Abstract
This is a book on Growth Theory and on the numerical methods needed to fully characterize the properties of most Growth models. In this introductory chapter, we describe the main characteristics of different families of Growth models and their relevance for policy analysis, which is moving leading economic and financial institutions throughout the world to increasingly rely on their use for forecasting as well as for policy evaluation. In particular, we emphasize how the richer structure provided to Growth models by their Microeconomic foundations allows us to address a much broader set of policy issues than in more traditional structural dynamic models.
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Notes
- 1.
That amounts to constructing the forecast by application of the conditional expectation operator to the analytical representation of the future value being predicted, where the conditional expectation is formed with respect to the sigma algebra of events known at time t.
- 2.
This is the limit of a random variable, and an appropriate limit concept must be used. It suffices to say that the power of ρ going to zero justifies the zero limit for the product random variable.
- 3.
That is, if the innovation \(\varepsilon _{t}\) has zero variance.
- 4.
When working with several variables, responses can be obtained for impulses in more than one variable. To make the size of the responses comparable, each innovation is supposed to take a value equal to its standard deviation, which may be quite different for different innovations.
- 5.
Or significantly increasing the innovation variance. What are the differences between both cases in terms of the values taken by the process?
- 6.
The two polynomials can be written as \(1 - a_{1}B - a_{2}B^{2} = (1 - B)(1 -\lambda B),\) the second root being 1∕λ. The reader just need to find the value of λ in each case.
- 7.
We could have done otherwise, like starting the first-order autoregresisons at their mathematical expectation, and the second-order autoreegressions outside their expected values. The reader can experiment with these changes.
- 8.
As it would be obtained by a profit-maximizing competitive firm with a Cobb–Douglas technology, \(Y = a_{0}K^{a_{1}}L^{a_{2}},a_{1} + a_{2} \leq 1,\) represented in logs by the first relationship, with \(d_{0} =\ln (a_{0}a_{1}).\)
- 9.
A system with as many equations as endogenous variables.
- 10.
We assume here, for simplicity, that all random shocks are white noise. Extending the model to incorporate possible autoregressive structures for the shocks is straightforward.
- 11.
If we denote by p i the i-th row of the s × q matrix P, then \(\mathit{Var}(z_{i}) = p_{i}^{{\prime}}\varSigma _{\varepsilon }p_{i},\) \(\mathit{Var}(z_{j}) = p_{j}^{{\prime}}\varSigma _{\varepsilon }p_{j},\) \(\mathit{Cov}(z_{i},z_{j}) = p_{i}^{{\prime}}\varSigma _{\varepsilon }p_{j},\) and \(\mathit{Corr}(z_{i},z_{j}) = \frac{p_{i}^{{\prime}}\varSigma _{\varepsilon }p_{j}} {\sqrt{p_{i }^{{\prime} }\varSigma _{\varepsilon }p_{i}}\sqrt{p_{j }^{{\prime} }\varSigma _{\varepsilon }p_{j}}},\) with \(\varSigma _{\varepsilon }\) being the q × q variance–covariance matrix of vector \(\varepsilon.\)
- 12.
If, for instance, \(C_{t},C_{t-1}\) and C t−2 appear in the model, both, C t and C t−1 will form part of vector z t , while C t−1 and C t−2 will be included in vector z t−1. The representation could also be extended easily to accommodate lagged innovation values.
- 13.
Which is known as Cholesky identification strategy, from the way how a factor decomposition of the variance–covariance matrix of the original innovations is used to produce the linear transformation of the system of equations.
- 14.
Alternatively, we could have considered a process with some inertia for Government expenditures, or even change the model to make the value of Government expenditures to be related to the past level of output, for instance.
- 15.
The choice of the steady-state level as initial condition is arbitrary. However, in this stochastic version of the model that choice is as good as any other, since the economy is already going to experience fluctuations due to the stochastic component of government expenditures.
- 16.
Notice the difference between computing relative volatility by the ratios of standard deviations or through the ratios of the coefficients of variation, the latter option being preferable.
- 17.
The autocorrelation function is the sequence of values \(\mathit{Corr}(Y _{t},Y _{t-s}),\) for all s.
- 18.
This suggests no evidence of residual autocorrelation, a potential source of misspecification in the consumption equation.
- 19.
This is arbitrary. We should take an impulse of size equal to one standard deviation of the innovations estimated from actual time series data, since that is the likely single-period fluctuation in each variable.
- 20.
