollar to buy consumption, it obtains the benefit given by the increment in utility from consuming an extra unit of consumption today—the marginal utility of consumption:
u′(Ct).
• If it invests the dollar, it will receive the return Rt+1 at time t + 1. In terms of the utility increment generated by the extra consumption this allows the household to do at t + 1, this
translates into u′(Ct+1)Rt+1.
16
Notation Digression
• When we are dealing with functions of only one variable, we will denote the first derivative by using a superscript “′” and the second derivative by using a superscript “′′.”
• When we are dealing with functions of more than one variable, we will denote the first partial derivative with respect to a variable by having that variable indicated once as subscript and
the second derivative by having the variable indicated twice as subscript.
– Example: The first derivatives of the function f(x,y) with respect to x and to y are
denoted fx(x,y) and fy(x,y), respectively, and the second derivatives are fxx(x,y) and
fyy(x,y) (I am omitting cross derivatives, assuming things are clear).
• An alternative way of indicating partial first derivatives that may appear in these course slides will be to use numerical subscripts referring to the variable with respect to which we
are taking the derivative.
– Example: The first derivatives of the function f(x,y) with respect to x and to y are
denoted f1(x,y) and f2(x,y), respectively, and the second derivatives are f11(x,y) and
f22(x,y) (again omitting cross derivatives).
• Do not use “′” or “′′” superscripts when denoting derivatives of functions of more than one variable in this course.
• Why? Because f ′(x,y) does not tell anyone with respect to what variable the derivative is taken!
17
Solving the Household’s Problem: Intuitive Approach, Continued
• But the household does not know Ct+1and Rt+1 with certainty at time t, when it is taking its decision. Hence, it will compute its expectation of Rt+1u
′(Ct+1) based on the information it
has at time t: Et [u ′(Ct+1)Rt+1].
• Moreover, in comparing the benefit of consuming today to that of investing (and thus postponing consumption to the next period), the household will discount the future benefit
with the discount factor β.
• Hence, the household will compare u′(Ct) and βEt [u′(Ct+1)Rt+1].
• When is the household happy with the allocation of its resources across periods?
18
Solving the Household’s Problem: Intuitive Approach, Continued
• When it is indifferent between the two alternatives!
• In other words, for the household’s behavior to be optimized, it must be the case that:
u′(Ct) = βEt [u ′(Ct+1)Rt+1] .
• This optimality condition is known as Euler equation.
• In our model, the asset the household can invest in is capital, and the return that an extra unit of capital today generates at t + 1 is 1 + r̃t+1 − δ: the undepreciated portion of that unit of capital plus the rental rate that it generates.
• Hence, the Euler equation for optimal capital accumulation in our model is:
u′(Ct) = βEt [u ′(Ct+1)(1 + r̃t+1 − δ)] ,
or, given the assumed form of the period utility function,
C −γ t = βEt
[ C −γ t+1(1 + r̃t+1 − δ)
] . (5)
19
Solving the Household’s Problem: Doing the Math
• Now let us show how we can obtain this equation by doing math.
• The budget constraint (4) can be rearranged as:
Ct = −Kt+1 + (1 + r̃t − δ)Kt + wt. (6)
• Recall that the household faces a constraint like (4) in every period—put differently, it faces a sequence of constraints like (4) for time that goes from t to ∞.
• In the generic period s, it has to be:
Cs = −Ks+1 + (1 + r̃s − δ)Ks + ws. (7)
• We can substitute this constraint for Cs in the objective of the household, which will therefore be maximizing:
Et
{ ∞∑ s=t
βs−t [−Ks+1 + (1 + r̃s − δ)Ks + ws]1−γ
1 −γ
} . (8)
20
Solving the Household’s Problem: Doing the Math
• What does the household choose?
• The household takes the rental rate and the wage as given—as we mentioned above, they are determined in perfectly competitive markets in which all agents are price takers.
• Moreover, at any time s, Ḱs is predetermined : It is the capital stock with which the household begins the period. It was determined in the previous period.
• Having substituted investment and consumption out of the problem through our manipulation of constraints and substitutions (the substitution of (2) into (3), and the substitution of (7)
into (1)) leaves Ks+1 as the only variable that the household actually chooses at any time s.
21
Solving the Household’s Problem: Doing the Math
• Without loss of generality, focus on s = t. The first-order condition for the household’s optimal behavior follows from setting the derivative of (8) with respect to Kt+1: equal to 0.
• To find this derivative most transparently, note what happens if we write the summation in (8) explicitly. The household maximizes:
[−Kt+1 + (1 + r̃t − δ)Kt + wt]1−γ
1 −γ + βEt
{ [−Kt+2 + (1 + r̃t+1 − δ)Kt+1 + wt+1]1−γ
1 −γ
}
+β2Et
{ [−Kt+3 + (1 + r̃t+2 − δ)Kt+2 + wt+2]1−γ
1 −γ
} + ...
• Note also that everything in the first term is known at time t (Kt+1 is chosen at t). Therefore, we can drop the expectation operator from that term.
• As you see, Kt+1: appears in two consecutive terms of this expression. Hence, taking the derivative yields:
−(1 −γ) [−Kt+1 + (1 + r̃t − δ)Kt + wt]−γ
1 −γ
+βEt
{ (1 −γ) [−Kt+2 + (1 + r̃t+1 − δ)Kt+1 + wt+1]−γ
1 −γ (1 + r̃t+1 − δ)
}
22
Solving the Household’s Problem: Doing the Math
• If you simplify the 1 −γ terms, substitute (7) for s = t and s = t + 1, respectively, in the first and in the second term of this expression, and set it equal to 0, you immediately find:
−C−γt + βEt [ C −γ t+1(1 + r̃t+1 − δ)
] = 0,
or
C −γ t = βEt
[ C −γ t+1(1 + r̃t+1 − δ)
] ,
i.e., the Euler equation (5).
• A sequence of such equations (one for every s = t, ...,∞) must be satisfied for the household to be optimizing its consumption and investment behavior over time.
23
The Transversality Condition
• It turns out that the Euler equation is actually not the only optimality condition the household must satisfy:
• The Euler equation describes optimal behavior between any two consecutive periods (s and s + 1, for s = t, ...,∞), but the household is solving an infinite horizon problem that requires it to look beyond any pair of consecutive periods.
• The additional condition that must be satisfied is known as transversality conditions, and it has this form:
lim Et T−→∞
[ βTu′(ct+T )(1 + r̃t+T − δ)Kt+T
] = 0. (9)
• We are not going to do the math to show why this condition must hold.
24
The Transversality Condition, Continued
• Intuitively, if the expression on the left-hand side were strictly positive, the household would be overaccumulating capital, so that a higher expected lifetime utility could be achieved by
increasing consumption today.
• The counterpart to such non-optimality in a finite horizon model would be that the household dies with positively valued capital holdings: There is no bequest motive in our model for
which anyone would want to die with positively valued assets!
• Condition (9) cannot be violated on the negative side because the marginal utility of consumption is never negative, 0 < δ ≤ 1, and capital (a factor of production) must be positive.
25
Euler Equations and Transversality Conditions
• One way to look at Euler equations and transversality conditions is to observe that Euler equations rule out arbitrage opportunities between consecutive periods (when the Euler
equation holds, the household cannot increase its utility by changing consumption and
capital holdings between two consecutive periods).
• Transversality conditions rule out permanent/infinite-horizon arbitrages (the household cannot increase its utility by increasing consumption permanently).
• Euler equations represent short-run optimality conditions, which all candidate paths for optimality of consumption and investment must satisfy, while the transversality condition
gives an additional long-run optimality condition, which (under the assumptions we are
making on the shape of the period utility function) only one of the short-run optimal paths
satisfies.
– Concavity of the utility function ensures that we do not need to compute second-order
conditions for the household’s maximization problem.
26
The Rental Rate and Production
• Households rent capital to firms and, with competitive markets,
r̃t = ∂Yt ∂Kt
(marginal product of capital),
where Yt is output.
• We assume that output in the economy is given by a Cobb-Douglas production function. In aggregate per capita terms,
Yt = (AtNt) αK1−αt = At
αK1−αt (10)
where 0 < α < 1 and At denotes exogenous technology (which is subject to random
shocks).
• Therefore,
r̃t = (1 −α) ( At Kt
)α and the Euler equation (5) becomes:
C −γ t = βEt
{ C −γ t+1
[ (1 −α)
( At+1 Kt+1
)α + 1 − δ
]} (11)
27
Efficiency and the Planner’s Outcome
• There are no distortions in the model-economy we are considering (markets are perfectly competitive).
• Then, the decentralized, competitive equilibrium generated by market behavior coincides with the solution of the problem that a benevolent social planner would solve.
• Specifically, the planner would recognize that the following aggregate per capita resource constraint must be satisfied in each period:
Yt = Ct + It.
• Thus, from (3) and (??), Yt = r̃tKt + wt,
or
wt = Yt − r̃tKt (as implied by Euler’s output exhaustion theorem).
