Observing the Evolution in Macroeconomic Theory
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Munich Personal RePEc Archive Observing the Evolution in Macroeconomic Theory Podshivalov, Georgii The Victor Popov Academy 30 November 2019 Online at https://mpra.ub.uni-muenchen.de/97657/ MPRA Paper No. 97657, posted 18 Dec 2019 12:28 UTC Observing the Evolution in Macroeconomic Theory Georgii G. Podshivalov Abstract: The principal purpose of the given work is to summarize certain observations on the evolution of thought in macroeconomic theory with the original (rather than conventional) notation where appropriate. The observations are organized by topic and supplied with respective references. JEL Classification Numbers: B20, E12, E19, E21, E24, E31, E43, E49, E52 Keywords: Economic Though, Economic History, Macroeconomic Models, Models Derivation, Notation, Expectations, Growth, Consumption, Unemployment, Inflation, Random Walk Hypothesis, Money, Natural Rate Electronic copy available at: https://ssrn.com/abstract=3495882 2 1. Expectations Rational behavior suggests that economic agents intent to choose actions that maximize their benefits (minimize costs) in prolonged periods of time. To be able to evaluate the optimality of decisions affecting their future in present, agents need information about relevant conditions and trajectory of change in such conditions in future. Expectations are beliefs used by agents as substitutes to factual information about said conditions. Accounting for expectations leads to more detailed and correct reflection of agents’ behavior in economic models. Consider, for example, the CBOE Volatility index (VIX) futures fair value model FV=f(Pt,vart[FT])=100sqrt((365-30)(Pt-vart[FT]) where FV is the fair value of the futures contract on VIX, Pt – forward price of de- annualized variance in the 30 days after the futures expiration, and vart[FT] - concavity adjustment (“Additional Features of VIX Futures”). Since the settlement is conditioned on the “spot” value of VIX, it seems reasonable to expect actual futures prices to follow the model quite closely, as deviations will create arbitrage opportunity. In reality, however, this is not always the case (Rhoads 2015). Speculating on the nature of such deviations, one can observe that although VIX itself is influenced by expectations about the underlying market, there is no easy way to “spot” trade and directly affect its value. Expectations about VIX, though not reflected in its value, can nevertheless be observed in its derivatives’ prices. Therefore, CBOE futures model could be modified from FV=f(Pt,vart[FT]) to FV=f(Pt,vart[FT],Et), where Et stands for expectations over the “spot” value of VIX at time t. Augmented by expectations, such model is expected to be reflect the instrument more accurately. The early use of expectations can be found in Cobweb modeling of agriculture markets, essentially adding the expected price at time t+1 terms to conventional supply and demand functions (Kaldor 1934). Later, the adaptive expectations model was introduced by Philip Cagan and further developed by Milton Friedman to reflect permanent or expected income Yp (Chow 2011): Electronic copy available at: https://ssrn.com/abstract=3495882 3 2 Yp-Yp(t-1)=b(Y-Yp(t-1))=bY+(1-b)Yp(t-1)=bY(t)+b(1-b)Y(t-1)+b(1-b) Y(t-2)+… Dissatisfaction with the way the existed models reflected the dynamics, John Muth (1961) and later Robert Lucas (1976) put emphasis on the information circulation and the mechanics of decision making within the system. The resulting rational expectations approach was derived directly from the concept of mathematical expectation. Consider the following form of the model (Nimark 2016): πt=βEtπt+1+k(yt- t) y =E y - -E t t t+1 σ(it tπ푦̅t+1 it=фπt 2 t=p t-1+ut: ut~N(0, σu ), 푦̅ 푦̅ where πt is inflation, yt – output, t – potential output and it – interest rate. 푦̅ t=b t a t=βap t+k(b t- t) a-βap-kb=-k 푦̅πt=a 푦̅t → b 푦̅t=bp 푦̅t- σ(фa푦̅ t-푦̅ap t) → b-bp+σфa-σap=0 → yt=b푦̅t 푦̅ 푦̅ 푦̅ 푦̅ 푦̅ → → ф ф 1 − βp 푘 푎 −푘 푎 1 − βp −푘 −푘 ( )( ) = ( ) ( ) = ( ) ^T ( ) = σ − σ푝 1 − 푝 푏 0 푏 σ − σ푝 1 − 푝 0 =( 푝−1 2 −푘 푝+βp−βp −kσф+푘σp−1 σф−σp 2 ) −푘 푝+βp−βp −kσф+푘σp−1 References: “Additional Features of VIX Futures.” CBOE Futures Exchange, CBOE, cfe.cboe.com/cfe-education/cboe-volatility-index-vx-futures/vix-primer/vix-features. Chow, Gregory. Usefulness of Adaptive and Rational Expectations in Economics. Working paper no. 221. Princeton University, Sept. 2011. Web. 16 Oct. 2017. Electronic copy available at: https://ssrn.com/abstract=3495882 4 Kaldor, Nicholas. “A Classificatory Note on the Determinateness of Equilibrium.” The Review of Economic Studies, vol. 1, no. 2, Feb. 1934, pp. 122–136. JSTOR. Lucas, Robert E. “Econometric policy Evaluation: A Critique.” Carnegie-Rochester Conference Series on Public Policy, vol. 1, 1976, pp. 19-46. ScienceDirect. Muth, John F. “Rational Expectations and the Theory of Price Movements.” Econometrica, vol. 29, no. 3, 1961, pp. 315–335. JSTOR. Nimark, Kristoffer P. “Solving Linear Rational Expectations Models.” Kristoffer Nimark's Economics etc, 2016, www.kris-nimark.net/Applied_Macro_2016/Solving_RE_notes.pdf. Rhoads, Russell. “The History of VIX Contango and Backwardation.” CBOE Blogs, CBOE, 30 Aug. 2015, www.cboe.com/blogs/options-hub/2015/08/30/the-history-of-vix- contango-and-backwardation. 