Monetary and Fiscal Dynamics

A macroeconomic shock induces an extended process of adjustment. Accordingly the present monograph focuses on the following questions: How does the process of adjustment look? What problems are involved in it? How can these problems be solved? The analysis will be carried out within the framework of an IS-LM model characterized by capital and wage dynamics.

In the current section, we shall present an overlapping generations model without bequests, confer Diamond 1965. It offers the monetary analysis of a stationary economy. The aim is to furnish a microfoundation for the (long-run equilibrium in the) basic model. Why do we postulate a stationary economy? For ease of exposition, without losing generality. And why overlapping generations? Because in a Solow model of a stationary economy, no steady state does exist.

The process of adjustment can be viewed as a sequence of temporary equilibria which converge to a permanent equilibrium. In the current section, we shall inaugurate the temporary equilibrium, where money wages and the stock of capital are given exogenously. Later on, in section 4, the permanent equilibrium will be installed, where money wages and the stock of capital have adjusted completely.

At first we shall establish the pertinent IS-LM diagram. Substitute (2), (4) and (5) into (1) from section 2, respectively, and solve for r to get the equation of the IS curve: 1$$r = \frac{{\alpha \lambda Y}}{{\beta \delta \mu Y + \lambda K - \mu K}}$$ If λ<μ, an increase in Y causes a reduction in r, which empirically seems to be sound. Conversely, if λ>μ,an increase in Y brings about an increase in r. Henceforth, it will be assumed throughout that: 2$$\lambda < \mu $$Accordingly, the IS curve is downward sloping, see figure 1. Over and above that, r rises as K goes up, so the IS curve shifts to the right. How is income affected by capital formation? Obviously, two counteracting forces are at work. On the one hand, capital accumulation means wealth accumulation. Therefore households save less and consume more, thus stimulating aggregate demand and output. On the other hand, the higher the stock of capital, the less firms do invest, thereby curbing aggregate demand and output. Under the premise λ< μ, the first channel appears to be stronger.

In the current section, we shall ascertain the long-run equilibrium of the basic model. This is a permanent equilibrium where the process of adjustment has been finished:1$$\dot{K}=I=0$$2$$\dot{W}=0$$ Firms do no longer invest, so the stock of capital is invariant. And money wages have settled down.

The short-run equilibrium can be condensed into a system of two differential equations: 1$$\dot{k} = f(K, w)$$2$$\dot{w} =g(k,w)$$ Here the question arises whether the long-run equilibrium is stable. By applying phase diagram techniques, we shall try to give an answer to this question.

In sections 6 until 10, we shall keep track of the process of adjustment generated by various macroeconomic shocks: The quantity of money declines spontaneously, the propensity to save rises on its own, labour supply increases autonomously, investment is being reduced, since sales expectations worsen, or money wages spring up exogenously. First of all, have a look at a monetary disturbance. Strictly speaking, the quantity of money contracts suddenly. As a reaction, how do aggregate demand, employment and capital develop over time?

Initially let the economy be in the long run equilibrium. In this situation, a savings shock happens: The preference for future consumption δ increases autonomously. In the phase diagram, both demarcation lines shift to the right such that the steady state level of K rises and the steady state of w falls, see figure 1. Once more the streamline suggests along which path the economy will move.

At the start, let the economy be in the steady state. Under these circumstances, a demographic shock takes place: Labour supply increases on its own. In the phase diagram merely the $$\dot{w}=0$$ line goes to the right, see figure 1. For the subsequent process of adjustment, have a look at the streamline. In the short run, the disturbance leaves no impact whatsoever on aggregate demand, output and labour demand. That is why the economy will suffer from unemployment.

In the current section, K* = αE/r will be substituted for K* = αY/r, where E denotes expected sales. Initially, the economy rests in the long-run equilibrium. Particularly, expected sales conform with actual sales E = Y. Against this background, an investment shock comes about: Sales expectations worsen exogenously. Then, after some time, expected sales will again agree with actual sales endogenously E = Y. In full analogy to section 3, the equation of the IS curve can be established: 1$$ r = \frac{{\alpha \lambda E}}{{\beta \delta \mu Y + \lambda K - \mu {\rm K}}} $$

Initially let the economy be in the steady state. In this situation, a wage shock happens: Money wages rise exogenously. In the phase diagram, both the $$ \dot{K}=0 $$ and $$ \dot{w}=0 $$ lines stay put, cf. figure 1. In the short run, firms are to increase prices, thereby contracting real balances. The interest rate jumps up, which reduces the desired stock of capital. For that reason, investment, aggregate demand and output drop. Firms have to lay off some workers, so unemployment comes into existence. In the phase diagram, money wages spring up, whereas the stock of capital remains unaffected.