28
Efficiency and the Planner’s Outcome, Continued
• So, (4) becomes: Ct + Kt+1 = (1 − δ)Kt + Yt,
or, taking (10) into account,
Ct + Kt+1 = (1 − δ)Kt + AtαK1−αt . (12)
• A planner would recognize that the gross return at t + 1 from investing one unit of the consumption good at t in capital would be:
Rt+1 ≡ (1 −α) ( At+1 Kt+1
)α + 1 − δ,
i.e., the marginal product of capital at t + 1 plus undepreciated capital.
29
Efficiency and the Planner’s Outcome, Continued
• Now, maximizing (1) subject to (12) yields:
C −γ t = βEt
{ C −γ t+1
[ (1 −α)
( At+1 Kt+1
)α + 1 − δ
]} ,
or
C −γ t = βEt
( C −γ t+1Rt+1
) , (13)
i.e., at an optimum, the cost of investing one unit of consumption today in capital
accumulation (the marginal utility of one unit of consumption today) must be equal to the
expected discounted marginal utility value of the gross return from investing one unit of
consumption good in capital accumulation.
• As expected, (13) (the Euler equation from the solution of the planner’s problem) and (11) (the Euler equation for the decentralized, market solution) are identical once the definition of
Rt+1 is taken into account.
30
Steady-State Growth
• Let us look for a steady-state, or balanced growth path of the model, in which technology, capital, and consumption all grow at a constant common growth rate.
• We denote this growth rate as:
G ≡ At+1
At (overbars denote steady-state levels).
• In steady-state, the gross rate of return on capital, Rt+1, becomes a constant R, while the first-order condition (13) becomes:
Gγ = βR, (14)
or, in logs (letting r ≡ log R and g ≡ log G):
g = log β + r
γ = σ log β + σr.
• This condition, relating the equilibrium growth rate of consumption to the intertemporal elasticity of substitution times the real interest rate, is a standard result of models with power
utility.
31
Steady-State Growth, Continued
• The definition of R and equation (14) imply that, in steady state, the constant technology- capital ratio is:
At
Kt =
[ Gγ/β − (1 − δ)
1 −α
]1/α .
• A higher rate of technology growth leads to a lower capital stock for a given level of technology.
– The reason is that, in steady state, faster technology growth must be accompanied by
faster consumption growth.
– Agents will accept a steeper consumption path only if the rate of return on capital is
higher, which requires a lower capital stock.
• Setting Gγ/β = R ≈ 1 + r yields:
At
Kt ≈ ( r + δ
1 −α
)1/α . (15)
32
Steady-State Growth, Continued
• It is possible to solve for various ratios of variables that are constant along a steady-state growth path.
• These ratios can be expressed in terms of four underlying parameters:
– g, the log technology growth rate;
– r, the log real return on capital;
· strictly speaking, r is an endogenous variable of our model, but we treat is as a parameter as we recognize that it must satisfy r = − log β + g
σ = − log β + γg.
– α, the exponent on labor and technology in the production function, or equivalently,
labor’s share of output;
– and δ, the rate of capital depreciation.
33
Steady-State Growth, Continued
• For purposes of “calibration,” interpreting periods as quarters, benchmark values of these parameters may be:
g = .005 (2% at annual rate),
r = .015 (6% at annual rate),
α = .667,
δ = .025 (10% at annual rate).
• These are all plausible numbers for the U.S. economy.
• Given r = .015 and g = .005, r = − log β + γg defines the pairs of values for γ and β such that r = .015 and g = .005.
34
Steady-State Growth, Continued
• Using the production function and (15), we find the constant steady-state output capital ratio:
Y t
Kt =
( At
Kt
)α ≈ r + δ
1 −α . (16)
• Similarly, in steady state, the consumption-output ratio is constant at (see below for Ct/Kt):
Ct
Y t = Ct/Kt
Y t/Kt ≈ 1 −
(1 −α)(g + δ) r + δ
. (17)
• At the benchmark parameter values above, it must be:
Y t
Kt = .118 (.472 at annual rate) and
Ct
Y t = .745,
fairly reasonable values.
35
A Non-Linear Model of Fluctuations
• Outside the steady state, the model we laid out is a system of non-linear equations for consumption, capital, output, and technology.
• Nonlinearities are caused by incomplete capital depreciation (δ < 1 in (12) and in Rt+1 = (1 −α)
( At+1 Kt+1
)α + 1 − δ) and by time variation in the consumption-output ratio (or the
savings rate).
• Exact analytical solution of the model is possible only in the unrealistic special case in which capital depreciates fully in one period (δ = 1) and agents have log utility (γ = 1), so that the
consumption-output ratio (and therefore the savings rate) is always constant.
• You find the details on how this special case works in Appendix B.
• The problem is that δ = 1 and γ = 1 are extremely restrictive hypotheses. In all other cases, the model features a mixture of multiplicative and additive terms that make an exact solution
impossible.
• How do we proceed?
36
Log-Linearization
• Our solution approach will be to seek an approximate analytical solution by transforming the model into a system of approximate log-linear difference equations.
• In doing so, we are going to rely on the following result: For small deviations of the variable Xt from the steady state Xt, it is:
dXt
Xt = Xt −Xt Xt
≈ d log Xt = log Xt − log Xt,
and we are going to define:
xt ≡ dXt
Xt .
• Now, interpret all lower-case variables below as zero-mean percentage deviations from the steady state of the model that we obtained above.
37
Log-Linearization, Continued
• From the production function, yt = αat + (1 −α)kt. (18)
This one is easy: Just take logs of the production function and remember that Nt = 1; (18)
holds exactly, it is not an approximation.
38
Log-Linearization, Continued
• Things are harder for equations that are not log-linear.
• For example: Ct + Kt+1 = (1 − δ)Kt + Yt. (19)
• John Campbell of Harvard University used Taylor expansions to approximate the model in a 1994 article in the Journal of Monetary Economics that is a standard reference on how to
find the approximation below.
• I find it more transparent and efficient to proceed as follows.
39
Log-Linearization, Continued
• The differential of (19) is:
dCt + dKt+1 = (1 − δ)dKt + dYt.
• Thus,
Ct dCt
Ct + Kt+1
dKt+1
Kt+1 = (1 − δ)Kt
dKt
Kt + Yt
dYt
Yt ,
or
Ct
Kt ct +
Kt+1
Kt kt+1 = (1 − δ)kt +
Yt
Kt yt. (20)
40
Log-Linearization, Continued
• Now, we know that Kt+1
Kt = G ≈ 1 + g.
Also,
Yt
Kt ≈ r + δ
1 −α
• Then, a steady-state version of (19) implies:
Kt+1
Kt = (1 − δ) +
Yt
Kt − Ct
Kt .
• Using Kt+1 Kt ≈ 1 + g and Yt
Kt ≈ r+δ
1−α and solving for Ct Kt
yields:
Ct
Kt ≈ r + δ
1 −α − (g + δ)
41
Log-Linearization, Continued
• Therefore, substituting these results into (20), we can rewrite it as:( r + δ
1 −α − (g + δ)
) ct + (1 + g)kt+1 = (1 − δ)kt +
r + δ
1 −α yt,
or:
(1 + g)kt+1 = (1 − δ)kt + r + δ
1 −α yt +
( g + δ −
r + δ
1 −α
) ct.
This is a linear equation in the variables kt+1, kt, yt, and ct—the percentage deviations of the
variables Kt+1, Kt, Yt, and Ct from their steady-state levels!
• Moreover, substituting yt = αat + (1 −α)kt, we have:
kt+1 = 1 + r
1 + g kt +
α (r + δ)
(1 −α) (1 + g) at +
[ g + δ
1 + g −
r + δ
(1 + g) (1 −α)
] ct.
42
Log-Linearization, Continued
• Let
λ1 ≡ 1 + r
1 + g and λ2 ≡
α (r + δ)
(1 −α) (1 + g) .
Then:
kt+1 = λ1kt + λ2at + (1 −λ1 −λ2) ct. (21) At the benchmark parameter values,
λ1 = 1.01, λ2 = .08, and 1 −λ1 −λ2 = −.09.
• To understand these coefficients, note that
1 −λ1 −λ2 = − Ct
Y t
Yt
Kt (1 + g)
−1 = −(.118)(.745)(1.005)−1.
– This is the negative of the steady-state ratio of this period’s consumption to next period’s
capital stock:
· A $1 increase in consumption today lowers tomorrow’s capital stock by $1, but a 1 percent increase in consumption this period lowers next period’s capital stock by only
.09 percent because in steady state one period’s consumption is only .09 times as big
as the next period’s capital stock.
43
Log-Linearization, Continued
• Now focus on the Euler equation:
C −γ t = βEt
( C −γ t+1Rt+1
) .
• Assume that the variables on the right-hand side are jointly log-normal and homoskedastic.
– The first assumption means that the log-variables are normally distributed and the
second means that they have constant second moments (variances and covariances).
– The assumptions are consistent with a log-normal productivity shock being the source of
fluctuations and with the approximations we use.
44
Log-Linearization, Continued
• Taking logs of both sides of the Euler equation:
−γ log Ct = log β + log [ Et ( C −γ t+1Rt+1
)] . (22)
• But log-normality implies the following property:
log [Et (Xt+1)] = Et [log (Xt+1)] + 1
2 vart [log (Xt+1)] .
