Appendix to Skill-Biased Technical Change, Educational Choice, and Labor Market Polarization: The U.S. versus Europe

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1 Appendix to Skill-Biased Technical Change, Educational Choice, and Labor Market Polarization: The U.S. versus Europe Ryosuke Okazawa April 21, 2012 A. Multiple Pooling Equilibria In Section 3, although we assume that equation (12) has a unique solution, this assumption is not crucial to the main result. Here, we consider the more general case in which equation (12) may have multiple solutions. Let Φ p (η) be the set of solutions of (12) when the relative productivity is η. We define ϕ(η) and ϕ(η) by τ(ϕ(η)) = ω P (0, η), τ(ϕ(η)) = ω P (1, η). Since ω P (ϕ, η) is increasing in ϕ and ω P (1, η) < ω S (η), we have η > 1 ϕ Φ P (η) ϕ(η) < ϕ < ϕ(η) < ϕ S (η). As in Section 3, we define η and η by η = Γ(ϕ(η)) and η = Γ(ϕ(η)). Note that η > η > η because ϕ(η) < ϕ(η) < ϕ S (η) for any η. By the same argument as that in Section 3, we have ϕ Φ P (η) η η = Γ(ϕ(η)) Γ(ϕ(η)) < Γ(ϕ) if η η. Therefore, the pair of any solution of equation (12) and the pooling strategy is a Nash equilibrium if η η. On the other hand, we have ϕ Φ P (η) η > η = Γ(ϕ(η)) > Γ(ϕ(η)) > Γ(ϕ) if η > η. Thus, we have no when η exceeds η. By summarizing these results, we have the following proposition. Proposition 1. A Nash equilibrium always exists even if we allow the possibility that more than one exists. When the relative productivity of skilled workers is sufficiently high, η > η, there is a. When the relative productivity is sufficiently low, η η, there is at least one. If the level of relative productivity is intermediate, η (η, η], both types of equilibria exist. There is no pooling equilibrium if η > η. 1

2 B. Proof of Lemma 2 We consider the optimal investment problem of the firm given the probability of adoption. When a firm hires both types of workers, i.e., x H = x L = 1, the amount of physical capital that maximizes Ṽ (k, 1, 1) is given by Then, the expected profit is k P = a Ṽ P V ( k P, 1, 1) = ϕη + (1 ϕ). 1 (1 β)aϕη + (1 ϕ) When the firm hires only skilled workers, i.e., x H = 1 and x L = 0, the amount of physical capital that maximizes Ṽ (k, 1, 0) is given by Then, the expected profit is Ṽ S V ( k S, 1, 0) = k S = aη. (1 β)ϕaη 1 Since we can show that the other hiring strategies are not optimal, as shown in Lemma 1 in Acemoglu (1999), we can determine the optimal strategy of the firm by comparing Ṽ P and Ṽ S. Since Ṽ P Ṽ H if and only if η Γ(ϕ), the result is proved. C. Numerical Analysis We parameterize the model as follows. In macroeconomic models, output elasticity of labor is usually set to equal the labor share. However, output elasticity is not equal to labor share in our model as in standard search model, because the wage is determined by bargaining. Therefore, instead of output elasticity of labor, we make the bargaining power of worker β correspond with the labor market share, so we set β = 0.66 and set the output elasticity of labor so that the return of capital in the model is close to its actual value. We regard the second period of the model as the term from the start of production until full depreciation of capital and assume that the length of the second period corresponds to about 11.5 years, following Hornstein et al. (2007). Since the return of capital in is given 1, we set = 0.44, which makes the annual rate of return of capital roughly equal to 7%. The gap of the ( log ) wage between skilled and unskilled workers in the w benchmark economy is given by ln H pool = ln η. We assume that half of the workers are by Ṽ ( k p,1,1) = c k p w L pool skilled in the benchmark economy and set η so this gap equals 0.5, which is roughly equal to the 75/25 gap, 90/50 gap, and 50/10 gap of residual wage in 1976 in the U.S. (Juhn et al. 1993). The mean cost of skill acquisition τ is set so that the equilibrium fraction of skilled workers in the benchmark economy is equal to 0.5. It is difficult to find an adequate empirical target for the dispersion of skill acquisition cost among workers σ, but predictions.. 2