In the preceding sections, as a rule, we assumed slow money wages. Now in sections 11 and 12, as an exception, we shall consider the polar cases of flexible and fixed money wages. Let us begin with flexible money wages. As a response to a shock, money wages adjust continuously so as to clear the labour market all the time.

In the current section, money wages are supposed to be fixed. Apart from this, we shall take the same approach as before. More precisely, w = const will be substituted for the Phillips curve $$\dot{w}=\varepsilon w\left( \operatorname{N}\sqrt{\operatorname{N}}-1 \right)$$, cf. (7) in section 2. Accordingly, the short—run equilibrium can be described by a system of eight equations: 1$$Y=\operatorname{C}+\operatorname{I}$$2$$C=\left( 1-\beta \delta \mu \right)\operatorname{Y}+\mu K$$3$$Y={{K}^{\alpha }}{{N}^{\beta }}$$4$${{K}^{*}}=\alpha Y/r$$5$$\operatorname{I}=\lambda \left( {{\operatorname{K}}^{*}}=\operatorname{K} \right)$$6$$\dot{K}=\operatorname{I}$$7$$w/p=\beta Y/N$$8$$\operatorname{M}/\operatorname{P}=\operatorname{Y}/{{\operatorname{r}}^{\eta }}$$ Here α, β, δ, η, λ, μ, w, K and M are given exogenously, while p, r, C, I, K*, $$\dot{K}$$, N and Y are endogenous variables. It is worth noting that the short—run equilibrium does not depend on labour supply. In addition, the IS—LM equation coincides with that acquired for slow money wages: 9$$\frac{\alpha \lambda Y}{\beta \delta \mu Y+\lambda K-\mu K}={{\left\{ \left. \frac{\operatorname{w}{{\operatorname{Y}}^{1/\beta }}\operatorname{K}{{-}^{\alpha /\beta }}}{\beta M} \right\} \right.}^{1/\eta }}$$

In the preceding section, money wages were supposed to be fixed, which appears to be a limiting case. In the current section, we shall come back to the premise that money wages are a slow variable, which seems better to suit facts. Given a macroeconomic shock, monetary policy can aim at three targets at least. First, the central bank restores full employment now, thereby incurring underemployment (or overemployment) later on. Second, the monetary authority brings back full employment in the long run. During the process of adjustment, however, unemployment, persists. And third, the central bank maintains full employment at all times. This calls for a kind of dynamic monetary policy. On that grounds, the third avenue will be taken here. More accurately: As a response to a shock, the central bank continuously adjusts the quantity of money so as to always keep up full employment. As a corollary, money wages do not move. Can this strategy be sustained?

Take for example a monetary shock. In the preceding sections, we considered a monotonic adjustment, which seems to be the most simple case. In the current section, instead, we shall study a cyclical adjustment, which may occur, too.

The investigation will be conducted within an overlapping generations model without bequests (Diamond 1965). The current section offers the real analysis of a stationary economy. The purpose is to deliver a microfoundation for (the long-run equilibrium in) the extended model.

At first have a look at the goods market. Firms produce as much as households, firms and the government want to buy Y = C + I + G, hence the goods market clears. In doing this, firms employ capital K and labour N. For ease of exposition, let technology be of the Cobb-Douglas type Y = Kα Nβ with α > 0, β > 0 and α + β = 1. Firms maximize profits Π = pY − rpK − wN under perfect competition. From this follows r = ∂Y/∂K = αY/K. As a consequence, the interest rate accords with the marginal product of capital. This in turn yields the desired stock of capital K* = αY/r. Likewise one obtains w/p = ∂Y/∂N = βY/N. That means, real wages harmonize with the marginal product of labour.

The long-run equilibrium is defined by the fact that money wages, public debt and the stock of capital do no longer move:1$$\dot{W}=0$$2$$\dot{D}=B=0$$3$$\dot{K}=I=0$$ As an implication, the budget is balanced, and firms stop to invest.

In sections 4 until 6, we shall discuss the stability of the long-run equilibrium. In addition, we shall trace out the process of adjustment released by various macroeconomic shocks. In doing this, it will prove useful to distinguish between flexible, fixed and slow money wages. To begin with, in the current section, we shall postulate flexible money wages. Strictly speaking, as a response to a shock, money wages adjust continuously so as to clear the labour market at all times.

