Steady solutions before and after the policy shock.

## Abstract

Thus far involuntary unemployment does not occur in Diamond-type Overlapping Generations models. In line with Keynesian macroeconomics, involuntary unemployment is traced back to aggregate demand failures. While macro-economists majority refers aggregate demand failures to sticky prices, a minority attributes lacking aggregate demand to not perfectly flexible aggregate investment. The chapter investigates how an independent aggregate investment function causes involuntary unemployment under perfectly flexible competitive wage and interest rates in a Diamond-type neoclassical growth model with public debt and human capital accumulation. Moreover, it is shown that a higher public debt to output ratio enhances output growth and reduces involuntary unemployment.

### Keywords

- involuntary unemployment
- overlapping generations models
- inflexible aggregate investment
- public debt
- human capital accumulation

## 1. Introduction

Involuntary unemployment in Diamond-type Overlapping Generations (OLG) models seem to be a contradiction in terms. As in Solow [1]‘s neoclassical growth model, Diamond assumed full employment of the workforce for the OLG economy with production and capital accumulation. Thus, in this economy, unemployment is purely voluntary. Moreover, fiscal policy does not impact the steady-state growth rate of gross domestic product (GDP) since the output growth is exogenously determined. Both, voluntary unemployment and growth ineffectiveness of fiscal policy, do not accord well with the current state of the pandemic-affected world economy which is characterized by high involuntary unemployment and enormous government expenditures to compensate lockdown-related private losses. To be capable to address the effectiveness of fiscal policy to reduce involuntary unemployment, an additional extension of Diamond [2]‘s seminal OLG model towards endogenous growth becomes inevitable.

As is well-known, involuntary unemployment is usually associated with Keynesian macroeconomics [3, 4]. Involuntary unemployment is traced back to lacking aggregate demand. But on the reasons why aggregate demand remains below full employment output in a perfectly functioning market economy, there is no consensus among mainstream macro-economists to this date. The majority view follows the New-Keynesian approach in which prices and wages adapt sluggishly to market imbalances due to imperfect competition and other market failures (for a survey see [5]). In contradistinction to the majority view, a macroeconomists’ minority follows [6] and more recently [7] who trace back aggregate demand failures to inflexible aggregate investment demand governed by (pessimistic) “animal spirits” of investors independently from aggregate savings of households. In contrast to the imperfectly flexible-price approach, [6, 7] presume perfectly flexible and perfectly competitive output prices, wage rates and interest rates. Despite this perfect-market setting employees do not become fully employed because an independent investment function makes the general equilibrium equations’ system over-determinate. Over-determinacy disappears only if at least one market-clearing condition is cancelled, and it is the labor market clearing condition that is deleted.

Magnani [7] without noting precursor Morishima [6] incorporates a macro-founded investment function into Solow [1]‘s neoclassical growth model without public debt. Since the chapter intends to study the effects of public debt on private capital accumulation, GDP growth and unemployment in the long run, the present author switches to Diamond [2]‘s OLG model with non-neutral internal public debt. Long-run GDP growth in Diamond [2]‘s OLG model is, however, exogenous precluding the analysis of how larger public debt impacts GDP growth and unemployment. Hence, there is a need for a mechanism that endogenizes GDP growth. To this end, we stick to human capital accumulation à la [8, 9].

This chapter pursues several purposes: Firstly, it will be shown how in a log-linear utility and Cobb–Douglas production function version of Diamond [2]‘s OLG model with internal public debt, the intertemporal equilibrium dynamics based on household’s and firm’s first-order conditions, on government’s budget constraint and intertemporal market-clearing conditions is modified when aggregate investment demand is governed by a savings-independent investment function. Transcending [7] we secondly intend to rigorously prove the existence and dynamic stability of a steady-state of the equilibrium dynamics in a model which closely follows Farmer [10]‘s model setting. Our third purpose is to investigate the effects of a higher public debt to output ratio on the output growth rate, on the capital-output ratio, on the interest factor and on the wage tax rate in a steady state of the Diamond OLG model extended by human capital accumulation which is financed by public human capital investment expenditures as in Farmer [11] and in Lin [9]. In extending [9, 11] by an independent aggregate investment function we are capable of exploring analytically and numerically the steady-state effects of a higher public debt to output ratio on the unemployment rate. In particular, we will demonstrate on which factors it depends whether a higher public debt to output ratio raises the output growth rate and decreases the unemployment rate. In contradistinction to the author’s contributions in Farmer [10] and Farmer [12] this chapter exhibits the OLG model presented there more completely and succeeds in deriving the steady-state effects of larger public debt more succinctly. This chapter together with Farmer and Farmer [10, 12] can be seen as our contribution to the recent macroeconomic literature.

