^{1}

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In this study, we introduce nominal wage stickiness into an endogenous growth model based on R & D. This study examines how money growth affects long-run economic growth. We find that there exists a unique balanced growth path for sufficiently high rates of money growth, and that the economy exhibits sustained growth based on sustained R & D. Faster money growth results in greater employment and faster economic growth along such a balanced growth path. Furthermore, under some parameter restrictions, no balanced growth path exists for low rates of money growth; the economy is trapped in a steady state without long-run growth. These results suggest that money growth may be an important factor for long-run economic growth.

This study proposes a new monetary growth model involving price stickiness and endogenous R & D. Short-run macroeconomic models usually consider price stickiness, as in new Keynesian models. In this study, we introduce nominal wage stickiness into a long-run growth model based on R & D and investigate how money growth affects long-run output, employment, and economic growth.

We base the dynamics of our model on the new Keynesian Phillips curve (NKPC), under which money is not super neutral, even in the long run^{1}. [

We focus on the steady-state economic growth and employment. For sufficiently high money growth rates, there is a unique balanced growth path, and the economy exhibits sustained growth based on sustained R & D. Faster money growth causes greater employment and faster economic growth along the balanced growth path. Furthermore, under some parameter restrictions, there is no balanced growth path for low money growth rates, and the economy is trapped in a steady state without long-run growth. These results suggest that money growth may be an important factor for long- run economic growth. That is, financial authorities are required to maintain high money growth rates to achieve sustained and faster economic growth.

Most of the preceding theoretical studies on money and endogenous growth have concluded that a higher money growth is associated with a lower rate of long-run growth, which is contrary to the conclusion of this study. See for example [^{2}. This study proposes a new channel attributed to nominal rigidities and endogenous R & D through which money growth influences the long-run economic growth.

Some empirical studies argued that inflation has a negative impact on economic growth ( [^{3}. Our study provides a theoretical explanation for these empirical results.

The remainder of this paper is organized as follows. Section 2 sets up the model used in our theoretical investigation. Section 3 derives the law of motion and the steady state, which characterize the equilibrium path of the economy. It also investigates the existence and the uniqueness of the steady state. Section 4 concludes the paper.

We consider the continuous-time version of the dynamic model based on [

The manufacturing and R & D sectors regard each household's labor as an imperfect substitute for any other household's labor. To simplify the analysis, we assume that an employment agency combines differentiated labor forces into a composite labor force

according to the Dixit-Stiglitz function,

posite labor to the intermediate goods and the R & D sectors.

Cost minimization of the employment agency yields the demand functions for dif- ferentiated labor j,

We assume that perfect competition prevails in the final goods market. The final goods firm produces the quantity y according to the Dixit-Stiglitz function,

where

Cost minimization by the final-goods producing firm yields the following demand functions for intermediate goods

where

Each intermediate good is produced using one unit of composite labor; thus, marginal cost is equal to the nominal wage level, W. Because patents have an infinite life, all intermediate goods are supplied monopolistically. Maximization of the monopoly pro- fit,

where

From (2), the market equilibrium levels of output, y, and the price of the final good, p, are obtained as

We can rewrite (5) as

The number of intermediate goods, N, expands according to the following equation:

where

In equilibrium, the following free-entry condition must be satisfied:

The right-hand side is the nominal unit cost of R & D. V represents the value of the patent, which is given by the discounted stream of the monopoly profit:

where R is the nominal interest rate. Differentiating both sides with respect to time, t, yields the following no-arbitrage condition:

Household j possesses nominal money balances,

where

when

where

Household j obtains utility from consumption,

where ^{4}. If

Summarizing the above, household j faces the following dynamical optimization problem:

where

where

On the other hand, when

Financial authorities are assumed to change money supply, M, at a constant rate

When the nominal wage is sticky (

where

When R,

If the law of motion (11) through (13) has fixed points, they are derived as follows:

where

The steady-state values of

However, to guarantee that

If it is the case that

We refer to the output and employment level in the flexible-price economy (i.e., when

NKPC (9). Then, substituting (4), (19),

When

When

These results may be summarized as follows:

Proposition 1. Let

When the R & D sector is sufficiently productive and the parameters satisfy

If

Some algebra shows that

When the parameters satisfy

^{5}. To sum up these findings, we can see four cases as shown in Figures 2(a)-(d)^{6}.

At first, in the cases of

In the case of

The following proposition summarizes the above properties.

Proposition 2.

1. If the parameters satisfy

2. Let the parameters satisfy

On the other hand, in the case of ^{7}. To put it more precisely, we can state the following proposition.

Proposition 3. Let

Letting

The arguments of Propositions 1 through 3 are summarized in

Let a unique BGP exist. Then, we obtain the following proposition.

Proposition 4. Let

This proposition can be proved as follows. As shown in

no BGP | no BGP | a unique BGP | |

dual BGPs or no BGP | a unique BGP | a unique BGP | |

- | - | a unique BGP |

Note: “―” shows that no such combinations of parameters exist because

fore, an increase in ^{8}. As a result, since^{9}.

Furthermore, consider the following two facts. First, the growth acceleration effect of money growth is attributed purely to nominal wage stickiness. A small value of ^{10}. Second, even if financial authorities add 1% to the money growth rate, the rise in the long-run inflation rate is smaller than 1% because of the rise in the long-run growth rate

As for dual BGPs, we can prove the following proposition in a similar way to that of Proposition 4.

Proposition 5. Let dual BGPs exist. At the BGP with lower employment level, an increase in the money growth rate raises employment and the balanced-growth rate. Whereas, at the BGP with a higher employment level, an increase in the money growth rate depresses employment and the balanced-growth rate.

There exists a different steady state from the BGP at which no labor is allocated to R & D and long-run growth never occurs. We refer to such a steady state as the no-growth steady state. At the no-growth steady state, since the free-entry condition (7) does not hold with an equality, (14), (15), and (16) are not fulfilled, and

The value of each variable at this steady state is derived as follows:^{11}

If and only if^{12}. When

This study developed a R & D-based endogenous growth model by introducing money growth and a price adjustment process. This study assumed that nominal wage is adjusted stickily because of adjustment cost and derived the new Keynesian Phillips curve, under which money is not super neutral even in the long-run.

When the money growth rate is sufficiently high, the economy has a unique balanced growth path, and can sustain long-run positive growth based on sustained R & D. Fur- thermore, faster money growth brings greater employment and faster economic growth along a unique balanced growth path. In contrast, under some parameter restrictions, when the money growth rate is sufficiently low, there is no balanced growth path, and the economy is trapped in a no-growth steady state. These results suggest that money growth may be an important factor for long-run economic growth.

To highlight the effect of nominal wage stickiness, this study adopted the money-in- utility-function approach, under which money is supernatural in a flexible-price eco- nomy. One interesting extension would be to analyze a model with another specifi- cation for money demand, for example cash-in-advance approach. In such case, the super neutrality of money may not hold even in a flexible-price economy, and the growth acceleration effect which is argued in this study might be weakened or strengthened. Such topic will be the subject of future research.

We thank the Editor and the referee for their comments. This paper is a part of the outcome of research performed under a Waseda University Grant for Special Research Projects (Project number: 2015B-014).

Shinagawa, S. and Inoue, T. (2016) R & D-Based Growth Model with Nominal Wage Stickiness. Theoretical Economics Letters, 6, 854-867. http://dx.doi.org/10.4236/tel.2016.65089