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Erschienen in:

2019 | OriginalPaper | Buchkapitel

1. Long-Run Growth: The Basic Framework

verfasst von : Jin Cao, Gerhard Illing

Erschienen in: Money: Theory and Practice

Verlag: Springer International Publishing

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Abstract

This chapter focuses at the real economy, introducing important concepts for inter-temporal analysis which play a key role in modern monetary policy analysis. We start with a simple two-period endowment economy with one representative agent. We illustrate the concepts of wealth constraint, inter-temporal optimization with time preferences, marginal rate of substitution (MRS), and the real rate interest. We derive optimality conditions for the inter-temporal path of consumption (the Euler equation) and the natural rate of interest as key equilibrium condition. We show that our endowment can be interpreted as a base line model for a growing economy. As next step, we introduce government spending and taxation and extend our base line model by introducing the labor market, making production endogenous. We derive optimal price setting under monopolistic competition in an economy with heterogeneous firms for the case that all firms can adjust their prices optimally without cost. The general equilibrium conditions for this economy provide an important reference point for later analysis, characterizing the natural level of employment, potential output, and the corresponding natural real rate of interest in the absence of nominal rigidities.

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Nächstes Kapitel Money and Long-Run Growth

Standard references are Christiano, Eichenbaum, and Evans (2005) and Smets and Wouters (2007). See also Christiano, Eichenbaum, and Trabandt (2018).

That is, the good perishes completely if it is not consumed in the same period. Check to what extent the analysis must be modified when we allow for goods to be stored with some decay rate $ \delta \in \left[0,1\right] $.

In standard microanalysis, consumers choose the optimal bundle, given their budget constraint. In general equilibrium, only relative prices are determined—the real rate of exchange between different commodities. The absolute price level is in determined and irrelevant. So the price of some arbitrary good can be normalized to one. In nearly all real economies, however, most prices are quoted in terms of a specific numeraire—the particular commodity which is used as money or medium of exchange. Modern economies use fiat money—a medium of exchange which has no intrinsic value. A key challenge for monetary economics is to provide a theory explaining how the price level is determined in a fiat money regime. We will come back to this issue in later chapters.

Nothing depends on this specific mechanism (the quantity theory of money). Crucial is that the price level (and thus the rate of inflation) is determined by the central bank. It may control the price level by other means, such as a strategy of inflation targeting. Later chapters discuss challenges to implement the target in different ways like a money growth rule or an interest rate rule.

Indexed (inflation-linked) bonds are not frequently used. One reason may be that it is hard to specify all contingencies in detail, so such payments characterize an incomplete contract. It is not easy to specify exactly how to index payments to changes in the “true” rate of inflation. What commodity basket should be used? How to allow for adjustments in the way changes in the price level are calculated? Contractual arrangements with indexation are used mainly for long-term rent contracts. Governments also issue to some extent indexed bonds for financing debt (see the Focus Box on Inflation Linked Bonds).

This has to be modified when the assets have different risk characteristics.

When the Fisher relation holds, we get the equivalent formulation merging the nominal budget constraints:

$$ {C}_t+\frac{1}{\left(1+{i}_t\right)\cdot {P}_t/{P}_{t+1}}{C}_{t+1}={Y}_t+\frac{1}{\left(1+{i}_t\right)\cdot {P}_t/{P}_{t+1}}{Y}_{t+1}. $$

We can derive the $ MRS $ by total differentiation of inter-temporal preferences, holding overall utility constant:

$$ dV={U}_{C_t}\left({C}_t\right)\ d{C}_t+\beta\ {U}_{C_{t+1}}\left({C}_{t+1}\right)\ d{C}_{t+1}=0. $$

That is, when the second derivative is negative $ {U}_{C_t\ {C}_t}\left({C}_t\right)<0. $

Usually, the Euler equation is derived using complex dynamic techniques, such as dynamic programming, calculus of variation, or the maximum principle. For our purposes, however, it is sufficient to derive the Euler equation in the following straightforward way (For a more general case, we need to use the Lagrangian—see the Appendix A in the Instructor’s Manual for Money: Theory and Practice). The wealth constraint defines a relation between consumption tomorrow and today:

$ {C}_{t+1}\left({C}_t\right)={Y}_{t+1}+\left(1+{r}_t\right)\left({Y}_t-{C}_t\right) $ with $ \partial {C}_{t+1}/\partial {C}_t=-\left(1+{r}_t\right) $. Inserting this constraint into the utility function, allows us to state overall utility simply a function of current consumption $ {C}_t $:

$ \hat{V}\left({C}_t\right)=U\left({C}_t\right)+\beta U\left({C}_{t+1}\left({C}_t\right)\right)= $$ U\left({C}_t\right)+\beta U\left({Y}_{t+1}+\left(1+{r}_t\right)\left({Y}_t-{C}_t\right)\right) $

Differentiating $ \hat{V}\left({C}_t\right) $ with respect to$ {C}_t $, the first-order condition for an interior optimum is:

$ \frac{\partial \hat{V}\left({C}_t\right)}{\partial\ {C}_t}=\frac{\partial U\left({C}_t\right)}{\partial\ {C}_t}+\beta \frac{\partial U\left({C}_{t+1}\right)}{\partial\ {C}_{t+1}}\frac{\partial\ {C}_{t+1}}{\partial\ {C}_t}= $$ {U}_{C_t}\left({C}_t\right)-\left(1+{r}_t\right)\beta {U}_{C_{t+1}}{\left({C}_{t+1}\right)}_{=}^{!}\ 0 $

Reformulating this first-order condition gives the Euler equation in the text.