The structure of the chapter is as follows. In the next section (2.) the model set-up will be presented. In section 3, temporary equilibrium relations and the intertemporal equilibrium dynamics are derived from intertemporal utility maximization, atemporal profit maximization, government’s budget constraint and the market-clearing condition in each period. In section 4, the existence of a steady-state solution of the equilibrium dynamics and its local dynamic stability is investigated. Section 5, is devoted to the analysis of the comparative steady-state effects of larger public debt. Section 6, concludes.

## 2. The model set-up

As in Farmer and Farmer [10, 12], we consider an economy of the infinite horizon which is composed of infinitely lived firms, finitely lived households and an infinitely lived government. In each period

As mentioned above, to be able to address the question of how fiscal policy impacts long-run growth we extend Diamond [2]‘s basic OLG model by introducing human capital accumulation. To point out the growth-enhancing effects of human capital accumulation most clearly, it is assumed here that there is no population growth

Each household consists of one agent and the agent acts intergenerationally egoistic: The old agent does not take care of the young agent and the young agent does not take care of the old agent. They live two periods long, namely youth (adult) and old age. In youth age, each household starts with human capital

Any unemployed young-age household, denoted by superscript

In old age, the employed household supplies inelastically

A log-linear intertemporal utility function slightly generalized in comparison to Diamond ([2], p. 1134]‘s leading example represents the intertemporal preferences of all two-period lived households. As usual, this simple specification aims at closed-form solutions for the intertemporal equilibrium dynamics (see e.g. [13, pp. 181–184]).

The typical younger, employed household maximizes the following intertemporal utility function subject to the budget constraints of the active period (i) and the retirement period (ii):

subject to:

Here,

A strictly positive and finite solution to maximizing the intertemporal utility function subject to the constraint (1) requires that the following no-arbitrage condition holds:

The no-arbitrage condition (2) implies that

Solving equations (3) and (4) for

Since

The typical younger, unemployed household maximizes the following intertemporal utility function subject to the budget constraints of the active period (i) and the retirement period (ii):

subject to:

Again,

The no-arbitrage condition (2) implies that

Solving equations (9) and (10) for

Since

All firms are endowed with an identical (linear-homogeneous) Cobb–Douglas production function which reads as follows:

Here,

Maximization of

However, since the number of employed workers is

Finally, the GDP function can be rewritten as follows:

As in Diamond [2], the government does not optimize, but is subject to the following constraint period by period:

where

In line with Glomm and Ravikumar [8] human capital in period

whereby

The economy grows, even in the absence of population growth and exogenous progress in labor efficiency. Using the GDP growth factor

As Magnani [7], Morishima [6], Salotti and Trecroci [14] rightly states, aggregate investment in Solow [1]‘s neoclassical growth model is not micro- but macro-founded since it is determined by aggregate savings. The same holds in Diamond [2]‘s OLG model of neoclassical growth where perfectly flexible aggregate investment is also determined by aggregate savings of households. Deviating from those neoclassical growth models, Morishima [6] and more recently Magnani [7] and Salotti [14] claim that “investments are determined by an independent investment function.” This function is specified in discrete time as follows:

The positive parameter

In addition to the restrictions imposed by household and firm optimizations and the government budget constraint, markets for labor, capital services and assets, ought to clear in all periods (the market for the output of production is cleared using Walras’ Law^{1}).

## 3. Temporary equilibrium and intertemporal equilibrium dynamics

As a first step, the unemployment rate in period

Starting with identity (28), we insert equations (19), equation (5) and constraint (ii) from the employed household’s optimization problem for period

On dividing equation (29) into both sides by

Rewriting profit maximization condition (15) as

and using the definitions

The capital-output ratio

which implies:

By use of the no-arbitrage condition (2) as well as of equations (33) and (34)equation (32) turns out to be:

Next, it is apt to specify how the government determines its intertemporal policy profile. To this end, we assume that government consumption expenditures per GDP,

Equation (36) implies that the wage-tax rate ought to become endogenous and is determined by the following equation:

Inserting

Collecting terms and simplifying the resulting expression yields the following equation for

In terms of the transformed variables, the growth factor of human capital reads as follows:

The GDP growth factor in terms of the capital-output ratio can be rewritten as follows:

By using the intertemporal equilibrium condition

The final steps needed to arrive at the equation of motion for the capital-output ratio entail; first, inserting the GDP growth factor equation (41) into equation (42). This procedure yields:

Next, after inserting the growth factor equation (41) into equation (39) and rearranging, we arrive at the following intermediate result:

Solving equation (44) for

Reinserting the dynamic equation (45) into equation (43) and solving for

## 4. Existence and dynamic stability of steady states

The steady states of the equilibrium dynamics depicted by the difference equations (45) and (46) are defined as

To this end, for given structural and policy parameters (except

Using the steady-state version of equation (39) with

Whereby, to ensure a strictly larger than zero

For the proof of the existence of at least one

By substituting

Using the short cut

Hereby,

** Proposition 1**. Suppose there exist

** Proof**. For

_{.}Moreover, for a broad set of feasible parameters the solution is unique. Q.E.D.