• Therefore:
log [ Et ( C −γ t+1Rt+1
)] = Et
[ log ( C −γ t+1Rt+1
)] +
1
2 vart
[ log ( C −γ t+1Rt+1
)] = −γEt (log Ct+1) + Et (log Rt+1)
+ γ2
2 σ2t,logCt+1 +
1
2 σ2t,logRt+1 −γσt,logCt+1,logRt+1,
where for any variable Xt+1, σ 2 t,logXt+1
denotes the conditional variance at time t of log Xt+1,
and σt,logCt+1,logRt+1 denotes the conditional covariance at time t of log Ct+1 and log Rt+1.
45
Log-Linearization, Continued
• Hence, (22) becomes:
−γ log Ct = log β + Et (−γ log Ct+1 + log Rt+1) + γ2
2 σ2t,logCt+1 +
1
2 σ2t,logRt+1 −γσt,logCt+1,logRt+1.
• Now differentiate this equation to obtain:
−γd log Ct ≈−γEt (d log Ct+1) + Et (d log Rt+1) .
• Why did second moments disappear?
• Remember: We are assuming that variables are homoskedastic. Hence, conditional second moments are constant, and they drop out when we differentiate!
46
Log-Linearization, Continued
• Given
xt ≡ dXt
Xt = Xt −Xt Xt
≈ d log Xt = log Xt − log Xt,
it finally follows that we can write the Euler equation in log-linear form as:
γEt (ct+1 − ct) ≈ Etrt+1
where rt+1 = d log Rt+1.
• Or, recalling σ = 1 γ ,
Et (ct+1 − ct) ≈ 1
γ Etrt+1 = σEtrt+1. (23)
– The intertemporal elasticity of substitution σ measures the responsiveness of consump-
tion to a change in the return to asset accumulation.
47
Log-Linearization, Continued
• Now,
Rt+1 = (1 −α) ( At+1 Kt+1
)α + 1 − δ.
• Recall d log Rt+1 ≈ dRt+1Rt+1 . Then:
dRt+1 = (1 −α) α (dAt+1) Kt+1 − (dKt+1) At+1
K 2 t+1
( At+1
Kt+1
)α−1 ,
or:
Rt+1rt+1 ≈ α (1 −α) At+1
Kt+1 (at+1 −kt+1)
( At+1
Kt+1
)α−1 = α (1 −α) (at+1 −kt+1)
( At+1
Kt+1
)α .
48
Log-Linearization, Continued
• Recall also: At+1
Kt+1 ≈ ( r + δ
1 −α
)1/α .
• Then:
Rt+1 ≈ (1 −α) r + δ
1 −α + 1 − δ = 1 + r.
49
Log-Linearization, Continued
• So,
(1 + r) rt+1 ≈ α (1 −α) (at+1 −kt+1) r + δ
1 −α ,
and
rt+1 ≈ λ3 (at+1 −kt+1) , (24) with:
λ3 ≡ α (r + δ)
1 + r .
– The same result can be obtained by taking the differential of
log Rt+1 = log
[ 1 − δ + (1 −α)
( At+1 Kt+1
)α] .
You should try doing it as an exercise.
50
Log-Linearization, Continued
• At the benchmark parameter values, λ3 = .03.
– This coefficient is very small. One way to understand this is to note that changes in
technology have only small proportional effects on the one-period return on capital
because capital depreciates only slowly, so most of the return R is undepreciated capital
rather than marginal output from the Cobb-Douglas production function.
– Alternatively, we can note that rt+1 ≈ Rt+1 − 1 ≈ (1 −α) ( At+1 Kt+1
)α when δ is negligible.
In this case, a 1 percent increase in the technology-capital ratio raises rt+1 by about α
percent. But α percent of rt+1 is only αrt+1 percentage points.
• Equations (23) and (24) together imply:
Et (ct+1 − ct) ≈ σλ3Et (at+1 −kt+1) . (25)
51
Log-Linearization, Continued
• To close the model, we only need to specify a process for the technology shock at, the percentage deviation of At from its steady-state level At: at =
At−At At
.
• We assume an AR(1) process:
at = φat−1 + εt, − 1 ≤ φ ≤ 1. (26)
• We assume that the innovations to technology, εt, are normally distributed and such that Et−1 (εt) = 0.
• The AR(1) coefficient φ measures the persistence of technology shocks, with the extreme case φ = 1 being a random walk for technology.
52
Log-Linearization, Continued
• We did a ton of math and it looked awfully complicated, but look what we have now: Equations (21), (25), and(26) form a system of linear expectational difference equations in
(the percentage deviations from the steady state of) technology, capital, and consumption!
• In other words, we boiled down the non-linear model we started from to the linear system:
kt+1 ≈ λ1kt + λ2at + (1 −λ1 −λ2) ct, Et (ct+1 − ct) ≈ σλ3Et (at+1 −kt+1) ,
at = φat−1 + εt. (27)
• The parameter of these equations include λ1,λ2 and λ3 (where λ1 ≡ 1+r1+g, λ2 ≡ α(r+δ)
(1−α)(1+g), and
λ3 ≡ α(r+δ)1+r ), σ = 1 γ , the AR(1) coefficient φ that measures the persistence of technology
shocks, and the variance of the technology innovations εt.
53
The Calibration Approach to RBC Analysis
• In Campbell’s interpretation, the “calibration” approach to real business cycle analysis takes λ1, λ2, and λ3 as known, and searches for values of σ and φ (and a variance for the
technology innovation, ε) to match the moments of observed macroeconomic the series.
– If λ1, λ2, and λ3 are not taken as given, one can search for values of all the structural
parameters of the model–including those of which λ1, λ2, and λ3 are functions–to match
moments of observed data.
• One can then verify if the values of parameters such that the model matches moments of the data are reasonable.
• An alternative interpretation of the calibration approach runs in the opposite direction and asks the following questions:
– Given reasonable values of the structural parameters of the model and a process for
at that is roughly consistent with the data, how far are the moments of endogenous
variables implied by the model from the moments of actual data?
– How far are the impulse responses (the responses of endogenous variables to exogenous
shocks) from those implied by the data?
54
Determinacy of the Solution
• To compute impulse responses (i.e., the responses of consumption and capital to technology innovations) or the second-moment properties (i.e., the variances or covariances of
consumption and capital implied by assumptions on the variance of technology innovations)
to compare them to properties of the data, we must solve the system (27).
• But whenever we solve a system of linear, expectational, difference equations such as (27), in principle we need to check that there is a unique solution, i.e., that the solution (if it exists)
is determinate.
• If the solution is indeterminate, the economy is subject to fluctuations that are not caused by changes in the fundamentals–sunspot fluctuations.
• We will not study how to prove that the system (27) has a unique solution. You find the information in Appendix C. For our purposes, trust that the system does have a unique
solution.
55
Determinacy of the Solution, Continued
• Important: A very interesting branch of macroeconomics that time limitations do not allow us to study focuses precisely on what happens in model-economies that do not have a unique
solution (for dynamics around the steady state or even for the steady state itself).
• These models are best suited to capture John Maynard Keynes’ idea of animal spirits, fluctuations in sentiment that trigger economic fluctuations and that are not captured by
so-called fundamental-driven fluctuations we are focusing on. (Fundamental in the sense
that technology is among the “fundamentals” of our model as opposed to, say, the color of
sunspots.)
• This is a fascinating, very important branch of macroeconomics. Roger Farmer of the University of Warwick, UK, and Karl Shell of Cornell University are among the major
contributors.
56
The Method of Undetermined Coefficients
• Once we trust that the system (27) has a unique solution, it can be solved with a method known as the method of undetermined coefficients.
• Let ηzx denote the partial elasticity of variable z with respect to variable x.
• Guess that the solution for consumption takes the form:
ct = ηckkt + ηcaat, (28)
where ηck and ηca are unknown but assumed constant.
• We are going to verify the guess by finding values of ηck and ηca that satisfy the restrictions of the approximate model.
57
The Method of Undetermined Coefficients, Continued
• Note 1: The guess (28) is consistent with the logic of a technique for solving dynamic models known as dynamic programming: Optimal behavior maps the state of the economy
at time t (described in our model by capital—kt—and technology—at) into the variables that
are endogenous during that period (in this case, consumption during that period—ct).
– Think about it: From the perspective of households and firms in the economy, what can
summarize the state (the initial condition) of the economy at the time when they take a
decision?
– Well, that’s the capital stock they entered the period with and the current realization of
the available technology.
– So, given a set of linear equations that we are trying to solve, we are going to guess that
the solution is a linear function of the endogenous state (capital) and the exogenous one
(technology).
58
The Method of Undetermined Coefficients, Continued
• Note 2: Having verified determinacy allows us to be confident that the solution we are guessing (referred to also as the minimum state vector solution, since it depends on the
smallest number of state variables) is the unique solution of the system (27).
– If the solution were indeterminate, (28) would be only one of the possible solutions of
(27).
– Alternative solutions would exist—among them, solutions that map so-called non-
fundamental states (such as the color of sunspots...) into the endogenous variables.
59
The Method of Undetermined Coefficients, Continued
• Given our guessed solution for consumption and equation (21), we can write the guessed solution for capital as:
kt+1 = ηkkkt + ηkaat, (29)
with:
ηkk ≡ λ1 + (1 −λ1 −λ2) ηck, ηka ≡ λ2 + (1 −λ1 −λ2) ηca.
