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Public Economics
This chapter presents the Ramsey model. It is the benchmark model for most dynamic macroeconomic models that study growth and business cycle phenomena. We first study the deterministic Ramsey model in which the total factor productivity is certain. We contrast the effects of a onceandforall change with those of a temporary change in productivity on investment, output, and labor supply. In addition, we distinguish the effects of this change when it is known in advance or only observed at the beginning of the period, t, when the shock occurs. Finally, we also introduce uncertainty with respect to the technology level and discuss the real business cycle (RBC) model.
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One way to justify this assumption is that a household with a finite lifetime also cares about the utility of its descendants and applies the same discount factor
β to their (representative) lifetime utility.
Take care to distinguish between the discount factor
β and the discount rate
θ > 0 that is given by
$$\displaystyle \begin{aligned} \frac{1}{1+\theta} = \beta \;\; \Leftrightarrow \;\; \theta=\frac{1}{\beta}1.\end{aligned} $$
Why have we added ‘− 1’ in the nominator of the utility function in (
2.2) in the case
σ≠1? First notice that the additive constant − 1∕(1 −
σ) does not change the solution of the utility maximization problem and, therefore, does not affect optimal consumption. Furthermore, we know from calculus that
$$\displaystyle \begin{aligned} \lim_{x\to 0} \left( \frac{a^x1}{x}\right) =\ln a.\end{aligned} $$
Therefore,
\(\ln c\) is just the limit of the function (
c
^{1−σ} − 1)∕(1 −
σ) for
σ → 1.
In order to derive the limit formula above, notice that from the
L’Hôspital rule—which states that if the functions
f(
x) and
g(
x) in the nominator and denominator have the limit equal to zero, lim
_{x→∞}
f(
x) = 0 and lim
_{x→∞}
g(
x) = 0, the value of the limit
\(\lim _{x\to 0} \left (f(x)/g(x)\right )\), if it exists, is given by
\(\lim _{x\to 0} \left (f'(x)/g'(x)\right )\)—implies
$$\displaystyle \begin{aligned} \lim_{x\to 0} \left( \frac{a^x1}{x}\right) = \lim_{x\to 0} \left( \frac{a^x \ln a}{1}\right) = \ln a.\end{aligned}$$
Appendix 2.1 derives the IES in a simplified twoperiod model.
The elasticity of substitution
σ
_{p} is defined as follows:
$$\displaystyle \begin{aligned} \sigma_p = \frac{\frac{d\left( \frac{K}{L}\right)}{\frac{K}{L}}}{\frac{d\left( \frac{w}{r}\right)}{\frac{w}{r}}},\end{aligned}$$
where
w and
r denote the marginal products of labor and capital.
You are asked to compute the dynamics for the case in which
σ
_{p} = 3∕4 in Problem
2.1.
More generally, an
Euler equation is the intertemporal firstorder condition for a dynamic choice problem and is usually formulated as a difference of differential equation. Equation (
2.12) is also referred to as the
KeynesRamsey rule that describes the growth rate of consumption as a result of intertemporal utility maximization.
By local stability we mean that if we perturb the initial condition slightly, then the system stays in the neighborhood of that steady state. If we use the term global stability, the system returns to the steady state even if the starting point is not very close to the steady state.
A recommendable introduction to the methods of calibration is provided by deJong and Dave (
2011).
Of course, we should check whether this number of periods is sufficient to guarantee a smooth approximation of the new steady state. If not, we should increase the number of periods. For example, I first used 40 periods and found the number to be insufficient. Use the computer code and test for different values of the number of periods.
Appendix 2.2 provides an overview of how this numerical problem can be solved. The MATLAB/Gauss programs
Ch2_ramsey1.m/
Ch2_ramsey1.g compute the solution presented in Figs.
2.1,
2.2,
2.3, and
2.4 and can be downloaded from my homepage with all the other programs used in this book.
The argument for this result is straightforward: The central planner could also choose to behave exactly the same in the case of an expected change as in the case of an unexpected change. Since he chooses a different policy, this must be superior, and it yields a higher value of the objective function.
You are asked to compute the Jacobian and its value in Problem
2.2.
