Heterogeneous entrepreneurs from occupational choices in economies with minimum wages

Abstract

This paper examines how the distribution of skills, organizational diseconomies of size, and the introduction of a minimum wage affect the total output of the economy and the relative size and average per capita income of employees, solo self-employed, and employers, in a market equilibrium from occupational choices of individuals with different skills. The model explains the heterogeneity observed among occupational groups beyond employers and employees, and predicts differences in the sing of the association between entrepreneurs’ rates and economic development depending on the entrepreneurs’ type. The introduction of a minimum wage leads to involuntary solo self-employed, more skilled voluntary solo self-employed, and fewer employers. A minimum wage lowers total output and has income redistribution effects: income per capita decreases in the individuals in the two extremes of the skills’ distribution and increases in the middle class.

Appendices

Appendix 1: The market equilibrium solution for the power production function and the uniform distribution of skills

This appendix shows the calculations used to obtain the market equilibrium solution for the occupational choice problem formulated in Sect. 3. We recall that the skills of the individuals are uniformly distributed in the interval [e _m , e _M ], and the production function of an employer with skills e combined with E units of operational skills from salaried employees is given by

$$ Q = \theta eEf\left( {e/E} \right) = \theta eE\left( {e/E} \right)^{\beta } = \theta e^{1 + \beta } E^{1 - \beta } $$

(19)

The profit-maximizing demand for skills from (3) and (19) is equal to

$$ E^{*} = \left( {\frac{{\theta \left( {1 - \beta } \right) e^{1 + \beta } }}{w}} \right)^{{\frac{1}{\beta }}} $$

(20)

Substituting back into the profit function, we obtain

$$ \varPi^{*} \left( e \right) = w^{{\frac{{ - \left( {1 - \beta } \right)}}{\beta }}} \theta^{{\frac{1}{\beta }}} \beta \left( {1 - \beta } \right)^{{\frac{1 - \beta }{\beta }}} e^{{\frac{1 + \beta }{\beta }.}} $$

The market equilibrium conditions with only two occupational choices, employers and employees, (Eqs. (4) and (5) in the main text) are

$$ w^{{\frac{\beta - 1}{\beta }}} \theta^{{\frac{1}{\beta }}} \beta \left( {1 - \beta } \right)^{{\frac{1 - \beta }{\beta }}} e^{{*\frac{1 + \beta }{\beta }}} = we^{*} $$

(21)

$$ \mathop \smallint \limits_{{e^{*} }}^{{e_{M} }} \left[ {\frac{{\theta \left( {1 - \beta } \right) e^{1 + \beta } }}{w}} \right]^{{\frac{1}{\beta }}} {\text{d}}H\left( e \right) = \frac{1}{{2 \left( {e_{M} - e_{m} } \right)}}\left[ {1 - \frac{1}{{\left( {2 m - 1} \right)^{2} }}} \right]e^{*2} $$

(22)

Based on the Eq. (21), we can solve for skill price (as a function of e ^*):

$$ w\left( {e^{*} } \right) = \theta \beta^{\beta } \left( {1 - \beta } \right)^{1 - \beta } e^{*} $$

(23)

Substituting into Eq. (22), we obtain the equilibrium level of skills that makes the choice between being an employee or an entrepreneur–manager indifferent (e ^*):

$$ e^{*} = \left[ {\frac{{\frac{{\left( {2 m - 1} \right)^{2} }}{{\left( {2 m - 1} \right)^{2} - 1 }}2 \left( {1 - \beta } \right)}}{{\frac{{\left( {2 m - 1} \right)^{2} }}{{\left( {2 m - 1} \right)^{2} - 1}}2\left( {1 - \beta } \right) + \left( {1 + 2 \beta } \right)}}} \right]^{{\frac{\beta }{1 + 2 \beta }}} e_{M} $$

(24)

There are $\frac{{e_{\text{M}} - e^{*} }}{{e_{\text{M}} - e_{\text{m}} }}$ employers and $\frac{{e^{*} - e_{0} }}{{e_{\text{M}} - e_{\text{m}} }}$ employees in the equilibrium. The remainder $\frac{{e_{0} - e_{\text{m}} }}{{e_{\text{M}} - e_{\text{m}} }}$ are unemployed (when e ₀ > e _m or e ^* > (2 m − 1)e _m , from (15)). The equilibrium average of employees per employer, as a span of control (ASC ^*), is equal to

$$ {\text{ASC}}^{*} = \frac{{e^{*} - e_{0} }}{{e_{M} - e^{*} }} = \left\{ {\begin{array}{*{20}c} {\frac{{e^{*} }}{{e_{M} - e_{m} }}\frac{{2 \left( {m - 1} \right)}}{2 m - 1}} & {{\text{if}}\;e_{0} > e_{m} } \\ {\frac{{e^{*} - e_{m} }}{{e_{M} - e^{*} }}} & {{\text{if}}\;e_{0} = e_{m} } \\ \end{array} } \right. $$

(25)