• Note 3: As expected (again, consistent with the logic of dynamic programming), the solution maps the state at time t into the choice of assets entering t + 1 (kt+1).
60
The Method of Undetermined Coefficients, Continued
• Note 4: As noted above, the solution in (28) and (29) is also called the minimum state variable solution, in the sense that it is the solution that expresses the endogenous variables
as functions of the minimum state vector—the vector consisting of the endogenous,
predetermined state variable kt and of the exogenous state variable at.
– The concept of minimum state variable (MSV) solution has been proposed by Bennett
McCallum of Carnegie-Mellon University as the only solution that is relevant in practice
even in situations in which there is indeterminacy, and, therefore, the MSV solution is not
the unique rational expectation equilibrium of the model.
– Many scholars see determinacy of the equilibrium (which ensures that the MSV solution
is the unique equilibrium) as an important, desirable property of macroeconomic models.
– Others, like Roger Farmer and Karl Shell, view allowing for multiple possible solutions
(and therefore multiple possible equilibria) as central to understanding fluctuations.
61
The Method of Undetermined Coefficients, Continued
• Substituting the conjectured solution into Et (ct+1 − ct) = σλ3Et (at+1 −kt+1) yields:
ηck (kt+1 −kt) + ηcaEt (at+1 −at) = σλ3Etat+1 −σλ3kt+1 (30)
(kt+1 is known at time t, when it is determined).
• Substitute (29) into (30) and use Etat+1 = φat. The result (taking the definitions of ηkk and ηka into account) is an equation in only the two state variables, kt and at:
ηck [λ1 − 1 + (1 −λ1 −λ2) ηck] kt + ηck [λ2 + (1 −λ1 −λ2) ηca] at + ηca (φ− 1) at (31) = σλ3φat −σλ3 [λ1 + (1 −λ1 −λ2) ηck] kt −σλ3 [λ2 + (1 −λ1 −λ2) ηca] at.
62
The Method of Undetermined Coefficients, Continued
• To solve this equation, we first equate coefficients on kt to find ηck and then equate coefficients on at to find ηca, given ηck.
• Equating coefficients on kt gives the quadratic equation:
Q2η 2 ck + Q1ηck + Q0 = 0, (32)
with:
Q2 ≡ 1 −λ1 −λ2, Q1 ≡ λ1 − 1 + σλ3 (1 −λ1 −λ2) , Q0 ≡ σλ3λ1.
63
The Method of Undetermined Coefficients, Continued
• The quadratic formula gives two solutions to (32).
• With the benchmark set of parameter values, one of these is positive.
– Equation (21), with λ1 > 1, shows that ηck must be positive for the steady state to be
locally stable.
· If ηck < 0, then λ1 + (1 −λ1 −λ2) ηck > 1, which implies ηkk > 1 in (29), or an unstable steady state to which the economy never returns after shocks.
– Hence, the positive solution is the appropriate one:
ηck = 1
2Q2
( −Q1 −
√ Q21 − 4Q0Q2
) .
– Intuitively: It makes economic sense that, if the economy invested more during period
t − 1, and therefore it enters period t with more capital, consumption during period t is higher.
• Note that ηck depends only on σ and the λ parameters and is invariant to the persistence of the technology shock, φ.
64
The Method of Undetermined Coefficients, Continued
• The solution of the model is then completed by finding ηca as:
ηca = −ηckλ2 + σλ3 (φ−λ2)
φ− 1 + (1 −λ1 −λ2) (ηck + σλ3) .
• To obtain this, equate coefficients on at at the left and right side of equation (31), substitute the solution for ηck, and solve the resulting equation for ηca.
65
The Method of Undetermined Coefficients, Continued
• To summarize, we have:
at = φat−1 + εt,
kt+1 = ηkkkt + ηkaat,
ct = ηckkt + ηcaat,
as solution of the model, with the parameters obtained above.
• These equations make it possible to study the dynamics of consumption, capital, and technology following an innovation to the latter.
• In other words, the equations can be used to analyze the response of the economy to technology shocks, i.e., to perform impulse response analysis.
66
The Method of Undetermined Coefficients, Continued
• Given numerical values for parameters, we can use the equations to compute the paths of technology, capital, and consumption over time in response to an initial innovation to
technology ε0 = 1 at time t = 0 (assuming that the economy was in steady state until—and
including—period t = −1).
– We will want to use the fact that k0 = 0 (since it was determined at t = −1, before the innovation to technology).
• We will explore how to do this using a simple Excel spreadsheet that will illustrate how the consumption elasticities, ηck and ηca, and the capital elasticities, ηkk and ηka, derived from
them determine the dynamic behavior of our model economy.
• For those of you who have taken time series econometrics, Appendix D goes over some time series implications of the model.
67
A Summary of the Dynamic Properties of the Model
• Three characteristics of the fixed-labor model deserve note.
• First, analysis of impulse responses shows that capital accumulation has an important effect on the dynamics of the economy only when the underlying technology shock is persistent,
lasting long enough for significant changes in capital to occur.
• The stochastic growth model–or at least this version–is unable to generate persistent effects from transitory shocks.
68
A Summary of the Dynamic Properties of the Model, Continued
• To understand this point, recall the solution equations:
ct = ηckkt + ηcaat,
yt = ηykkt + ηyaat,
kt+1 = ηkkkt + ηkaat.
• Persistence in the dynamics of consumption and other endogenous variables follows from their dependence on the exogenous state at and on the endogenous, predetermined state
kt.
• The persistence of technology (φ) is an exogenous parameter.
• Therefore, the persistence of dynamics that comes from dependence on at is exogenous.
• Instead, we refer to the persistence that arises as a consequence of dependence on the endogenous state kt as endogenous persistence.
• However, if φ is small, the technology shock does not last long enough to generate significant changes in capital, and the effect of capital dynamics on the economy is consequently small,
so that the deviation of consumption and output from the steady state becomes very small
once at has returned to the steady state.
69
A Summary of the Dynamic Properties of the Model, Continued
• Second, technology shocks do not have strong effects on realized or expected returns on capital.
• The reason is that the gross rate of return on capital largely consist of undepreciated capital rather than the net output that is affected by technology shocks.
• The realized return on capital equals λ3, and λ3 = .03 at benchmark parameter values.
• Thus, a 1 percent technology shock changes the realized return on capital on impact by only 3 basis points (12 at annual rate).
• The expected return on capital is even more stable (constant if the representative agent is risk neutral) because capital accumulation lowers the marginal product of capital one period
after a positive technology shock occurs, partially offsetting any persistent effects of the
shock.
70
A Summary of the Dynamic Properties of the Model, Continued
• Third, capital accumulation does not generate a short or long-run “multiplier” in the sense of an output response to a technology shock that is larger (in percentage terms) than the
underlying shock itself.
• This means that slower-than-normal technology growth can generate only slower-than- normal output growth and not actual declines in output.
• The model with fixed labor supply can explain output declines only by appealing to implausible declines in the level of technology.
71
Variable Labor Supply
• We now move to a model in which labor supply is allowed to vary over time and is determined endogenously.
• The production function is unchanged:
Yt = (AtNt) α K1−αt . (33)
• Also the law of motion for capital remains:
Kt+1 = (1 − δ) Kt + Yt −Ct. (34)
• However, we now assume that the period utility function is:
u (Ct, 1 −Nt) = log Ct + θ (1 −Nt)1−γn
1 −γn . (35)
72
Variable Labor Supply, Continued
u (Ct, 1 −Nt) = log Ct + θ (1 −Nt)1−γn
1 −γn .
• The total amount of time available to agents in each period is normalized to 1. Thus, 1 −Nt is leisure in period t.
• Utility is additively separable in consumption and leisure. Robert King of Boston University, Charles Plosser of the Hoover Institution, and Sergio Rebelo of Northwestern University
showed in a 1988 Journal of Monetary Economics article that log utility from consumption is
required to obtain constant steady-state labor supply (i.e., balanced growth) when utility is
additively separable over consumption and leisure.
– The balanced growth requirement does not restrict the form of the utility function for
leisure.
• Power utility nests several cases in the literature (for example, log when γn = 1, linear when γn = 0). Let σn ≡ 1γn denote the elasticity of intertemporal substitution for leisure.
73
Variable Labor Supply, Continued
• The Euler equation for consumption is still:
C−1t = βEt ( C−1t+1Rt+1
) . (36)
• But now:
Rt+1 = (1 −α) ( At+1Nt+1 Kt+1
)α + 1 − δ. (37)
• And the key new feature of the model is that there is now a static first-order condition for the optimal choice of leisure relative to consumption at each point in time.
74
Variable Labor Supply, Continued
• We refer to this first-order condition as the labor-leisure tradeoff : the agent needs wage income to consume, but labor has a utility cost.
• Intuitively, it must be that, for the household to be optimizing, the marginal utility of leisure equals the real wage evaluated in terms of the marginal utility of consumption.
– i.e., the marginal utility of leisure must equal how much marginal utility of consumption
the real wage earned by supplying an extra unit of labor generates:
θ (1 −Nt)−γn = C−1t wt. (38)
– Ceteris paribus, if consumption increases, its marginal utility (and thus the marginal utility
of wage income to buy consumption) decreases, and so does labor supply.