3 of the model highly depend on this parameter since the elasticity of the supply of skilled workers depends on σ. When the distribution of skill acquisition cost is small, many workers acquire skill in response to an increase in skill premium. Conservatively, we examine four values of dispersion: σ = 0.15, 0.20, 0.30, and 0.40, which imply the elasticities of skill supply with respect to a one percent increase of skill premium equal to 1.30, 0.86, 0.52, and 0.37, respectively. Table 1 shows the impacts of an increase in relative productivity on the share of skilled workers and wages. It reveals that the response of skill supply to a skill-biased technical shock and wage growth is larger in the case in which the variance of skill acquisition cost is small. 1 For instance, 10% growth of relative productivity increases the wage of skilled and unskilled workers by 16% and 11% respectively in the case of σ = 0.10, but it increases it by only 10% and 6% in the case of σ = While the ratio between the wages of skilled and unskilled workers does not depend on the value of σ, the growth rate of each wage depends on the dispersion of skill acquisition cost. This is because an increase in relative productivity encourages capital investment not only directly but also only indirectly through an increase of the skill supply in. Since a small dispersion of skill acquisition cost makes the supply of skill more elastic, it enhances the indirect effect. On the other hand, the small dispersion of skill cost accelerates labor market polarization and thus is harmful for unskilled workers in the long run. Figure 1 shows the impact of tax rates on the share of skilled workers at the initial technology level. Since an increase in the tax rate reduces the incentive of human capital investment, there is a negative relationship between the fraction of skilled workers and the tax wedge. The response of skill supply to tax rate is higher in the case that the cost of skill acquisition is lower. Figure 2 shows the relationship between the tax wedge and the condition under which arises. An increase in the tax wedge substantially tightens the condition under which labor market polarization arises. Figure 3 shows the relationship between the tax wedge and the condition under which pooling equilibrium can survive. The relationship between σ and threshold η is reversed at t = If the initial fraction of skilled workers is the same, a smaller dispersion of the cost of skill acquisition accelerates labor market polarization. However, higher tax rates reduce the fraction of skilled workers at the initial productivity level and this negative impact is larger in the case that σ is lower, as Figure 1 shows. If the tax wedge is large enough, the latter effect dominates the former effect, which reverses the relation between σ and η. References Hornstein, A., Krusell, P., and Violante, G.L. (2007) Technology-Policy Interaction in Frictional Labor Markets. Review of Economic Studies, 74, Juhn, C., Murphy, K.M., and Pierce, B. (1993) Wage Inequality and the Rise in Returns to Skill. Journal of Political Economy, 101, Since the wage growth of skilled workers in a does not depend on σ, those in the case σ = 0.20, 0.30, 0.40 are the same as that when σ =

4 Table 1: Quantitative impacts of skill biased technical change growth of relative productivity 10% 20% 30% 50% 70% σ = 0.15 share of skilled wage growth of skilled wage growth of unskilled share of skilled wage growth of skilled σ = 0.2 share of skilled wage growth of skilled wage growth of unskilled share of skilled σ = 0.3 share of skilled wage growth of skilled wage growth of unskilled share of skilled σ = 0.4 share of skilled wage growth of skilled wage growth of unskilled share of skilled

5 0.7 = 0.15 = 0.2 = 0.3 = 0.4 Share of skilled workers Tax wedge Figure 1: The share of skilled workers and the tax wedge (η = η 0 ) 5

6 = 0.15 = 0.2 = 0.3 = Tax wedge Figure 2: The necessary condition for ( η η 0 η 0 ) and the tax wedge = 0.15 = 0.2 = 0.3 = Tax wedge Figure 3: The sufficient condition for ( η η 0 η 0 ) and the tax wedge 6