This is, in fact, very important, since the structure and implications of a model may significantly change by just a change in assumptions on the timing of decisions, the arrival of information, or the opening and closing of markets.
- 21.
This modelling approach is now commonplace in Macroeconomics. Dynamic models with microeconomic foundations for aggregate economies are often used in Public Finance, Monetary Theory, Labour Economics or International Economics, as they are used in Growth theory. The main difference for the latter is their focus on characterizing the main determinants of short- and long-run growth.
- 22.
A standard result in intermediate Microeconomics courses.
- 23.
Of course, different utility functions could give raise to different functional forms for the way how current consumption relates to future consumption and interest rates.
- 24.
With δ being the percent per-period depreciation rate of capital.
- 25.
In consistency with the utility maximization problem above, we can either assume that there is a single consumer or household in the economy, or interpret labor and capital stock in this equation in per-capita terms.
- 26.
And also on conditional expectations of nonlinear functions of future state and decision variables, in the case of stochastic growth models, as we will see in the next paragraph.
- 27.
Under endogenous prices, optimization problems solved by economic agents do not have a linear-quadratic structure, implying that their decision rules are non-linear. Since these decisions are part of the system summarizing the model, that system ends up being nonlinear as well.
Sargent’s Macroeconomic Theory (1979) contains a variety of partial equilibrium models in which, with exogenous prices, optimization problems have a linear-quadratic structure. In that simple setup, decision rules are linear functions.
- 28.
Expectations of future variables or functions of variables appearing in a model need to be treated as new variables, so that a model that includes an explicit role for expectations is not complete without incorporating some kind of assumption on the way agents form their expectations. The assumptions on the expectations formation mechanism play the role of additional equations. They are a crucial part of a stochastic model, as important as the assumptions on the functional form of the utility function or the aggregate production function, and affect the model implications regarding the time behavior for the endogenous variables.
- 29.
These expectations mechanisms are said to be backward-looking, since they are substituted by a function of past variables, agents’ views about the future not playing any role.
- 30.
Alternative specifications for limited rationality, in which agents are assumed to form expectations which are partially rational, have been shown to be useful to explain some regularities in actual time series data.
- 31.
- 32.
This summary is intended to provide an overview to readers unfamiliar with Growth theory. We do not have any pretension of being fully comprehensive.
- 33.
As mentioned, the model also had implications regarding the convergence of economies in terms of per-capita income, which developed a huge empirical literature aiming to test such implications that is still very much alive, now in reference to more sophisticated growth models that have been developed since then. Along this line of reasoning, growth theory would not be very different from other areas of economic theory that imply more or less tight restrictions among the joint behavior of variables, that can be reduced to parameter testing in relatively simple econometric models.
- 34.
Kydland and Prescott [23] point out: “In other words, modern business cycle models are stochastic versions of neoclassical growth theory. And the fact that business cycle models do produce normal-looking fluctuations adds dramatically to our confidence in the neoclassical growth theory model - including the answers it provides to growth accounting and public finance questions.”
- 35.
In any event, like in any other Growth model, the relationships among per capita variables emerging from the model will generally be non-linear, and a linear econometric model might be too poor an approximation to them.
- 36.
Constant returns to scale in the single cumulative input as a reason for positive long-term growth is the characteristic of the AK economy, introduced by Rebelo [29]. An explicit role for public capital as a productive input was proposed by Barro [2]. The model with a variety of intermediate goods is due to Spence [38], Dixit and Stiglitz [15], Ethier [17] and Romer [31, 32]. Uzawa [41], Lucas [25] and Caballé and Santos [6] assigned an explicit role to the stock of human capital in the production of the final good.
- 37.
As shown in its Web page: http://www.ecb.int/home/html/researcher.en.html.
- 38.
- 39.
These equations can be fitted to data by recently developed econometric methods (Generalized Method of Moments). The idea is that analogous sample moments should not be very different from the theoretical moments implied by the model,
For instance, the stochastic moment condition in page 37, under time-varying taxes, can be written:
$$\displaystyle{ E_{t}\left [ \frac{1} {C_{t}} -\beta \left (1 + (1 -\tau _{t+1})r_{t+1}\right ) \frac{1} {C_{t+1}}\right ] = 0, }$$which, for any variable Z t in the information set on which the conditional expectations E t is formed, it implies:
$$\displaystyle{ E\left [Z_{t}\left ( \frac{1} {C_{t}} -\beta \left (1 + (1 -\tau _{t+1})r_{t+1}\right ) \frac{1} {C_{t+1}}\right )\right ] = E\left [h(Z_{t},X_{t},\theta )\right ] = 0, }$$suggesting that we estimate by solving the optimization problem:
$$\displaystyle{ \mathop{\mathit{Min}}\limits_{\beta,\delta }\sum _{t=0}^{\infty }\left [Z_{t}\left ( \frac{1} {C_{t}} -\beta \left [1 + (1 -\tau _{t+1})r_{t+1}\right ] \frac{1} {C_{t+1}}\right )\right ]^{2}, }$$where we have one such condition for each chosen Z t -variable and each function with a zero conditional expectation.