• Condition (38) also states that the marginal rate of substitution between leisure and consumption has to be equal to the real wage.
• You can obtain this condition by solving a properly modified version of the household’s maximization problem with period utility (35) and Nt 6= 1. Maximizing with respect to Nt gives (38).
75
Variable Labor Supply, Continued
• With competitive markets, the real wage equals the marginal product of labor:
wt = αA α t
( Kt Nt
)1−α . (39)
• Thus, in an efficient economy, the marginal rate of substitution between leisure (or labor) and consumption in household utility has to be equal to the marginal rate at which labor is
transformed into output in firm production.
• Combining (38) and (39) yields the labor market clearing condition that determines equilibrium employment:
θ (1 −Nt)−γn = α Aαt Ct
( Kt Nt
)1−α . (40)
76
The Steady State with Variable Labor Supply
• It turns out that the analysis of the steady state from the model with fixed labor carries over directly to the variable-labor model.
• It is still the case that, in a steady state with At+1 At
= G, Gγ = βR, or G = βR, as γ = 1 in this
model. Thus, g = log β + r.
• The steady-state values of the ratios At Kt
, Yt Kt
, and Ct Yt
can be obtained following similar steps
to those above.
– See Appendix E slides for some details.
77
The Log-Linear Model with Variable Labor Supply
• We can linearize the model’s equations around the steady sate as with did for the fixed-labor model, using d log Xt ≈ dXtXt = xt.
• The log-linear version of the capital accumulation equation, using
yt = α (at + nt) + (1 −α) kt,
is:
kt+1 ≈ λ1kt + λ2 (at + nt) + (1 −λ1 −λ2) ct, (41) with λ1 and λ2 the same as in the fixed-labor model. (See the Appendix F slides.)
• The interest rate is: rt+1 ≈ λ3 (at+1 + nt+1 −kt+1) , (42)
with λ3 the same as before.
• Linearizing the Euler equation, using the log-normality and homoskedasticity assumptions, and using (42) yields:
Et (ct+1 − ct) ≈ λ3Et (at+1 + nt+1 −kt+1) . (43)
78
The Log-Linear Model with Variable Labor Supply, Continued
• Now focus on (40). Taking logs:
log θ −γn log (1 −Nt) = log α + α log At − log Ct + (1 −α) log Kt − (1 −α) log Nt.
• Then:
−γnd log (1 −Nt) = αd log At −d log Ct + (1 −α) d log Kt − (1 −α) d log Nt.
• Observe that:
d log (1 −Nt) = − dNt
1 −N = −
N
1 −N dNt
N = −
N
1 −N nt.
• Thus,
γn N
1 −N nt ≈ αat + (1 −α) (kt −nt) − ct,
or:
nt ≈ 1 −N N
σn [αat + (1 −α) (kt −nt) − ct] . (44)
79
The Log-Linear Model with Variable Labor Supply, Continued
• N solves a non-linear equation shown in Appendix E.
• If we assume that households allocate on average one-third of their time to market activities, then N = 1
3 and 1−N
N = 2.
• We take this as benchmark (and the equation in Appendix E can be used to solve for the combinations (γn,θ) such that N =
1 3
with the other parameters at their benchmark values).
• Equation (44) can be rewritten as:
nt
[ 1 +
(1 −α) ( 1 −N
) N
σn
] ≈
1 −N N
σn [αat + (1 −α) kt − ct] ,
or:
nt
[ N + (1 −α)
( 1 −N
) σn
N
] ≈
1 −N N
σn [αat + (1 −α) kt − ct] ,
which implies:
nt ≈ µ [(1 −α) kt + αat − ct] , (45) with
µ = µ (σn) ≡ ( 1 −N
) σn
N + (1 −α) ( 1 −N
) σn .
80
The Log-Linear Model with Variable Labor Supply, Continued
• µ measures the responsiveness of labor supply to shocks that change the real wage or consumption, taking into account the fact that, if labor supply increases, the real wage is
driven down.
• To see this, observe that
wt = αA α t
( Kt Nt
)1−α ⇒
ωt = αat + (1 −α) (kt −nt) = (1 −α) kt + αat − (1 −α) nt,
where ωt ≡ dwtw .
• Because of this effect, even when utility from leisure is linear (σn →∞), µ is finite and equal to 1
1−α (as implied by ωt = (1 −α) kt + αat − (1 −α) nt).
• As the curvature of the utility function for leisure increases, µ falls and becomes zero when γn →∞ (σn → 0). This corresponds to the fixed-labor-supply model we studied before.
• Note that the value of N affects only the relation between σn and µ and not any other aspect of the model.
81
The Log-Linear Model with Variable Labor Supply, Continued
• Substituting equation (45) into equations (41) and (43) and maintaining the assumption at = φat−1 + εt, Et−1 (εt) = 0, returns a system of equations in capital, consumption, and
technology similar to the one we studied when labor was fixed at 1.
• The system can be solved using the same method of undetermined coefficients.
• Assuming that there is a unique solution, we can conjecture the MSV solution:
ct = ηckkt + ηcaat. (46)
– A good exercise for you would be to verify determinacy of the equilibrium as done in the
case of fixed labor supply.
• ηck solves a quadratic equation of the type:
Q2η 2 ck + Q1ηck + Q0 = 0,
where the coefficients Q2, Q1, and Q0 are now more complicated than before. (See Appendix
G.)
• As before, we pick the positive solution. The solution for ηca can be obtained straightforwardly from ηck and the other parameters. These solutions are the same as in the fixed-labor-supply
model when labor supply is completely inelastic, so that µ = 0.
82
Dynamics with Variable Labor Supply
• The dynamics of the economy take the same form as in the fixed-labor model. Once again:
kt+1 = ηkkkt + ηkaat. (47)
(The definitions/expressions for ηkk and ηka are in Appendix G.)
• Substituting (46) into (45) yields:
nt = µ [(1 −α) kt + αat −ηckkt −ηcaat] = ηnkkt + ηnaat, (48)
with ηnk ≡ µ (1 −α−ηck) and ηna ≡ µ (α−ηca).
• Increases in capital raise the real wage by a factor 1 − α (recall ωt = (1 −α) kt + αat − (1 −α) nt).
• This stimulates labor supply.
• But capital also increases consumption by a factor ηck, and this can have an offsetting effect.
• Similarly, increases in technology raise the real wage by a factor α, but the stimulating effect on labor supply is offset by the effect ηca of technology on consumption.
83
Dynamics with Variable Labor Supply, Continued
• Finally, using yt = α (at + nt) + (1 −α) kt and substituting for nt from (48) gives:
yt = ηykkt + ηyaat, (49)
with ηyk ≡ 1 −α + αµ (1 −α−ηck), ηya ≡ α + αµ (α−ηca).
• As before, if we use the lag operator notation, it turns out that output is an ARMA(2, 1) process.
• However, capital and technology now affect output both directly (with coefficients α and 1 −α, respectively) and indirectly through labor supply.
• The initial response to a technology shock is now α + αµ (α−ηca) rather than α.
• Thus, the variable-labor model can produce an amplified output response to technology shocks, even in the very short run.
84
Dynamics with Variable Labor Supply, Continued
• At the benchmark parameter values, if σn = 0, the model reduces to the fixed-labor case: ηnk = ηna = 0.
• As σn increases, ηnk becomes increasingly negative, while ηna becomes increasingly positive.
• Thus, an increase in capital lowers the work effort because it increases consumption more than the real wage.
• A positive technology shocks increases the work effort.
• ηnk is independent of the persistence parameter φ, but ηna declines with φ.
• The reason is that, if an innovation has persistent effects on technology, it increases consumption more than a transitory shock (ηca increases with φ),
• The increase in consumption lowers the marginal utility of income and reduces the work effort.
• Put another way, transitory shocks produce a stronger intertemporal substitution effect in labor supply (if φ is small, ηna is large).
85
Dynamics with Variable Labor Supply, Continued
• Campbell analyzes a number of special cases and properties of the model, along with the responses of the return on capital and the real wage to shocks.
• It turns out that the responses of the return on capital to capital and technology are λ3 (ηnk − 1) and λ3 (1 + ηna), respectively.
• These responses remain very small also in the variable-labor model.
• It is possible to check that: ωt = yt −nt, (50)
so that ηωk = ηyk −ηnk and ηωa = ηya −ηna. (Verify this as an exercise.)
• ηωa is smallest when utility is linear in leisure (γn = 0). In this case, ηωa = ηca, because linear utility from leisure generates a constant wage-consumption ratio.
• ηωa rises as labor supply becomes less elastic (i.e., as γn increases, or σn decreases).
86
Dynamics with Variable Labor Supply, Continued
• Variable labor supply has important implications for the short-run elasticity of output with respect to technology, ηya.
• When labor supply is fixed, ηya = α = .667.
• With variable labor supply, ηya = α + αµ (α−ηca), which can exceed 1 (it falls with φ, however, and it cannot exceed .99 when φ = 1).
• This is important, because an elasticity greater than 1 allows absolute declines in output to be generated by positive but slower-than-normal growth in technology, which is surely more
plausible than the notion of absolute declines in technology.