So far we assumed either slow or flexible money wages. Now we shall suppose that money wages are fixed. Relying on this premise, the short-run equilibrium can be described by a system of twelve equations: 1$$\text{Y=C+I+G} $$2$$Y={{K}^{\alpha }}{{N}^{\beta }} $$3$$\text{K*= }\!\!\alpha\!\!\text{ Y/r}$$4$$I=\lambda \left( K*-K \right)$$5$$\dot{K}=I $$6$$T=t\left( Y+rD \right) $$7$$B=G+rD-T $$8$$\dot{D}=B $$9$$Y+rD=C+S+T $$10$$S=\beta \delta \mu \left( 1-t \right)Y-\mu \left( D+K \right) $$11$$w/p=\beta Y/N $$12$$M/p=Y/{{r}^{\eta }}$$ Here α, β, δ, η, λ, μ, t, w, D, G, K, M and $$\bar{N}$$ are given exogenously, while p, r, B, C, $$\dot{D}$$, I, K*, $$\dot{K}$$, N, S, T and Y are endogenous variables.

In the current section, we shall assume slow money wages. The short-run equilibrium and the long-run equilibrium have been derived above, cf. sections 2 and 3. As a fundamental result, under both flexible and fixed money wages, the long-run equilibrium is unstable, as has been demonstrated in sections 4 and 5. As a consequence, under slow money wages, the long-run equilibrium will be unstable, too. This outcome is in sharp contrast to the conclusions drawn in the basic model, where the long-run equilibrium turned out to be stable. That is to say, the public sector creates long-run instability.

So far we started from the premise that government purchases and the tax rate are given exogenously. In this situation, the government budget constraint equals B = G + rD − t(Y + rD). And the budget deficit adds to public debt $$\dot{D}$$ = B. Now, as an exception, we shall posit that the government continuously adjusts public consumption so as to always balance the budget. Under these circumstances, the government budget constraint simplifies to G = tY. As a corollary, the budget balances B = 0, and there will be no public debt D = 0.

In the preceding sections of part II we learned that macroeconomic shocks involve problems like underemployment or fatal crowding out. In the current section, the limelight will be directed at monetary policy, which offers a radical change of perspective. More accurately, as a response to a shock, the central bank continuously adjusts the quantity of money so as to maintain full employment at all times. Accordingly there is no reason why money wages should move. Can this strategy be sustained? By the way, we return to the standard assumption that public consumption and the tax rate are given exogenously, thereby abandoning continuous budget balance.

In the literature, throughout, fiscal policy has been defined by the fact that the government increases public consumption once and for all. Here, in sharp contrast, this will be called a fiscal shock. Instead, we shall define fiscal policy in the following way: As a response to a shock, the government continuously adjusts public consumption so as to maintain full employment at all times. On that account, there is no need for money wages to move. As a consequence, will public debt displace private capital?

How does monetary policy perform as compared to fiscal policy? The answer seems to depend on the type of shock. To begin with, consider a monetary contraction. Under monetary policy, the disturbance has no real effects, neither in the short nor in the long period. Under fiscal policy, on the other hand, the disturbance reduces the stock of capital to a large extent. Clearly this outcome is not optimal. For that reason, monetary policy wins over fiscal policy.

In the current section, as a frame of reference, the Solow model will be sketched out briefly, offering the real analysis of a growing economy. Firms manufacture a single product by making use of capital and labour. For ease of exposition, consider a Cobb-Douglas technology Y = Kα Nβ with α > 0, β > 0 and α + β = 1. Output can be devoted to consumption and investment Y = C + I. Households save a certain fraction s = const of income S = sY. Savings are invested I = S, thereby adding to the stock of capital $$ \dot{K} $$ = I. Moreover let labour grow at the natural rate $$ \dot{N} $$/N = n = const. Now it is convenient to state this in per capita terms. Output per head y = Y/N is a well-known function y = kα of capital per head k = K/N. Next take the time derivative of k = K/N and rearrange adequately $$ \dot{k}\ $$ = $$ \dot{K} $$/N − (K/N)($$ \dot{N} $$/N). Then substitute $$ \dot{K} $$ = I = S = sY and $$ \dot{N} $$/N = n, observing y = kα, to arrive at: 1$$ \dot{k}=s{{k}^{\alpha}}-nk $$

First of all have a look at the goods market. Output is determined by consumption plus investment Y = C + I, hence the goods market is in equilibrium. Firms produce a homogenous commodity with the help of capital and labour. Assume a Cobb-Douglas technology Y = Kα Nβ with α > 0, β > 0 and α+ β = 1. Firms maximize profits II = pY − rpK − wN under perfect competition. For that reason, the interest rate corresponds to the marginal product of capital r = ∂Y/∂K = αY/K. From this one can deduce the desired stock of capital K* = αY/r. Likewise real wages agree with the marginal product of labour w/p = ∂Y/∂N = βY/N. Properly speaking N denotes labour demand which adjusts endogenously. On the other hand $$\overline{N}$$ symbolizes labour supply which is given exogenously. Let labour supply $$\overline{N}=\overline{{{N}_{0}}}{{e}^{n\Gamma }}$$ expand at the natural rate n = const, where τ stands for time.