This is an approximation for discrete time analysis. It holds exactly for small time intervals. See the Focus Box on “Constant Elasticity of Inter-temporal Substitution.”

Note, however, that this reasoning no longer holds when financial frictions prevent smooth intertemporal trade. If so, we cannot rule out $ r<\gamma. $ The natural real rate of interest may even get negative as in OLG models as Eggertsson, Mehrotra, and Robbins (2017) or in infinitely lived agent models with non-satiation in wealth—see Illing, Ono, and Schlegl (2018).

In contrast, the overlapping generations model (OLG), by construction, has heterogeneous agents with limited trade across different generations.

No rational consumer is willing to buy any bonds in the final period $ t+1 $ (compare the Focus Box on the No-Ponzi Game Constraint). After all, the world is ending at $ t+1 $. So rather than buying bonds at that time, the representative consumer will always be better off by consuming her resources. Obviously, just as for private agents, we need to impose a No-Ponzi game constraint also for the government: Since public debt at the end of the world cannot be positive, the condition $ {B}_{t+1}\le 0 $ must hold. With the government’s No-Ponzi game condition being binding, we get the transversality condition $ {B}_{t+1}/\left(1+{i}_t\right)=\left\{{B}_{t+1}/{P}_{t+1}\right\}\cdot 1/\left(1+{r}_t\right)=0 $. Of course, this condition needs to be modified if there is no definite final period. In that case, debt may indeed never be repaid. But nevertheless, we still get as feasibility condition that the present value of government spending plus the initial debt must be covered by the present value of revenues (see Sect. 3.4).

Write $ \frac{C_t}{Y_t}=1-\frac{G}{Y_t}=1-{g}_t $. Taking logs on both sides and use $ \log \left(1+Z\right)\approx Z $, we get $ {c}_t-{y}_t=-{g}_t $ or $ {y}_t={c}_t+{g}_t $ (Note that $ {g}_t $ is here the share of government spending to real GDP, not the log of G!).

Baxter, M., & King, R. (1993). Fiscal policy in general equilibrium. American Economic Review, 83(3), 315–334.

Blanchard, O., & Kiyotaki, N. (1987). Monopolistic competition and the effects of aggregate demand. American Economic Review, 77, 647–666.

Christiano, L., Eichenbaum, M., & Evans, C. (2005). Nominal rigidities and the dynamic effects of a shock to monetary policy. Journal of Political Economy, 113(1), 1–45.CrossRef

Christiano, L., Eichenbaum, M., & Rebelo, S. (2011). When is the government spending multiplier large? Journal of Political Economy, 119(1), 78–121.CrossRef

Christiano, L., Eichenbaum, M., & Trabandt, M. (2018). On DSGE models? Journal of Economic Perspectives, 32(3), 113–140.CrossRef

Dixit, A., & Stiglitz, J. (1977). Monopolistic competition and optimum product diversity. American Economic Review, 67(3), 297–308.

Eggertsson, G., Mehrotra, N., & Robbins, J. (2017). A model of secular stagnation: Theory and quantitative evaluation. NBER working papers 23093.

Fisher, I. (1896). Appreciation and interest. Publications of the American Economic Association, First Series, 11(4), 1–110.

Guvenen, F. (2011). Macroeconomics with heterogeneity: A practical guide. Federal Reserve Bank of Richmond Economic Quarterly, 97(3), 255–326.

Illing, G., Ono, Y., & Schlegl, M. (2018). Credit booms, debt overhang and secular stagnation. European Economic Review, 108, 78–104.CrossRef

Monacelli, T., & Perotti, R. (2008). Fiscal policy, wealth effects, and markups. NBER Working Papers 14584.

Smets, F., & Wouters, R. (2007). Shocks and frictions in US business cycles: A Bayesian DSGE approach. American Economic Review, 97(3), 586–606.CrossRef

Titel: Long-Run Growth: The Basic Framework
verfasst von: Jin Cao
Gerhard Illing
Verlag: Springer International Publishing
Buch: Money: Theory and Practice
Print ISBN: 978-3-030-19696-7

Electronic ISBN: 978-3-030-19697-4

Copyright-Jahr: 2019
DOI: https://doi.org/10.1007/978-3-030-19697-4_1