The next step is to investigate the local dynamic stability of the unique steady-state solution. To this end, the intertemporal equilibrium equations (39), (41), and (42) are totally differentiated with respect to

with

The sign of

The sign of the trace turns out to be, in general, indeterminate, while the determinant of the Jacobian (54) is larger than zero. Moreover, the sign of

** Proposition 2**. Suppose the assumptions of Proposition 1 hold. Then, the calculation of the eigenvalues

In other words, the steady-state solution in the present endogenous growth model with involuntary unemployment represents a non-oscillating, monotone saddle point with

## 5. Comparative steady-state effects of a higher public debt to GDP ratio

Being assured of the existence and dynamic stability of a steady-state solution it is now apt to investigate how a larger government debt to GDP ratio impacts the steady-state GDP growth rate and the steady-state unemployment rate. A comparable OLG model with endogenous growth and full employment [9] finds that the GDP growth is raised by a higher government debt to GDP ratio if the GDP growth rate is larger than the real interest rate in the initial steady state. If the real interest rate is higher than the GDP growth rate in the initial steady-state, larger government debt to GDP ratio lowers the GDP growth rate. Because of these interesting results, it will be expedient to explore whether in our model with an independent aggregate investment function the GDP growth rate effect of more government debt will also depend on the difference between the initial GDP growth rate and the initial interest rate. In addition, of particular interest is how a larger public debt to GDP ratio affects the unemployment rate which could not be investigated by Lin [9].

To proceed, we now consider the intertemporal equilibrium equations (37), (39), (41) and (42) in a steady-state and differentiate the resulting static equation system totally with respect to

Solving simultaneous equations (58) and (61) for

The right-hand side of the differential quotient (62) shows that a higher public debt to GDP ratio affects the capital-output ratio unambiguously negatively if dynamic inefficiency prevails, i.e. the GDP growth factor is larger than the real interest factor since

The term on the right-hand side of equation (63) shows the response of one minus the unemployment rate to a higher government debt to GDP ratio. It transpires that when dynamic inefficiency prevails and moreover

The calculation of the marginal change of the steady-state wage tax rate and the GDP growth factor from equations (62) and (63) brings forth the following result:

A glance on the right-hand side of equation (64) shows that the response of the wage tax rate to a higher debt to GDP ratio is in general ambiguous. If, however, the GDP growth factor is sufficiently larger than the interest factor (precisely if

Because in the case of dynamic efficiency the response of the unemployment rate to higher public debt is in general ambiguous, we use a numerical parameter set that implies dynamic efficiency before the policy shock and which is in line with the assumptions of Proposition 1. To this end, the following parameter combination not untypical because of medium-term econometric parameter estimations is assumed:

Debt to GDP ratio | Capital-output ratio | GDP growth factor | One minus unemployment rate | Wage tax rate | Interest factor |
---|---|---|---|---|---|

0.14450 | 1.51422 | 0.905413 | 0.44784 | 2.07608 | |

0.134872 | 1.5212 | 0.941195 | 0.45870 | 2.22433 |

Although under the present parameter set dynamic efficiency prevails (= the interest factor is larger than the GDP growth factor in Table 1 with

## 6. Conclusions

This chapter aims to incorporate involuntary unemployment in an OLG growth model with internal public debt and human capital accumulation. Deviating from new-Keynesian macro models in which involuntary unemployment is traced back to inflexible wages, output prices and interest rates vis-à-vis market imbalances, real wages and real interest rates are perfectly flexible in our Diamond-type growth model with involuntary unemployment. Involuntary unemployment occurs in line with [6, 7] aggregate investment is inflexible due to investors’ animal spirits.

After presenting the model set-up temporary equilibrium relations and the intertemporal equilibrium dynamics are derived from intertemporal utility maximization, atemporal profit maximization, government’s budget constraint and the market-clearing conditions in each period. To arrive at determinate equilibrium dynamics, it is assumed that the government holds constant over time: the public debt to GDP ratio, the HCI-expenditure ratio, the non-HCI expenditure ratio and the unemployment benefit to GDP ratio. As a consequence, the wage tax rate becomes endogenous.