If you take the eigenvalues of the Jacobian provided in (
2.17), the eigenvalues are slightly different due to rounding errors. I used the value of the Jacobian with an accuracy of 10
^{−8} to compute the eigenvalues
ρ
_{1} and
ρ
_{2}.
In a twodimensional difference equation system, the steady state is a saddle if one of the eigenvalues has an absolute value below one and the other above. The steady state is locally saddlepath stable if one of the two variables is predetermined and the other is a jump variable (not predetermined). (In addition, divergent paths must be ruled out by boundary conditions.) To learn more about the stability analysis in systems of difference equations, consult Azariadis (
1993).
We used the condition “
c
_{t} =
c
_{t−1}” rather than “
c
_{t+1} =
c
_{t}” so that both functions which are graphed in Fig.
2.5 have the same argument
k
_{t} (and not
k
_{t+1} as in the case “
c
_{t+1} =
c
_{t}”).
To verify this statement, differentiate
c
_{t} with respect to
k
_{t} and solve for
\(k_t= \left (\frac {\alpha }{\delta +n}\right )^{1/(1\alpha )}\). For
β < 1, the value of
k
_{t} is above the steady state
\(k= \left ( \frac {\alpha }{(1+n)/\beta 1+\delta }\right )^{1/(1\alpha )}\).
One can show that all transition paths that start above S remain above and, similarly, that all paths that start below S remain below it. In addition, paths with the same initial capital stock
k
_{0} but with different consumption values
c
_{0} do not cross.
The closer we are to the steady state, the better the fit of our linear approximation will be.
In the case of a real matrix
\(\tilde T\), the inverse
\(\tilde T^{1}\) of a unitary matrix is just the transpose
\(\tilde T'\).
Standard software, such as MATLAB or Gauss, provides commands to compute the Schur factorization. MATLAB also provides a routine,
ordschur(.), that can change the order of the eigenvalues if needed.
For those readers interested in numerical linear algebra, a Givens rotation is represented by a matrix transformation. In our problem, we search for a matrix
$$\displaystyle \begin{aligned} G=\left( \begin{array}{cc} d & e\\ e & d \end{array}\right)\end{aligned}$$
that helps to transform
\(\tilde S\) into
\(S=G\tilde S\).
Notice that the coefficient of the firstorder difference equation is equal to the stable root of the Jacobian,
ρ
_{1} = 0.9242.
For a better illustration of the dynamics during the early periods 1–20, I only used 40 periods for the number of transition periods. Although the adjustment is not complete after 40 periods, the approximation is close during periods when the technology shock increases to
Z
_{t} = 1.1. The MATLAB/Gauss program
Ch2_ramsey2.m/
Ch2_ramsey2.g computes the solution presented in Fig.
2.9.
In 2004, Finn E. Kydland and Edward C. Prescott received the Nobel prize for their research on RBCs.
The
second theorem of welfare economics states that any efficient allocation can be sustained by a competitive equilibrium and, thus, constitutes the converse of the first theorem.
Two basic types of business cycle models are presented by RBC models, in which only real variables enter the model, and New Keynesian models, in which nominal variables enter the model and prices and/or wages are sticky. Both types of business cycle models are described in greater detail in Heer and Maußner (
2009), McCandless (
2008), and Cooley (
1995).
The notation of rational expectation was originally introduced by Muth (
1961).
Some time series are also available as monthly data, e.g., industrial production and employment. Some other economic variables such as distributional measures of income and consumption concentration in the form of their Gini coefficients, however, are only available on an annual basis, rendering the analysis of the shortterm distributional effects of economic policy more difficult.
In some studies, the technology level follows a unit root process with trend
$$\displaystyle \begin{aligned} \ln Z_t = \ln Z_{t1} + a +\epsilon_t, \;\;\; \epsilon_t \sim N(0,\sigma^Z),\end{aligned}$$
where
a denotes the drift or growth rate of total factor productivity. The modeling of the technology process (and, more generally, time series of macroeconomic variables) is not an innocuous assumption and affects businesscycle results. For example, Cogley and Nasan (
1995) demonstrate that if prefiltered series are firstorder integrated, then HPfiltering of the series may result in business cycles that do not exist in the original prefiltered data.
Basu, Fernald, and Kimball (
2006) construct a measure of technology change in the presence of variable capacity utilization and imperfect competition.