2. Solow Growth Model The demand for a new model of economic growth in the mid-1900s can be associated with criticism of once dominant Harrod-Domar model. In particular, this neo-classical model was criticized for its adjustment process depiction (Sato 1964). Solow model, at least in part, can be considered an answer to such demand. Assuming single commodity and implying closed economy with no government intervention, Solow (1956) treats flow of capital over time as augmented function of output saved (s), namely dK/dt=sF(K,L0e^(nt)) where the exogenous growth of initial labor force Lo is represented by n (the model was further extended to include multiplier A(t), so that Q=A(t)f(K,L)). Introducing capital-labor ratio r=K/L, K=rL=rL0e^(nt), we are led to dK/dt=L0e^(nt)(dr/dt)+nrL0e^(nt) and, consequently, to Electronic copy available at: https://ssrn.com/abstract=3495882 5 (dr/dt+nr)L0e^(nt)=sL0e^(nt)F(K/(L0e^(nt)),1) for constant returns to scale. This leads to the final differential equation dr/dt=sF(r,1)-nr where s is the exogenous savings rate. Steady-state can be found at the intersection nr=sF(r,1) where dr/dt=0 yields us r*; that is, the equilibrium ratio of capital to labor. Using constant return to scale that Solow (1957) considers “practically unavoidable”, such r* will be stable, so that for other r≠r* there will be tendency to converge to r* (unless the assumptions on the shape of productivity curve F(r,1) will allow other equilibria as well). The following Inada (1963) conditions are applied: f’(k)>0 (here and in other literature k is conveniently used for capital-labor ratio, that is, k=K/L), f”(k)<0, =0 (Inada originally used wage-rental ratio that can be skipped here for simplicity’s sake),lim푡→0 푓(푘) =∞ and =0. 푘→0lim 푓′(푘) 푘→∞lim 푓′(With푘) this, the equation can be rewritten as dk/dt=sf(k)-nk or, more extensively, as dk/dt=sAk(t)-(d+n)k(t). Solving, we set sAk*=(d+n)k* we are led to k*=(sA)/(d+n), s*=(d+n)/A. Returning to Solow’s original notation, one can define consumption as c=F(r,1)-(n+d)r, where d represents exogenous capital depreciation rate. Maximizing in c, one is led to F’(r)=n+d; hence the optimal “golden rule” savings rate s*=((n+d)r*)/F(r*). One important consideration of Solow model corresponds to the standard utility maximization logic. As additional output that is not consumed is impractical, we consider the set of conditions that maximizes consumption over time to be optimal. The model’s optimality was explored by Phelps (1961) by elaborating on the example of the Golden Age (hence the name) in the fictional Kingdom of Solovia. Phelps defines consumption as c=(1-s)f(s)e^(gt), where g stands for parametric “natural rate of output growth” (exogenous). The consumption function is further maximized to yield Electronic copy available at: https://ssrn.com/abstract=3495882 6 -f(s)e^(gt)+(1-s)f’(s)e^(gt)=0 → f(s)=(1-s)f’(s), which is indeed maximum by the second derivative test. One advantage of this form is that it clearly shows the consequences of excessive savings (so 1-s=0). The significance of the model was hard to overestimate at its time, as it attempted to answer questions about the disparities in economic growth between different countries and economic systems. The other significant features of the model are its non-linearity and dynamic nature. Later more sophisticated dynamic models of Rebelo, Romer and Lucas seem to be at least in part based on Solow’s findings. One major criticism of Solow model is its treatment of factors augmenting labor and capital as exogenous (the rate of technical progress). As the economy became less industrialized by the end of the century, the significance of ideas and knowledge demanded greater attention. Models of Endogenous growth theory attempted to solve this problem by expanding the understanding of such augmentation and inserting it inside the model (Romer 1986). References: Inada, Ken-Ichi. “On a Two-Sector Model of Economic Growth: Comments and a Generalization.” The Review of Economic Studies, vol.