More generally, the optimization problem:
$$\displaystyle{ \mathop{\mathit{Min}}\limits_{\theta }\left [H(Z_{t},X_{t},\theta )AH(Z_{t},X_{t},\theta )^{{\prime}}\right ] }$$is solved, where \(H(Z_{t},X_{t},\theta ) = \left (h_{1}(Z_{t},X_{t},\theta ),h_{2}(Z_{t},X_{t},\theta ),\ldots,h_{k}(Z_{t},X_{t},\theta )\right )\), with the h j (. ) functions being cross products of Z t -variables and expressions like the one inside the bracket above, and A is a kxk matrix of weights, which conditions the statistical efficiency of the implied estimates.
- 40.
For a discussion of analytical solution methods for lineal rational expectations models, see Whiteman [42].
- 41.
We do not pretend these methods to be superior in any sense to those not covered in the chapter. They have been chosen because of their relative simplicity. An introduction to more complex, but possibly more exact methods, is also provided in that chapter.
- 42.
Unless we work under the assumption of rational expectations, the model’s implications regarding the way agents’ expectations relate to state variables are generally hard to derive.
- 43.
What is called a bubble equilibrium.
- 44.
Welfare should be understood as the discounted time aggregate value of current and future utility. We are thinking here about a set of identical consumers, who live together forever, a usual assumption in growth models.
- 45.
However, the appropriate approach to use frequency distributions from the alternative models to evaluate in probability terms (or in likelihood terms) their ability to fit the data is still very much open to discussion. And so it is the selection of statistics whose value in actual data should be replicated by the theoretical models considered.
- 46.
Exogebous shocks could be modelled as shocks in productivity, as it is done often throughout the book.
- 47.
Alternatively, we could consider the possibility of maintaining tax rates unchanged and finance the fluctuations in expenditures by debt management or money injections. Appropriate conditions guaranteeing long-run solvency would then have to be imposed, as it is discussed at different points in this textbook.
- 48.
Of course, the type of results reached by Poole [27] in a static setup, that “in the presence of supply shocks it is better to implement a monetary policy aimed to maintaining a given growth rate of money, while leaving interest rates to be determined in the market, the opposite being true if randomness enters mainly through the demand side” is another result typical from the type of analysis described in these sections.
- 49.
And hence, in which per-capita variables display zero growth.
References
Aghion, P., and P. Howitt. 1992. A model of growth through creative destruction. Econometrica 80(2): 323–351.
Barro, R.J. 1990. Government spending in a simple model of endogenous growth. Journal of Political Economy 98(5): S103–S126.
Barro, R.J., and X. Sala-i-Martin. 1997. Technological diffusion, convergence, and growth. Journal of Economic Growth 2(1): 1–26.
Bayoumi, T., D. Laxton, and P. Pesenti. 2004. Benefits and spillovers of greater competition in Europe: A macroeconomic assessment. ECB Working Paper, No. 341, European Central Bank.
Blanchard, O., and C.M. Kahn. 1980. The solution of linear difference models under rational expectations. Econometrica 48(5): 1305–1311.
Caballe, J., and M. Santos. 1993. On endogenous growth with physical and human capital. Journal of Political Economy 101: 1042–1067.
Cagan, P. 1956. The monetary dynamics of hyperinflation. In Studies in the quantity theory of money, ed. M. Friedman, 25–117. Chicago: University of Chicago Press.
Calvo, G. 1983. Staggered prices in a utility maximizing framework. Journal of Monetary Economics 12: 383–398.
Canova, F. 2007. Methods for applied macroeconomic research. Princeton: Princeton University Press.
Cass, D. 1965. Optimum growth in an aggregative model of capital accumulation. Review of Economic Studies 32: 233–240.
Castañeda, A., J. Diaz-Gimenez, and J.V. Rios-Rull. 1998. Exploring the income distribution business cycle dynamics. Journal of Monetary Economics 42: 93–130.