The total produced output is given by $Q^{*} = \mathop \smallint \nolimits_{{e^{*} }}^{{e_{M} }} e^{1 + \beta } \left( {E^{*} \left( e \right)} \right)^{1 - \beta } {\text{d}}H(e)$:

$$ Q^{*} = \frac{{\theta e_{M}^{3} }}{{\left( {e_{M} - e_{m} } \right)}}\frac{{\beta^{\beta } \left( {1 - \beta } \right)^{1 - \beta } }}{1 + 2 \beta }\left( {\frac{{e^{*} }}{{e_{M} }}} \right)^{{\frac{{ - \left( {1 - \beta } \right)}}{\beta }}} \left[ {1 - \left( {\frac{{e^{*} }}{{e_{M} }}} \right)^{{\frac{1 + 2 \beta }{\beta }}} } \right] $$

(26)

1.1 Voluntary and involuntary solo self-employment

1.1.1 Voluntary solo self-employment

The voluntary solo self-employed are individuals who have sufficient skills to work as salaried employees but choose to be self-employed without employees. In the two-choice economy of employers and employees, an individual with skills e working as an employee makes an income of w(e ^*)e where w(e ^*) is the equilibrium price per unit of skill (23). An equilibrium of employers and employees is not sustainable if R(e), the income from solo self-employment, is higher than the salaried-employee income w(e ^*)e for some e ≤ e ^*. Because salary is increasing and linear in e and income from self-employment is increasing and convex in e, a necessary condition for an equilibrium with voluntary solo self-employed is that R(e) intersects w(e ^*)e from below at some value of e < e ^*. Conversely, the maximum value of k for which the equilibrium with employers and employees is sustainable is k _c for which the revenue of the highest skilled employee as self-employed, R(e ^*), is equal to the salary as employee, w(e ^*) e ^*: R(e ^*) = w(e ^*)e ^*, or θβ ^β(1 − β)^1−β e ^*2 = θk _c e ^*2. Therefore,

$$ k_{\text{c}} = \beta^{\beta } \left( {1 - \beta } \right)^{1 - \beta } $$

(27)

If k < k _c then for e < e ^*, income as a solo self-employed is always lower than income as a salaried employee, whereas if k > k _c , then at e ^*, income as a self-employed is higher, and therefore, the intersection between R(e) for this value of k and w(e ^*)e occurs at a value of e < e ^*.

1.1.2 Involuntary self-employment

The indifference condition for people with skills lower than e ₀ working as (involuntary) self-employed or staying unemployed is R(e _s ) = θke ² _s  = U. Solving this equation derives the following equation

$$ e_{s} = \left( {\frac{U}{\theta k}} \right)^{1/2} $$

(28)

1.1.3 Market equilibrium solution

The equations that determine the critical levels of skills e ₂ and e ₁ as a function of the price per unit of skill w are

$$ W(e_{1} ) = we_{1} = R(e_{1} ) = k\theta e_{1}^{2} $$

(29)

$$ \varPi (e_{2} ) = w^{{\frac{ - (1 - \beta )}{\beta }}} \theta^{{\frac{1}{\beta }}} \beta (1 - \beta )^{{\frac{1 - \beta }{\beta }}} e_{2}^{{\frac{1 + \beta }{\beta }}} = R(e_{2} ) = k\theta e_{2}^{2} $$

(30)

Solving Eqs. (29) and (30), we obtain

$$ w = k\theta e_{1} $$

(31)

and

$$ e_{1} = \left( {\frac{{\beta^{\beta } (1 - \beta )^{1 - \beta } }}{k}} \right)^{{\frac{1}{1 - \beta }}} e_{2} $$

(32)

The equilibrium value e ₂ and the equilibrium price per unit of skill obtained from this value were dependent on the condition that the supply of employee skills equals the demand of skills by employers

$$ \int\limits_{{e_{2} }}^{{e_{M} }} {E\left( {w(e_{1} ),e} \right){\text{d}}H(e) = \int\limits_{{e_{2} }}^{{e_{M} }} {\left( {\frac{1 - \beta }{{ke_{1} }}} \right)}^{1/\beta } } e^{{\frac{1 + \beta }{\beta }}} {\text{d}}H(e) = \int\limits_{{e_{0} }}^{{e_{1} }} {e{\text{d}}H(e} ) $$

Taking into account (31) and (32), the solution to this equation is

$$ \frac{{e_{2} }}{{e_{M} }} = \left( {\frac{{2k^{{\frac{3}{1 - \beta }}} \left( {\frac{{(2m - 1)^{2} }}{{(2m - 1)^{2} - 1}}} \right)}}{{(1 + 2\beta )(1 - \beta )^{2} \beta^{{\frac{3\beta }{1 - \beta }}} + 2k^{{\frac{3}{1 - \beta }}} \left( {\frac{{(2m - 1)^{2} }}{{(2m - 1)^{2} - 1}}} \right)}}} \right)^{{\frac{\beta }{1 + 2\beta }}} $$

(33)