87
RBC Wrapping Up
• This concludes our analysis of the real business cycle model.
• As we noted when we began, key assumptions and results of the framework are not supported by evidence, but the model gives us a methodological starting point and
conceptual foundation.
• Chapter 14 of Sanjay Chugh’s textbook gives you a diagrammatic analysis of the model and its properties.
• We will begin departing from this basic framework by introducing the consequences of monopoly power next.
88
Appendix A: Risk Aversion and the Equity Premium Puzzle
• Consider the utility function u = c 1−γ
1−γ, which we are using in the RBC model and we will use
again in many other models. γ = −cu ′′(c)
u′(c) is called coefficient of relative risk aversion.
• Interpretation (Pratt, 1964, Econometrica): Suppose we offer two alternatives to a consumer who starts off with risk-free consumption level c: (s)he can receive c − π with certainty or a lottery paying c−y with probability .5 and c + y with probability .5.
• For given values of c and y, we want to find the value of π = π(y,c) that leaves the consumer indifferent between the two choices (the maximum amount the consumer is willing to pay in
order to avoid the bet).
• That is, we want to find π(y,c) such that:
u [c−π (y,c)] = .5u (c + y) + .5u (c−y)
• For given c and y, this non-linear equation can be solved for π.
89
Appendix A: Risk Aversion and the Equity Premium Puzzle, Continued
• Alternatively, for small y, use Taylor expansions and a local argument:
– Expansion of u (c−π):
u (c−π) = u (c) −πu′ (c) + O ( π2 ) , (51)
where we let c−π be the variable x in our expansion of f (x) = u (c−π) around x0 = c and O (·) means terms of order at most (·).
– Expansion of u (c + ỹ):
u (c + ỹ) = u (c) + ỹu′ (c) + 1
2 ỹ2u′′ (c) + O
( ỹ3 ) ,
where ỹ is the random variable that takes value y with probability .5 and −y with probability .5, and we let c + ỹ be the variable x in our expansion of f (x) = u (c + ỹ)
around x0 = c.
– We consider a second-order expansion here due to the randomness of ỹ, which requires
us to include second moments in the expansion.
90
Appendix A: Risk Aversion and the Equity Premium Puzzle, Continued
• Taking expectations of both sides of this equation yields:
Eu (c + ỹ) = u (c) + 1
2 y2u′′ (c) + o
( y2 ) , (52)
where o (·) means terms of smaller order than (·).
• Equating (51) and (52) and ignoring higher-order terms gives:
π (y,c) ≈ 1
2 y2 [ −U ′′ (c) U ′ (c)
] ,
or, if u (c) = c 1−γ
1−γ,
π (y,c) ≈ 1
2 y2 γ
c ,
which can be rearranged to
π (y,c)
y ≈
1
2 γ y
c .
• This tells us that the premium that the consumer is willing to pay to avoid a fair bet of size y is (approximately) equal to 1
2 γ times the ratio between the size of the bet and the consumer’s
initial level of consumption. γ characterizes the consumer’s attitude toward uncertainty and
is key to determine the premium (s)he is willing to pay to avoid it.
91
Appendix A: Risk Aversion and the Equity Premium Puzzle, Continued
• Now, think of confronting someone with initial consumption of $50, 000 per year with a 50-50 chance of winning or losing y dollars.
• Consider y = 10, 100, 1000, 5000. How much would the person be willing to pay to avoid that risk?
• Based on π = 1 2 γy
2
c :
γ
y 10 100 1000 5000
2 .002 .2 20 500
5 .005 .5 50 1250
10 .01 1 100 2500
• A common reaction to these premia is that for γ as high as 5, they are too big. This motivates most macroeconomists’ view that γ should not be much higher than 2 or 3.
92
Appendix A: Risk Aversion and the Equity Premium Puzzle, Continued
• Mehra and Prescott (1985, Journal of Monetary Economics) consider data on average yields on relatively riskless bonds and risky equity in the U.S. for the period 1889 -1978.
• The average real yield on the S&P 500 index was 7 percent. The average yield on short-term debt was only 1 percent, i.e., there was an equity premium of 6 percent.
• Let 1 + rit+1 denote the real rate of return on asset i between t and t + 1, i = b for bonds, i = s for stocks and look at the summary statistics below:
Mean Var-Cov
1 + rst+1 1 + r b t+1
ct+1 ct
1 + rst+1 1.070 .0274 .00104 .00219
1 + rbt+1 1.010 .00308 -.000193 ct+1 ct
1.018 .00127
• The presence of an equity premium is consistent with the theory: Stocks are riskier than bonds and therefore agents require a premium in order to hold them.
• But is a 6 percent spread justifiable within basic models given actual riskiness of stocks and bonds?
• No—and addressing this puzzle resulted in its own literature in macro-finance.
93
Appendix B: A Log-Linear Model of Fluctuations
• To see what happens when γ = 1 and δ = 1, observe that, with γ = 1, the Euler equation becomes:
C−1t = βEt ( C−1t+1Rt+1
) (53)
• Recall that the economy we are modeling is such that:
Yt = Ct + It, or 1 = Ct Yt
+ It Yt .
• Let It Yt ≡ s̃t = saving rate.
• Then, Ct Yt
= 1 − s̃t, or Ct = (1 − s̃t)Yt.
94
Appendix B: A Log-Linear Model of Fluctuations, Continued
• Thus, (53) implies:
− log(1 − s̃t) − log Yt = log β + log { Et
[ Rt+1
(1 − s̃t+1)Yt+1
]} (54)
• Yt = AtαK1−αt and δ = 1 imply:
Rt+1 = (1 −α) ( At+1 Kt+1
)α = (1 −α)
Yt+1 Kt+1
.
• Also, δ = 1 implies Kt+1 = Yt −Ct = s̃tYt. Hence,
Rt+1 = 1 −α st
( Yt+1 Yt
) .
95
Appendix B: A Log-Linear Model of Fluctuations, Continued
• Thus, (54) reduces to:
− log(1 − s̃t) − log Yt = log β + log { Et
[ (1 −α)
s̃t(1 − s̃t+1)Yt
]} = log β + log(1 −α) − log s̃t − log Yt + log
[ Et
( 1
1 − s̃t+1
)] ,
which implies:
log s̃t − log(1 − s̃t) = log β + log(1 −α) + log [ Et
( 1
1 − s̃t+1
)] . (55)
• Because technology and capital do not enter (55), there is a constant value of s̃t, ŝ, that satisfies (55).
96
Appendix B: A Log-Linear Model of Fluctuations, Continued
• To verify this, note that, if s̃t = ŝ ∀t, then
Et
( 1
1 − s̃t+1
) =
1
1 − ŝ ,
and (55) becomes:
log ŝ = log β + log(1 −α), or
ŝ = β(1 −α).
• Now, if st = ŝ = β(1 −α), it follows that: Ct Yt
= 1 −β(1 −α),
and:
Kt+1 = β(1 −α)Yt.
97
Appendix B: A Log-Linear Model of Fluctuations, Continued
• Given the production function Yt = AtαK1−αt , Kt = β(1 −α)Yt−1 yields:
Yt = At α [β(1 −α)]1−α Y 1−αt−1 ,
or
log Yt = (1 −α) log ŝ + (1 −α) log Yt−1 + α log At. (56)
• Equation (56) implies that, given assumptions on the process for log At, it is the possible to obtain exact solutions for the paths of all endogenous variables:
– Given assumptions on log At, we can use (56) to reconstruct the exact path of log Yt.
– We can then use log Kt+1 = log [β(1 −α)] + log Yt and log Ct = log [1 −β(1 −α)] + log Yt to reconstruct the exact paths of capital and consumption.
· Since capital is predetermined (capital at t + 1 is chosen at t), Kt+1 is a function of Yt.)
· Recall that the assumptions on utility and constraints that ensure a unique solution for the competitive equilibrium/planner’s problem are satisfied here.
· Hence, ŝ is the unique optimal saving rate when δ = 1 and γ = 1, so that we are assured that the model can be solved exactly under these assumptions.
98
Appendix C: Determinacy of the Solution
• To verify determinacy, we proceed as follows.
• Focus on the endogenous variables (consumption and capital) and on a perfect foresight version of (27).
• We do not need to worry about the exogenous shock variable at and about the expectation operator when verifying determinacy. Use the symbol = instead of ≈ to simplify notation.
99
Appendix C: Determinacy of the Solution, Continued
• We can write:
kt+1 = λ1kt + (1 −λ1 −λ2) ct, ct+1 = ct −σλ3kt+1 = ct −σλ3 [λ1kt + (1 −λ1 −λ2) ct]
= −σλ1λ3kt + [1 − (1 −λ1 −λ2) σλ3] ct.
• Or, in matrix form:[ kt+1 ct+1
] = M
[ kt ct
] , M ≡
[ λ1 1 −λ1 −λ2
−σλ1λ3 1 − (1 −λ1 −λ2) σλ3
] . (57)
100
Appendix C: Determinacy of the Solution, Continued
• Blanchard and Kahn (1980, Econometrica) showed that, for a system of linear, expectational difference equations such as (57) to have a unique solution, the number of eigenvalues
of the matrix M that lie (strictly) outside the unit circle must be equal to the number of
non-predetermined variables in the vector [ kt ct
]′ .