Coenen, G., and V. Wieland. 2000. A small estimated euro area model with rational expectations and nominal rigidities. European Economic Review 49: 1081–1104.
Coenen, G., P. McAdam, and R. Straub. 2008. Tax reform and labor-market performance in the Euro area: a simulation-based analysis using the new area-wide model. Journal of Economic Dynamics and Control 32(8): 2543–2583.
De Jong, D.N., and C. Dave. 2007. Structural macroeconometrics. Princeton: Princeton University Press.
Dixit, A.K., and J. Stiglitz. 1977. Monopolistic competition and optimum product diversity. American Economic Review 67: 297–308.
Erceg, C.J., L. Guerrieri, and C. Gust. 2005. SIGMA: A new open economy model for policy analysis. International Finance Discussion Papers No. 835. Board of Governors of the Federal Reserve System, July.
Ethier, W.J. 1982. National and international returns to scale in the modern theory of international trade. American Economic Review 72: 389–405.
Gali, J., and M. Gertler. 1999. Inflation dynamics: A structural econometric analysis. Journal of Monetary Economics 44(2): 195–222.
Howitt, P., and P. Aghion. 1998. Capital accumulation and innovation as complementary factors in long-run growth. Journal of Economic Growth 3: 111–130.
Jones, L.E., and R. Manuelli. 1990. A convex model of economic growth. Journal of Political Economy 98(5): 1008–1038.
King, R.G., C.I. Plosser, and S. Rebelo. 1988. Production, growth, and business cycles: II. New directions. Journal of Monetary Economics 21: 309–341.
Koopmans, T.C. 1965. On the concept of optimal economic growth. In The economic approach to development planning. North-Holland, Amsterdam.
Kydland, F.E., and E.C. Prescott. 1996. The computational experiment: An econometric tool. Journal of Economic Perspectives 10(1): 69–85.
Lucas, R.E. 1976. Econometric policy evaluation: A critique. Carnegie-Rochester Conference Series on Public Policy.
Lucas, R.E. 1988. On the mechanism of economic development. Journal of Monetary Economics 122: 3–42.
Marcet, A., and W.J. den Haan. 1990. Solving nonlinear stochastic models by parameterizing expectations. Journal of Business and Economic Statistics 8: 31–34.
Poole, W. 1970. Optimal choice of monetary policy instruments in a simple stochastic macro model. Quarterly Journal of Economics 84(2): 197–216.
Ramsey, F. 1928. A mathematical theory of saving. Economic Journal 38: 543–559.
Rebelo, S. 1991. Long-run policy analysis and long-run growth. Journal of Political Economy 99(3): 500–521.
Rios-Rull, J.V. 1996. Life-cycle economies and aggregate fluctuations. Review of Economic Studies 63: 465–490.
Romer, P.M. 1987. Growth based on increasing returns due to specialization. American Economic Review 77(2): 56–62.
Romer, P.M. 1990. Endogenous technological change. Journal of Political Economy Part II 98(5): S71–S102.
Schumpeter, J.A. 1934. The theory of economic development. Cambridge: Harvard University Press.
Sidrauski, M. 1967. Rational choice and patterns of growth in a monetary economy. American Economic Revenue 57(2): 534–544.
Sims, C.A. 2001. Solving linear rational expectations models. Journal of Computational Economics 20: 1–20.
Smets, F., and R. Wouters. 2003. An estimated dynamic stochastic general equilibrium model of the Euro area. Journal of the European Economic Association 1(5): 1123–1175.
Solow, R.M. 1956. A contribution to the theory of economic growth. Quarterly Journal of Economics 70(1): 65–94.
Spence, M. 1976. Product selection, fixed costs, and monopolistic competition. Review of Economic Studies 43(2): 217–235.
Swan, T.W. 1956. Economic growth and capital accumulation. Economic Record 32: 334–361.
Uhlig, H. 1999. A toolkit for analyzing nonlinear dynamic stochastic models easily. In Computational methods for the study of dynamic economics, ed. R. Marimon and A. Scott, 30–61. Oxford: Oxford University Press.
Uzawa, H. 1964. Optimal growth in a two sector model of capital accumulation. Review of Economic Studies 31(1): 1–24.
Whiteman, C.H. 1983. Linear rational expectations models: A user’s guide. Minneapolis: University of Minnesota Press.
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Novales, A., Fernández, E., Ruiz, J. (2014). Introduction. In: Economic Growth. Springer Texts in Business and Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54950-2_1
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