Ranking and Long-Term Unemployment in a Model with Efficiency Wages

Abstract

This chapter considers the long-run consequences of ranking job applicants on the basis of their unemployment durations by using a general equilibrium model, in which the wages paid by firms not only motivate their employees but also induce jobless workers to preserve their employability. Ranking and long-term unemployment become actual when the cost of establishing a new firm is so large that firms cannot pay high wages to their employees. By subsidizing newly established firms, the government can guide the economy to a more efficient equilibrium, in which every job seeker can find a new job by experiencing one period of unemployment, and thus firms’ distaste for the long-term unemployed is effectively nullified.

Appendices

Appendix 1: Equilibrium Values

Tables 3.12, 3.13, 3.14, 3.15, 3.16 and 3.17 report, respectively, the equilibrium values of $\hat{w}$, E, $U_O$, a, $L^*$, $E/L^*$, $w^1$, and $w^{2+}$. Under the parameter configuration of Table 3.1, they are computed for such values of $\overline{n}$ and F as

$$ \overline{n}=-1, -2, -3, -4, -5, -10, -20, -30, -\infty , $$

$$ F=\left\{ \begin{array}{cl} 0.1i &{} \text {for} \quad i=1, 2,\cdots , 8\\ F_1^{1-0.1j}F_3^{0.1j} &{} \text {for} \quad j=0, 1, 2,\cdots , 10 \end{array} \right. \quad \text {and} \quad F>F_3. $$

Table 3.12 Aggregate employment (E)

Table 3.13 Population of unemployable workers ($U_O$)

Table 3.14 Per-period job finding rate for employable workers (a)

Table 3.15 Number of operating firms ($E/L^{*}$)

Appendix 2: Alternative Interpretations of Workers’ and Investors’ Surpluses

1.1 Workers’ Surplus

We can show that the WS equals the discounted sum of the lifetime utilities of the workers constituting the labor force in the present and future periods,

$$\begin{aligned} \begin{array}{rcl} WS&{}=&{}\displaystyle E[(\theta +b-\theta b)V_E^1+(1-\theta )(1-b)V_E]+\sum _{n=\overline{n}}^{-1}U_n V_U^n+U_OV_O\\ &{}&{}\displaystyle +\frac{\theta N}{1-\theta }\sum _{t=0}^\infty \frac{V(0,1)}{(1+r)^t}, \end{array} \end{aligned}$$

(3.82)

where V(0, 1) is the lifetime utility of newly born workers and $V_E^1$, $V_E$, $V_U^n$, $V_O$, E, $U_n$, and $U_O$ are defined as in the text.

Table 3.16 Starting wage ($w^1$)

Table 3.17 Wage paid in and after the second period of service ($w^{2+}$)

Proposition 3.5

The RHS of (3.82) can be expressed as

$$\begin{aligned} RHS=\sum _{t=0}^\infty \left( \frac{1}{1+r}\right) ^t\left[ \tilde{w}E+\overline{w}(N-E)-e(N-U_O)\right] , \end{aligned}$$

(3.83)

where

$$\begin{aligned} \tilde{w}\equiv w^1-(1-\theta )(1-b)(w^1-w^{2+}). \end{aligned}$$

(3.84)

Proof

Since $V_O$, $V_{U}^n$, $V_{E}^1$, $V_{E}$, E, $U_O$ and $U_n$ satisfy (3.50), (3.54)–(3.56) and (3.58), and since V(0, 1) is given by

$$\begin{aligned} V(0,1)=\frac{1-\theta }{1+r}[aV_E^1+(1-a)V_U^{-1}], \end{aligned}$$

(3.82) can be rewritten as

$$\begin{aligned} RHS= & {} \displaystyle E[(\theta +b-\theta b)V_E^1+(1-\theta )(1-b)V_E]+\sum _{n=\overline{n}}^{-1}U_n V_U^n+U_OV_O\\&\displaystyle +\frac{\theta N}{1-\theta }\sum _{t=0}^\infty \frac{V(0,1)}{(1+r)^t},\\= & {} \displaystyle (\theta +b-\theta b)E\left\{ w^1-e+\frac{1-\theta }{1+r}[(1-b)V_E+bV_U^{-1}]\right\} \\&+\displaystyle (1-\theta )(1-b)E\left\{ w^{2+}-e+\frac{1-\theta }{1+r}[(1-b)V_E+bV_U^{-1}]\right\} \\&+\displaystyle U_{\overline{n}}\left\{ \overline{w}-e+\frac{1-\theta }{1+r}[aV_E^1+(1-a)V_O]\right\} \\&+\displaystyle \sum _{n=\overline{n}+1}^{-1}U_n\left\{ \overline{w}-e+\frac{1-\theta }{1+r}[aV_E^1+(1-a)V_U^{n-1}]\right\} \\&+\displaystyle U_O\left( \overline{w}+\frac{1-\theta }{1+r}V_O\right) +\frac{\theta N}{1+r}[aV_E^1+(1-a)V_U^{-1}]\\&+\frac{\theta N}{1-\theta }\sum _{t=1}^\infty \frac{V(0,1)}{(1+r)^t}. \end{aligned}$$