• Capital at time t was chosen at time t − 1. Hence, kt is a predetermined variable. Consumption–ct–is not predetermined. Therefore, we need an eigenvalue of M outside the
unit circle and one inside for the system (57) (and (27)) to have a determinate solution.
• To calculate the eigenvalues of M, we must solve:
det
[ λ1 − q 1 −λ1 −λ2 −σλ3λ1 1 − (1 −λ1 −λ2) σλ3 − q
] = q2 − [1 + λ1 − (1 −λ1 −λ2) σλ3] q + λ1 = 0. (58)
101
Appendix C: Determinacy of the Solution, Continued
• We can try to solve equation (58) by brute force or we can be smart. The latter method is best. :-)
• Consider: J (q) = q2 − [1 + λ1 − (1 −λ1 −λ2) σλ3] q + λ1.
• J (q) is a parabola. It is strictly convex, since J′′ (q) = 2 > 0.
• Graph J (q). If the parabola intersects the q-axis once inside the unit circle and once outside, we are done: The matrix M has an eigenvalue outside and an eigenvalue inside the unit
circle, and our system of expectational difference equations has a unique solution.
– The solution is stable (the model displays the desired property that variables return to
the steady state after temporary shocks) if the eigenvalue inside the unit circle is strictly
inside.
102
Appendix C: Determinacy of the Solution, Continued
• To graph J (q), compute:
J (0) = λ1 > 0,
J (1) = (1 −λ1 −λ2) σλ3, J(−1) = 2 (1 + λ1) − (1 −λ1 −λ2) σλ3,
lim q→−∞
J (q) = lim q→+∞
J (q) = +∞.
• Using the expressions for λ1, λ2, and λ3, you can verify that:
J (1) = − σα (r + δ) [r + αδ −g (1 −α)]
(1 + r) (1 −α) (1 + g) < 0 ⇔ r + αδ > g (1 −α) .
• To simplify our analysis, assume that structural parameter values are such that r + αδ > g (1 −α). Note that:
J (1) < 0 ⇒ J (−1) > 0.
103
Appendix C: Determinacy of the Solution, Continued
• Compute also: J′ (q) = 2q − [1 + λ1 − (1 −λ1 −λ2) σλ3] .
• Hence:
J′ (0) = − [1 + λ1 − (1 −λ1 −λ2) σλ3] , J′ (1) = 1 −λ1 + (1 −λ1 −λ2) σλ3,
J′ (−1) = −3 −λ1 + (1 −λ1 −λ2) σλ3.
• Note that J (1) = (1 −λ1 −λ2) σλ3 < 0 implies
J′ (0) < 0 and J′ (−1) < 0.
104
Appendix C: Determinacy of the Solution, Continued
• Therefore, the graph of J (q) is strictly positive and decreasing at q = −1 and q = 0.
• It crosses the q-axis once between 0 and 1 (this is a consequence of J (0) > 0, J′ (0) < 0, and J (1) < 0).
• At q = 1, J (q) may be increasing or decreasing, but, regardless of the sign of J′ (1), the second intersection of J (q) with the q-axis must happen to the right of 1.
– The fact that J (q) is a parabola, i.e., it switches from decreasing to increasing only once,
and J (1) < 0 rule out a second intersection to the left of 1.
• It follows that the roots of J (q) = q2 − [1 + λ1 − (1 −λ1 −λ2) σλ3] q + λ1 = 0 lie one inside and one outside the unit circle.
• Therefore, the eigenvalues of M are one inside and one outside the unit circle, and the system of expectational difference equations (27) has a determinate solution.
– If J (q) never intersects the q-axis, the eigenvalues of M are complex. In this case,
verifying determinacy would involve checking the norm of the eigenvalues. We focus on
the real case for simplicity.
105
Appendix D: Some Time Series Implications of the Basic Model
• Now define the lag operator L by: Lxt = xt−1.
• Using the lag operator, the log-linearized solution for capital, kt+1 = ηkkkt + ηkaat, can be written as:
kt+1 = ηka
1 −ηkkL at. (59)
• Using the same notation, the AR(1) technology process that we have assumed can be written as:
at = 1
1 −φL εt. (60)
106
Appendix D: Some Time Series Implications of the Basic Model, Continued
• Equations (59) and (60) imply that capital follows an AR(2) process:
kt+1 = ηka
(1 −ηkkL) (1 −φL) εt, (61)
or:
(1 −ηkkL) (1 −φL) kt+1 = ηkaεt ⇒[ 1 − (φ + ηkk) L + φηkkL
2 ] kt+1 = ηkaεt ⇒
kt+1 − (φ + ηkk) kt + φηkkkt−1 = ηkaεt ⇒
kt+1 = (φ + ηkk) kt −φηkkkt−1 + ηkaεt.
107
Appendix D: Some Time Series Implications of the Basic Model, Continued
kt+1 = (φ + ηkk) kt −φηkkkt−1 + ηkaεt.
• Two points on this:
– (a) The roots of the capital stock process are ηkk and φ, which are both real numbers.
· Thus, the model does not produce oscillating responses to shocks (which would happen with complex roots).
– (b) The shock to capital at t + 1 is the technology innovation realized at time t.
· The capital stock is known one period in advance as it is an endogenous state variable, determined by lagged investment and a non-stochastic depreciation rate.
108
Appendix D: Some Time Series Implications of the Basic Model, Continued
• Recall yt = (1 −α)kt + αat.
• With fixed labor supply, ηyk = 1 −α and ηya = α. Substitute (60) and (61) into
yt = (1 −α)Lkt+1 + αat.
• It follows that:
yt = (1 −α)ηkaL
(1 −ηkkL) (1 −φL) εt +
α
1 −φL εt
= α + [(1 −α)ηka −αηkk] L
(1 −ηkkL) (1 −φL) εt. (62)
• Technology innovations affect output both directly ( α 1−φLεt) and indirectly, through their
impact on capital accumulation ( (1−α)ηkaL
(1−ηkkL)(1−φL) εt).
109
Appendix D: Some Time Series Implications of the Basic Model, Continued
• The sum of the two effects is what we call an ARMA(2, 1) process for output:
(1 −ηkkL) (1 −φL) yt = αεt + [(1 −α)ηka −αηkk] Lεt,
or
yt = (φ + ηkk) yt−1 −φηkkyt−2 + αεt + [(1 −α)ηka −αηkk] εt−1, where (φ + ηkk) yt−1 − φηkkyt−2 is the AR(2) component of the process and αεt + [(1 −α)ηka −αηkk] εt−1 is the MA(1) part.
• The process for consumption comes from substituting (59) and (60) into ct = ηckkt + ηcaat:
ct = ηckηkaL
(1 −ηkkL) (1 −φL) εt +
ηca 1 −φL
εt
= ηca + (ηckηka −ηcaηkk) L
(1 −ηkk) (1 −φL) εt. (63)
• This too is an ARMA(2, 1) process:
ct = (φ + ηkk) ct−1 −φηkkct−2 + ηcaεt + (ηckηka −ηcaηkk) εt−1.
110
Appendix D: Some Time Series Implications of the Basic Model, Continued
• Note that capital, output, and consumption processes all have the same autoregressive roots ηkk and φ.
• Thus, we can reconstruct the entire path of the dynamic responses of k, y, and c to a technology innovation at an initial point in time (impulse responses).
• Generally, we will let the computer do this job for us and plot the responses.
– A set of Matlab codes written by Harald Uhlig of the University of Chicago in 1999
essentially implements the method of undetermined coefficients.
• However, there are cases—like the basic RBC model—in which models are sufficiently simple that we can solve for the elasticities η with pencil and paper, and, as noted above,
we can calculate impulse responses using Excel.
111
Appendix D: Some Time Series Implications of the Basic Model, Continued
• Of curse, the nature of the response (size of initial movement, shape, speed of return to the steady state–if this happens) depends on parameter values.
– If φ = 1, technology innovations have permanent effects, and the economy does not
return to the original steady state.
– Output converges to a new, permanently higher (or lower) steady-state path after a
one-time positive (or negative) technology shock with φ = 1.
• Campbell’s paper analyzes the consequences of different parameter values for the elasticities η and the characteristics of impulse responses.
– Note that Campbell allows for β > 1, which is not the usual assumption.
– Read this part of Campbell’s paper for more information.
112
Appendix D: Using the Time Series Process Equations to Obtain Impulse
Responses
• Suppose the economy was in steady state until time 0, and suppose ε0 > 0, εt = 0 ∀t > 0.
• We can use the equations we obtained above for the processes followed by capital, output, and consumption to compute the responses to the innovation ε0.
• At time t = 0:
k0 = 0 (because capital is predetermined),
y0 = αε0,
c0 = ηcaε0.