In the current section, we shall inaugurate the pertinent IS-LM and AD-AS diagrams. Let us start with the IS curve. Substitute (13), (15) and (16) from section 2 into (12) and solve for r: 1$$r=\frac{\alpha \lambda y}{sy+\left( \lambda -n \right)k}$$ This is the IS equation. If λ ≶ n, then dr/dy ≶ 0. In order to obtain a downward sloping IS curve, we posit 2$$\lambda <n,$$ see figure 1. And an increase in k causes an increase in r, so the IS curve goes to the right. Put another way, a rise in capital per head leads to a rise in income per head. Obviously two counteracting forces are at work. On the one hand, a rise in capital per head lifts that level of investment per head which is required to keep up capital per head. This in turn raises aggregate demand and income, in per capita terms, respectively. On the other hand, according to the flexible accelerator λ(k* − k), the rise in capital per head lowers investment, aggregate demand and income, again in per capita terms. Granted λ < n, the first channel of transmission will dominate.

In the steady state, the motion of capital per head and money wages comes to a halt: 1$$ \dot{k}=0 $$2$$ \dot{w}=0 $$ Get rid of $$ \dot{w} $$ in $$ \dot{w}=\varepsilon w(N/\bar{N}-1) $$ by making use of (2) to attain: 3$$ N=\bar{N} $$ Therefore full employment prevails. Now divide $$ Y={{K}^{\alpha }}{{\bar{N}}^{\beta }} $$ through by $$ \bar{N} $$, which yields: 4$$ y={{k}^{\alpha }} $$ Then observe (1)in $$ \dot{k}=i-nk $$: 5$$ i=nk $$ Moreover substitute this together with c = (1 − s)y into y = c + i to acquire: 6$$ sy=nk $$ Further combine (4) as well as (6) and reshuffle terms: 7$$ k={{(s/n)}^{1/\beta }} $$8$$ y={{(s/n)}^{\alpha /\beta }} $$

The short-run equilibrium can be condensed to a system of two differential equations: 1$$\dot{k}=\operatorname{f}\left( k,w \right)$$2$$\dot{w}=\operatorname{g}\left( \operatorname{k},\operatorname{w} \right)$$ This poses the question whether the long-run equilibrium will be stable or not. By adopting phase-diagram techniques, we shall try to answer this question. First of all, the short-run equilibrium can be ascertained by cutting the IS and LM curves, see (1) and (3) in section 3: 3$$\frac{\alpha \lambda \operatorname{y}}{sy+\left( \lambda -\operatorname{n} \right)\operatorname{k}}={{\left\{ \frac{w{{y}^{1/\beta }}}{\beta m{{k}^{\alpha /\beta }}} \right\}}^{1/\eta }}$$ From this follows the equilibrium income in per capita terms y.

Initially the economy rests in the long-run equilibrium. In particular, money per head is constant. All workers have got a job, so money wages do not change. And investment per head exactly sustains capital per head, thus output per head is uniform. In the phase diagram, the crossing of the demarcation lines indicates the steady state, see figure 1. Against this background, a monetary shock occurs: Money per head diminishes autonomously. In the phase diagram, the $$ \overset{\centerdot }{\mathop{k}}\, $$ = 0 line shifts to the left, while the $$ \overset{\centerdot }{\mathop{w}}\, $$ = 0 line travels rightwards. And the streamline exhibits how the economy develops over time. In the short run, the disturbance raises the interest rate, thereby lowering desired capital per head. That is why investment, aggregate demand and output are reduced, in per capita terms, respectively. What is more, labour demand falls short of labour supply, hence unemployment emerges.

At the beginning, the economy is in the long-term equilibrium. Under these circumstances, the savings ratio increases on its own. In the phase diagram, both demarcation lines move to the right such that capital per head goes up and money wages come down. For the streamline see figure 1. In the short term, this impulse reduces consumption, aggregate demand and output, in per capita terms, respectively. On that grounds, unemployment comes into existence. The decline in income calls forth a decline in money demand, which lowers the interest rate and raises desired capital per head. As a consequence, investment per head mounts.

Investment per head declines abruptly, since sales expectations worsen. This reduces aggregate demand and sales, in per capita terms, respectively. Accordingly the economy suffers from underemployment. Investment per head falls short of the required level, so capital per head diminishes. More precisely, sales drop by less than expected. Because of that investment and aggregate demand recover, in per capita terms, respectively. This in turn improves sales, hence the economy switches to overemployment. After some time, investment per head surpasses the critical level, thus replenishing capital per head. For the streamline see the phase diagram in figure 1. In due course, the economy gravitates towards a steady state. Ultimately both money wages and capital per head come back to their starting point.