Due to the complexity of the intertemporal equilibrium relations, an explicit steady-state solution is not possible. Thus, the simplest mathematical existence theorem, the intermediate value theorem is applied to prove the existence of a steady-state solution with a strictly positive capital-output ratio and an unemployment rate larger than zero and smaller than one. Contrary to intuitive expectations, there exists a finite limit to the public debt to GDP ratio even in the economy with involuntary unemployment. Public debt to GDP ratios higher than that limit implies negative unemployment rates which are infeasible. As Farmer [10] shows in a similar model context maximum public debt in a growth model with involuntary unemployment is not a purely theoretical notion but turns out to be empirically relevant in a numerically specified growth model with involuntary unemployment.

Besides the existence of a steady-state solution for the intertemporal equilibrium dynamics, its dynamic stability was shown. It turns out that for a broad set of feasible structural and policy parameters the dynamics is saddle-point stable. With the capital-output ratio as a sluggish variable historically given, the unemployment rate jumps initially suddenly onto the saddle-path along which both variables converge monotonically (non-oscillating) towards their steady-state values.

Being assured of the existence and dynamic stability of the unique steady-state solution we shocked it by a higher public debt to GDP ratio mimicking the pandemic-related larger public debt to GDP ratios in almost all countries of the world economy. In line with Keynesian policy expectations, we were able to show analytically that in case of dynamic inefficiency, i.e., the GDP growth rate is larger than the real interest rate, a higher public debt to GDP ratio (below the maximum debt to GDP ratio) unambiguously reduces both the capital-output ratio and the unemployment rate while raising the GDP growth rate in a dynamic market economy with perfectly flexible real wage and interest rates. In the case of dynamic efficiency, the responses to the policy shock become in general ambiguous. However, in a numerically specified version of the presented model, it was shown that qualitatively similar comparative steady-state effects occur even in the case of dynamic efficiency. The main reason for these results is that under inflexible aggregate investment higher public debt creates a positive wealth effect with old-age consumers which raises aggregate demand and hence reduces unemployment.

The limitations of the present research are obvious: First, micro-foundations for the aggregate investment function are lacking. Here, stock-market foundations for the aggregate investment function in line with Farmer [15]‘s investor’s belief function should be provided to overcome the purely macro-foundation of the aggregate investment function. Second, there is no impact of larger public debt to GDP ratios on aggregate investment. Here, Salotti [14]‘s an empirical specification of a negative relationship between public debt and aggregate investment may be incorporated in a future version of the present model. Both subjects are left to future research.

## Notes

- The proof of Walras’ law proceeds as follows: Denote by Pt>0 the nominal price (level) of production output (GDP). Then, the current period budget constraint of employed households in youth can be rewritten as follows: PtLt1−utct1,E+PtItD,E+PtBt+1D,E=1−τtPtwtht1−utLt. (F.1) The budget constraint of households in old age employed in youth reads as follows: PtLt−11−ut−1ct2,E=PtqtKtS,E+Pt1+itBtS,E.(F.2) The budget constraint of young unemployed households in period t is as follows: PtLtutct1,U+PtItD,U+PtBt+1D,U=Ltutςt. (F.3) Moreover, the budget constraint of the household in old age in period t, which was unemployed in youth, reads as follows: PtLt−1ut−1ct2,U=PtqtKtS,U+Pt1+itBtS,U. (F.4) In addition, maximum profits are zero, which implies: PtYt=PtwtNt+PtqtKt. (F.5) Finally, government’s budget constraint is rewritten as follows: PtBt+1=Pt1+itBt+PtΔt+PtΓt+PtLtutςt−PtτtwthtLt1−ut. (F.6) Adding up the left- and right-hand side of equations (F.1), (F.2), (F.3) and (F.4) yields: PtLt1−utct1,E+PtItD,E+PtLtutct1,U+PtItD,U+PtLt−11−ut−1ct2,E+PtLt−1ut−1ct2,U=1−τtPtwtht××1−utLt−PtBt+1D,E−PtBt+1D,U+Ltutςt+PtqtKtS,E+Pt1+itBtS,E+PtqtKtS,U+Pt1+itBtS,U. (F.7) Considering (25), (26) and (27) in (F.7), gives: PtLt1−utct1,E+ PtLt−11−ut−1ct2,E+PtItD,E+PtLtutct1,U+PtLt−1ut−1ct2,U+PtItD,U=PtwtNt−τtPtwtNt−PtBt+1+PtqtKt+Pt1+itBt.(F.8) Considering labor market clearing condition (24) when inserting (F.6) into (F.8), and taking account of (F.5) in (F.8) yields: PtLt1−utct1,E+PtLt−11−ut−1ct2,E+PtLtutct1,U+PtLt−1ut−1ct2,U+PtItD,E+PtItD,U+PtΓt=PtYt, which is production-output market clearing. Since this equation is always true, Pt is indeterminate and can be fixed as Pt=1.