• At time t = 1:
k1 = ηkaε0,
y1 = (φ + ηkk) y0 + [(1 −α)ηka −αηkk] ε0 = (φ + ηkk) αε0 + [(1 −α)ηka −αηkk] ε0 = [αφ + (1 −α)ηka] ε0,
c1 = (φ + ηkk) c0 + (ηckηka −ηcaηkk) ε0 = (φ + ηkk) ηcaε0 + (ηckηka −ηcaηkk) ε0 = (ηckηka + ηcaφ) ε0,
113
Appendix D: Using the Time Series Process Equations to Obtain Impulse
Responses, Continued
• At time t = 2:
k2 = (φ + ηkk) k1 −φηkkk0 = (φ + ηkk) ηkaε0,
y2 = (φ + ηkk) y1 −φηkky0, where the solutions for y1 and y0 are above, c2 = (φ + ηkk) c1 + φηkkc0, where the solutions for c1 and c0 are above.
• And so on...
• As I suggested above, you can calculate exactly the same impulse responses directly using the equations:
kt+1 = ηkkkt + ηkaat,
yt = (1 −α) kt + αat, ct = ηckkt + ηcaat,
at = φat−1 + εt.
• Try it as an exercise: Set ε0 > 0, εt = 0 ∀t > 0 in the equations above, figure out the paths of k, y, and c, and verify that they coincide with those in the Excel example that used the
process equations for these variables.
114
Appendix E: The Steady State with Variable Labor Supply
•
Rt+1 = R = (1 −α) ( At+1Nt+1
Kt+1
)α + 1 − δ = (1 −α)
( At+1N
Kt+1
)α + 1 − δ ⇒
[ G β − (1 − δ) 1 −α
]1 α
= At+1N
Kt+1 ⇒
At+1N
Kt+1 ≈ ( r + δ
1 −α
)1 α
⇒
At+1
Kt+1 ≈
1
N
( r + δ
1 −α
)1 α
.
• Now, (40) implies:
θ ( 1 −N
)−γn = α
( At
Kt
)α ( Kt
Ct
) N −(1−α)
. (64)
115
Appendix E: The Steady State with Variable Labor Supply, Continued
• So, recalling At Kt ≈ 1
N
( r+δ 1−α )1 α ,
θ ( 1 −N
)−γn ≈ αN−α r + δ 1 −α
( Kt
Ct
) N −(1−α)
. (65)
• Also, (34) implies:
Kt+1
Kt = 1 − δ +
Y t
Kt − Ct
Kt ,
1 + g = 1 − δ + ( AtN
)α K 1−α t
Kt − Ct
Kt , (66)
or:
Ct
Kt =
( At
Kt
)α N
α − (g + δ) .
116
Appendix E: The Steady State with Variable Labor Supply, Continued
• But, using At Kt ≈ 1
N
( r+δ 1−α )1 α ,
Ct
Kt ≈ (
1
N
)α r + δ
1 −α N
α − (g + δ) = r + δ − (1 −α) (g + δ)
1 −α
= r + δ −g − δ + αg + αδ
1 −α = r + αδ −g (1 −α)
1 −α , (67)
which shows that Ct Kt
is the same as in the fixed-labor-supply model, and using (66) shows
that Y t Kt
is also the same as before.
• Equations (65) and (67) imply:
θ ( 1 −N
)−γn ≈ αN−1 r + δ r + αδ −g (1 −α)
. (68)
• Thus, N solves the non-linear equation (68).
117
Appendix F: Log-Linearizing the Variable-Labor Supply Model
• From the production function:
yt = α (at + nt) + (1 −α) kt.
• The law of motion of capital implies:
dKt+1 = (1 − δ) dKt + dYt −dCt,
or:
Kt+1
Kt
dKt+1
Kt+1 = (1 − δ)
dKt
Kt + Y t
Kt
dYt
Y t − Ct
Kt
dCt
Ct ,
from which:
(1 + g) kt+1 = (1 − δ) kt + Y t
Kt yt −
Ct
Kt ct,
where we showed before that Y t Kt
and Ct Kt
are the same as in the fixed-labor model.
118
Appendix F: Log-Linearizing the Variable-Labor Supply Model, Continued
• Using the log-linear production function yields:
kt+1 ≈ λ1kt + λ2 (at + nt) + (1 −λ1 −λ2) ct,
with λ1 and λ2 the same as before.
• Finally,
dRt+1 = (1 −α) α ( dAt+1N + dNt+1At+1
) Kt+1 −dKt+1
( At+1N
) K 2 t+1
( At+1N
Kt+1
)α−1 = (1 −α) α
At+1N
Kt+1 (at+1 + nt+1 −kt+1)
( At+1N
Kt+1
)α−1 = (1 −α) α
( At+1N
Kt+1
)α (at+1 + nt+1 −kt+1) .
• Then, using our results above,
rt+1 ≈ (1 −α) αr+δ
1−α (at+1 + nt+1 −kt+1) (1 −α) r+δ
1−α + 1 − δ = λ3 (at+1 + nt+1 −kt+1) ,
where λ3 ≡ α(r+δ)1+r , as in the fixed-labor model.
119
Appendix G: Solving the Variable Labor-Supply Model
• Recall equations (41), (43), and (45):
kt+1 ≈ λ1kt + λ2 (at + nt) + (1 −λ1 −λ2) ct, Et (ct+1 − ct) ≈ λ3Et (at+1 + nt+1 −kt+1) ,
nt ≈ µ [(1 −α) kt + αat − ct]
• Substitute (45) into (41):
kt+1 ≈ λ1kt + λ2at + λ2µ (1 −α) kt + λ2µαat −λ2µct + (1 −λ1 −λ2) ct,
or:
kt+1 ≈ [λ1 + λ2µ (1 −α)] kt + λ2 (1 + µα) at + [1 −λ1 −λ2 (1 + µ)] ct. (69)
• Substitute (45) into (43):
Et (ct+1 − ct) ≈ λ3Et [at+1 + µ (1 −α) kt+1 + µαat+1 −µct+1 −kt+1] = λ3Et{[µ (1 −α) − 1] kt+1 + (1 + µα) at+1 −µct+1} . (70)
120
Appendix G: Solving the Variable Labor-Supply Model, Continued
• Guess ct = ηckkt + ηcaat and substitute into (69):
kt+1 ≈ [λ1 + λ2µ (1 −α)] kt + λ2 (1 + µα) at + [1 −λ1 −λ2 (1 + µ)] ηckkt + [1 −λ1 −λ2 (1 + µ)] ηcaat
= ηkkkt + ηkaat,
with
ηkk ≡ λ1 + λ2µ (1 −α) + [1 −λ1 −λ2 (1 + µ)] ηck, ηka ≡ λ2 (1 + µα) + [1 −λ1 −λ2 (1 + µ)] ηca.
• Substitute kt+1 = ηkkkt + ηkaat and ct = ηckkt + ηcaat into (70).
• Use at = φat−1 + εt, so that Et (at+1) = φat, and the fact that kt+1 is known at time t.
121
Appendix G: Solving the Variable Labor-Supply Model, Continued
• Then,
ηck (kt+1 −kt) + ηca (φat −at) ≈ λ3{[µ (1 −α) − 1] kt+1 + (1 + µα) φat −µηckkt+1 −µηcaφat} ,
or:
ηck (ηkk − 1) kt + ηckηkaat + ηca (φ− 1) at ≈ λ3{[µ (1 −α) − 1 −µηck] (ηkkkt + ηkaat) + (1 + µα−µηca) φat} ,
and ηck solves:
ηck (ηkk − 1) = λ3 [µ (1 −α) − 1 −µηck] ηkk.
122
Appendix G: Solving the Variable Labor-Supply Model, Continued
• Recalling ηkk ≡ λ1 + λ2µ (1 −α) + [1 −λ1 −λ2 (1 + µ)] ηck, this equation becomes:
ηck {λ1 + λ2µ (1 −α) + [1 −λ1 −λ2 (1 + µ)] ηck}−ηck = λ3 [µ (1 −α) − 1 −µηck]{λ1 + λ2µ (1 −α) + [1 −λ1 −λ2 (1 + µ)] ηck} ,
which has the form:
Q2η 2 ck + Q1ηck + Q0 = 0,
with:
Q2 ≡ (1 + λ3µ) [1 −λ1 −λ2 (1 + µ)] , Q1 ≡ (1 + λ3µ) [λ1 + λ2 (1 −α) µ] −λ3 [µ (1 −α) − 1] [1 −λ1 −λ2 (1 + µ)] − 1, Q0 ≡ −λ3 [µ (1 −α) − 1] [λ1 + λ2 (1 −α) µ] .
123
Appendix G: Solving the Variable Labor-Supply Model, Continued
• Finally, ηca solves:
ηckηka + ηca (φ− 1) = λ3 [µ (1 −α) − 1 −µηck] ηka + λ3 (1 + µα−µηca) φ,
or:
ηca (φ− 1) + λ3µφηca = λ3 [µ (1 −α) − 1 −µηck] ηka + λ3 (1 + µα) φ−ηckηka.
• But ηka ≡ λ2 (1 + µα) + [1 −λ1 −λ2 (1 + µ)] ηca. Hence, substituting and rearranging,
ηca = (1 + αµ){λ3φ−λ2 [ηck (1 + λ3µ) −λ3 [µ (1 −α) − 1]]}
[1 −λ1 −λ2 (1 + µ)]{ηck (1 + λ3µ) −λ3 [µ (1 −α) − 1]}− [1 −φ (1 + λ3µ)] .
124