Using $\tilde{w}$ to rearrange the above equation, we can obtain

$$\begin{aligned} RHS= & {} \tilde{w}E+\overline{w}(N-E)-e(N-U_O)+(1+r)^{-1}\\&\times \left\{ \begin{array}{l} a\left[ (1-\theta )\sum _{n=\overline{n}}^{-1}U_n+\theta N \right] V_E^1\\ +(1-\theta )(1-b)E V_E\\ +\left[ (1-\theta )bE+\theta (1-a)N\right] V_U^{-1}\\ +\sum _{n=\overline{n}}^{-2}(1-\theta )(1-a)U_{n+1} V_U^{n}\\ +\left[ (1-\theta )(1-a)U_{\overline{n}}+(1-\theta )U_O\right] V_O\\ +[\theta N/(1-\theta )]\sum _{t=0}^\infty V(0,1)/(1+r)^t \end{array} \right\} \\= & {} \tilde{w}E+\overline{w}(N-E)-e(N-U_O)+(1+r)^{-1}\\&\times \left\{ \begin{array}{l} E[(\theta +b-\theta b)V_E^1+(1-\theta )(1-b)V_E]\\ +\sum _{n=\overline{n}}^{-1}U_n V_U^{n}+U_O V_O\\ +[\theta N/(1-\theta )]\sum _{t=0}^\infty V(0,1)/(1+r)^t \end{array} \right\} \\= & {} \tilde{w}E+\overline{w}(N-E)-e(N-U_O)+(1+r)^{-1}RHS, \end{aligned}$$

the second equality of which is obtained from (3.69)–(3.72). The obtained result implies that WS must satisfy

$$\begin{aligned} WS=\frac{1+r}{r}\left[ \tilde{w}E+\overline{w}(N-E)-e(N-U_O)\right] , \end{aligned}$$

which is equivalent to (3.83). $\square$

1.2 Investors’ Surplus

We can also show that the IS equals the discounted sum of investors’ expected gains from past, current, and future investments,

$$\begin{aligned} \begin{array}{l} IS=\displaystyle \sum _{t=0}^\infty \left( \frac{1}{1+r}\right) ^t\cdot \frac{bE}{L^*}\cdot \left[ -F+\frac{(L^{*})^\alpha -w^{*}L^{*}}{r+b}\right] +\frac{E}{L^{*}}[(L^{*})^\alpha -\tilde{w}L^{*}]\\ \qquad \quad +\displaystyle \sum _{t=1}^\infty \left( \frac{1-b}{1+r}\right) ^t\cdot \frac{E}{L^{*}}\left\{ (L^{*})^\alpha -[\theta w^1+(1-\theta )w^{2+}]L^{*}\right\} , \end{array} \end{aligned}$$

(3.85)

where $w^{*}$, $L^{*}$, and $\tilde{w}$ are defined by (3.9) and (3.84). The first term on the RHS of (3.85) is the expected sum of net gains from establishing new firms in and after the current period. As already seen in the proof of Lemma 3.2, this term equals zero under free entry. The second and third terms are the expected sum of the profits that currently operating firms will distribute to their investors in future periods.

Proposition 3.6

The RHS of (3.85) can be expressed as

$$\begin{aligned} RHS=\sum _{t=0}^\infty \left( \frac{1}{1+r}\right) ^t\cdot \frac{E}{L^*}\cdot [(L^{*})^\alpha -\tilde{w}L^{*}-bF]. \end{aligned}$$

(3.86)

Proof

The second and third terms on the RHS of (3.85) can be reduced to

$$\begin{aligned} \frac{1+r}{r+b}\cdot \frac{E}{L^*}\cdot [(L^*)^\alpha -\tilde{w}L^*+b(w^1-w^*)L^*]\ (\equiv X). \end{aligned}$$

Since the first term equals zero, we can safely say that X gives the value of the RHS. Note that

$$\begin{aligned} 0= & {} b\left[ -F+\frac{(L^*)^\alpha -w^*L^*}{r+b}\right] \\= & {} (L^*)^\alpha -\tilde{w}L^*-bF+\frac{-r[(L^*)^\alpha -\tilde{w}L^*]+b(\tilde{w}-w^*)L^*}{r+b}\\= & {} (L^*)^\alpha -\tilde{w}L^*-bF-\frac{r[(L^*)^\alpha -\tilde{w}L^*+b(w^1-w^*)L^*]}{r+b}\\= & {} (L^*)^\alpha -\tilde{w}L^*-bF-\frac{r}{1+r}\cdot \frac{L^*}{E}X, \end{aligned}$$

the third equality of which is obtained from the fact that

$$\begin{aligned} \tilde{w}-w^*= & {} \tilde{w}-\hat{w}\\= & {} -[r/(1+r)](1-\theta )(1-b)(w^1-w^{2+})\\= & {} -r(w^1-\hat{w})\\= & {} -r(w^1-w^*), \end{aligned}$$

where $\hat{w}$ is as defined in (3.6). The obtained result implies that the RHS must satisfy

$$\begin{aligned} RHS=\frac{1+r}{r}\cdot \frac{E}{L^{*}}\cdot [(L^{*})^\alpha -\tilde{w}L^{*}-bF], \end{aligned}$$

which is equivalent to (3.86). $\square$