# Abstract

We develop a simple growth model featuring individuals’ choices between general and specific skills, endogenous technological innovation, and a government subsidy for education. The two types of skills differ by their productivity and transferability: general skills are transferable across firms, while each firm-specific skill has a productivity advantage in the firm. Firms face uncertainty in their innovation activities, and the resulting heterogeneity in their labor demand makes the transferability of general skill valuable. We theoretically show that as a country catch up to the world technology frontier, firms invest more in innovation activities. This rises firms’ technological uncertainty and, thus, their demands for general skills increases. As a result, especially in more advanced economies, education subsidies may enhance GDP by increasing the supply of general skills. Using aggregated data for 12 European OECD counties, we calibrate the model and compare the theoretical prediction with the data. In cross-country comparisons, we find that the returns on general skills and the impact of general education expenditure on GDP are higher in countries with higher total factor productivity. These findings support our theoretical argument of the positive relationship between firms’ demand for general skills and countries’ stages of development.

# Acknowledgement

I would especially like to thank Tatsuro Iwaisako for his invaluable comments and encouragement. I am grateful to Koichi Futagami, Masaru Inaba, Keiichi Kishi, Keigo Nishida, Ryosuke Okazawa, Yoshiyasu Ono, Kouki Sugawara, Katsuya Takii, participants in the DSGE conference 2016 at Ehime University, participants in the workshop on the economics of human resource allocation 2017 at Osaka University, participants in Nagoya Macroeconomics Workshop 2017 at Nagoya Gakuin University.

## Appendix

### Appendix A: proof of g L , t = 0

In this appendix, we prove that *g _{L,t}* = 0 holds for any equilibrium. Conversely, we assume that

*g*> 0 and

_{H,t}*g*> 0. Then, firms’ demand for general skilled workers is obtained as follows:

_{L,t}Using these expressions, the firms’ problem in the first stage can be expressed as follows:

The maximization problem is linear in *s _{t}* and, hence, the non-negative constraint

*g*≥ 0 has to be binding. Thus, we have

_{L,t}*g*= 0.

_{L,t}### Appendix B: proofs of Lemma 1 and Proposition 1

From (18), we have that

Combining (33) and (20), we have that *a _{t−1}* must satisfy the following inequality:

Solving the above inequality with respect to *a _{t−1}* yields

Further, in order to guarantee the existence of general skills,

### Appendix C: proof of Proposition 2

From (21), we have that the dynamics of *a _{t}* satisfy

*a*is increasing and convex in

_{t}*a*, and has a positive intercept. Thus, the dynamics of

_{t−1}*a*have at most two steady-state points. As shown in the figure below, if two steady-state points exist, the one with a lower value is stable and the larger one is unstable. A condition for the existence of a unique and stable steady state in the interval

_{t}*a*= 1. Substituting this condition into (21), and rearranging, yields

_{t}Finally, because *μ* must satisfy

The left-hand side of the above inequality decreases in *γ* and is equal to 0 when *γ* → 1. On the other hand, the right-hand side increases in *γ* and is equal to 0 when *γ* → *π*. That is, *γ* has to be large enough to ensure that

### Appendix D

First, we return to the second-stage problem of firms, and resolve *g _{H, t}* and

*x*. The second-stage problem of type-

_{i, t}*H*firms is as follows:

The first-order conditions with respect to *g _{H, t}* and

*x*are, respectively,

_{H, t}and

Using the above two first-order conditions, we obtain

Then, we have the technology level of type-*H* firms as

Next, the first-stage problem of firms can be written as

Note that *G*-skill workers. Combining this condition with (5), we have

We focus on the equilibrium that satisfies the condition given in (36), in which both *G*-skill and *S*-skill workers exist. Section 5 showed that all estimates obtained from our calibration satisfy (36). Finally, the first-order condition of this problem yields

Rearranging (34) and (37) by evaluating at the steady state and substituting

### Appendix E: proof of Proposition 3

In this appendix, we examine the steady-state characteristics using the system given in (27) and (29). We first show that the steady-state values, *g*^{*} and *a*^{*}, increase in *g*^{*}, as a function of *a*^{*} and *a*^{*}. Substituting

Here, *a*^{*} is determined from the above expression. The left-hand side (LHS) of the above expression is increasing and is a linear function of *a*^{*} with a negative intercept. At the same time, the right-hand side (RHS) increases in *a*^{*}, and passes through the origin. However, we cannot examine the number of intersections between the LHS and RHS curves because very little is known about the shape of the RHS function. However, we have that if the LHS and the RHS curves have one or more intersections, the value of the smallest steady-state increases in *a* increases in

Next, we investigate the stability of the steady state corresponding to the smallest solution of *a*^{*}. Let us denote as *g _{t}* the total number of

*G*-skill workers in equilibrium, but not in the steady state. Then, substituting

and

Moreover, using (22) together with *s _{t}* = 1 −

*g*and

_{t}The three endogenous variables, *g _{t}*,

*a*, and

_{t}*a*> 0 when

_{t}^{[21]}Therefore, the steady state determined by the first intersection of the dynamic equation of

*a*and the 45-degree line is stable, indicating that the dynamics are much the same as those described in Section 3 and in the figure in Appendix C. In the subsequent discussion, we continued the analysis by supposing that economies are on this stable steady state, although we did not prove its uniqueness. Instead of deriving the parameter conditions for the existence and uniqueness of the stable steady state, we confirmed them in the quantitative analysis in Section 5.

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**Published Online:**2019-11-20

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