The calibration of RBC models with respect to the characteristics of other industrialized countries employs similar values, e.g., Heer and Maußner (
2009) estimate
ρ
^{Z} = 0.90 and
σ
^{Z} = 0.0072 for the German economy.
Notice that we interchanged the derivative and the expectational operator to derive the firstorder conditions using:
$$\displaystyle \begin{aligned} \frac{d}{dx} \mathbb{E} f(x,Z) = \mathbb{E}\frac{d}{dx} f(x,Z).\end{aligned}$$
This condition holds if
f(
x,
Z) is integrable for all
x and
f is differentiable with respect to
x. Furthermore, the expected value of
Z is finite,
\(\mathbb {E}(Z)<\infty \). The above equation is a special application of the
Leibniz integral rule according to which
$$\displaystyle \begin{aligned} \frac{d}{dx} \left( \int_{a(x)}^{b(x)} f(x,Z)\; dZ \right)= f(x,b(x))\cdot \frac{d}{dx} b(x)f(x,a(x))\cdot \frac{d}{dx} a(x)+ \int_{a(x)}^{b(x)} \frac{\partial }{\partial x} f(x,Z) \;dZ. \end{aligned}$$
Therefore, our approximation is fairly close to the steady state but becomes increasingly inaccurate with increasing distance from the steady state. Linear approximation is a useful technique for the behavior of economies during tranquil times. During periods of severe crisis such as the Great Recession of 2007–2008, one should instead apply global approximation methods, as described in Chapters 5 and 6 in Heer and Maußner (
2009).
The computation of the policy functions is described in greater detail in
Appendix 2.3.
The MATLAB and Gauss programs,
Ch2_rbc.m and
Ch2_rbc.g, compute the policy functions, the impulse responses, and the time series statistics.
Empirical studies such as Galí (
1999) and Basu, Fernald, and Kimball (
2006) find that a positive technology shock led to a contraction of labor inputs. The standard RBC model is inconsistent with this observation. In Sect.
4.5.2, we present a New Keynesian model with sticky prices and adjustment costs of capital that is able to account for this fact.
Sometimes, the researcher also cuts the first 50 periods or similarly from the simulation so that the initialization of the state variables in the first period with their steady state values does not have any effect on the results.
The paper had already circulated as a discussion paper two decades earlier and was then introduced as a working paper by Hodrick and Prescott in 1980 that was published in 1997.
For a more detailed description of this filter and its computation, see for example, Chapter 12.4 in Heer and Maußner (
2009).
The data are described in more detail in
Appendix 2.4.
The statistics are computed with the help of the MATLAB or Gauss programs
Ch2_data.m and
Ch2_data.g.
For the riskfree rate and the real equity return rate, we restrict our attention to the period 1959:Q2–2015:Q2. To construct the inflation rate for the computation of the real return, we use the price index for Private Consumption Expenditures (Excluding Food and Energy), which is only available during the period 1959:Q1–2015:Q2.
The parameter
β = 0.99 in the RBC model is often calibrated to imply an annual real interest rate of 4%, which is a midpoint between the real returns of TBills and US equity.
In the case of the interest rate or equity return, which are already measured in percentage points, we do not take the log but rather apply the HP filter to the original series.
This approximation follows from a firstorder Taylor series expansion
$$\displaystyle \begin{aligned} f(x) \approx f'(x_0) (xx_0) \end{aligned}$$
with
\(f(x)=\ln (1+x)\) and
x
_{0} = 0 implying:
$$\displaystyle \begin{aligned} \ln (1+x) \approx \frac{1}{1+x_0} (xx_0)= x.\end{aligned}$$
In Problem
2.6, you are asked to analytically demonstrate this result.
The toolbox is available as the Gauss source file
toolbox.src from my homepage.
This appendix is intended to offer a short introduction to the ideas of solution methods using a simple example. A much more detailed technical description with a generalization to multidimensional problems is provided in Chapter 2.4 of Heer and Maußner (
2009) or in Chapter 6.8 of McCandless (
2008).
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 Title
 Ramsey Model
 DOI
 https://doi.org/10.1007/9783030009892_2
 Author:

Burkhard Heer
 Publisher
 Springer International Publishing
 Sequence number
 2
 Chapter number
 2