The solution e ₂ in (33) is substituted into (32) to obtain the equilibrium value of e ₁; we can then obtain $e_{0} = {\text{Max}}\left( {\frac{{e_{1} }}{2m - 1},e_{m} } \right)$. There are $\frac{{e_{M} - e_{2} }}{{e_{M} - e_{m} }}$ persons working as employers–managers in the equilibrium; $\frac{{e_{2} - e_{1} }}{{e_{M} - e_{m} }} + \frac{{e_{0} - e_{s} }}{{e_{M} - e_{m} }}$ own-account self-employed and $\frac{{e_{1} - e_{0} }}{{e_{M} - e_{m} }}$ salaried employees. The number of persons remaining out of work in the economy is $\frac{{e_{s} - e_{m} }}{{e_{M} - e_{m} }}$. The average span of control is $\frac{{e_{1} - e_{0} }}{{e_{M} - e_{2} }} = \frac{{e_{1} }}{{e_{M} - e_{2} }}\frac{2(m - 1)}{2m - 1}$.

The total produced output in the equilibrium with three occupational choices is

$$ {\text{QT}}^{*} = \int_{{e_{2}^{{}} }}^{{e_{M} }} {\theta e^{1 + \beta } \left( {E^{*} (w,e)} \right)^{1 - \beta } {\text{d}}H(e)} + \int_{{e_{1} }}^{{e_{2} }} {k\theta e^{2} {\text{d}}H(e)} + \int_{{e_{s} }}^{{e_{0} }} {k\theta e^{2} {\text{d}}H(e)} $$

In substituting the values of the demand for skills E ^*(w, e ^*), we obtain, after certain arrangements, the following solution

$$ {\text{QT}}^{*} = \frac{\theta k}{{e_{M} - e_{m} }}\left\{ {\frac{{e_{M}^{3} }}{(1 + 2\beta )}} \right.\left[ {\left( {\frac{{e_{2} }}{{e_{M} }}} \right)^{{1 - \frac{1}{\beta }}} - \frac{{e_{2}^{3} }}{{e_{M}^{3} }}} \right] + \frac{{e_{2}^{3} }}{3}\left( {\frac{{\beta^{\beta } (1 - \beta )^{1 - \beta } }}{k}} \right)^{{\frac{3}{1 - \beta }}} \left[ {\frac{1}{{\left( {2m - 1} \right)^{3} }} - 1} \right] + \left. {\frac{{e_{2}^{3} - e_{s}^{3} }}{3}} \right\} $$

(34)

Appendix 2: Comparative static analysis in Table 1

We will now outline the proof of the comparative static analysis in Table 1. In all cases, the comparative static results correspond to the case when e ₁ > (2 m − 1)e _m and e _s  = e _m ; the reservation income U is arbitrarily set equal to zero.

1.1 Number of employers

The number of employers in the equilibrium is equal to

$$ \frac{{e_{M} - e_{2} }}{{e_{M} - e_{m} }} = \frac{{1 - \left( {e_{2} /e_{M} } \right)}}{{1 - \left( {e_{m} /e_{M} } \right)}} $$

(35)

where (e ₂/e _M ) is given by (33).

1.1.1 Changes in the distribution of skills

We compare the equilibrium value in (35) for the original distribution of skills [e _m , e _M ] with the equilibrium value under the new distribution [e _m  + d, e _M  + d], where d is a positive constant for the mean increasing case and with an equilibrium value under the distribution [e _m  − d, e _M  + d] for the variance increase case.

Because (e ₂/e _M ) is independent of the skills distribution parameters in Eq. (34), all of the effects from the changes in the skills distribution parameters in the equilibrium number of employers are explained by the sign of the differences between the ratio $\frac{{e_{m} }}{{e_{M} }}$ and the ratios $\frac{{e_{m} + d}}{{e_{M} + d}}$ and $\frac{{e_{m} - d}}{{e_{M} + d}}$. The expression $\frac{{e_{m} + d}}{{e_{M} + d}}$ is higher than $\frac{{e_{\text{m}} }}{{e_{\text{M}} }}$, whereas the ratio of $\frac{{e_{m} - d}}{{e_{M} + d}}$ is lower. Therefore, a higher mean with an equal variance in skills distribution increases the number of employers in the equilibrium. A higher variance of the skills distribution with an equal mean reduces the equilibrium number of employers–managers.

1.1.2 Organizational size diseconomies

The equilibrium number of employers in (35) depends on the organizational diseconomies parameter β through the term (e ₂/e _M ) in (33). The sign of the derivative of (e ₂/e _M ) with respect to β is undetermined, so the change in the number of employers resulting from changes in the organizational size diseconomies parameter can be either positive or negative. As indicated in the text, the sign shown in Table 1 near the question mark is obtained from the simulations of the market equilibrium for a long list of combinations of values of the model parameters. The simulations indicate that higher values of β are expected to be associated with a higher number of employers.

1.1.3 Productivity parameter k and minimum wage

Based on (33), (e ₂/e _M ) increases with k and decreases with m. Therefore, the equilibrium number of employers decreases when the comparative disadvantage from the lack of specialization of the own-account self-employed is lower (a higher k). Higher values of m imply a lower endogenous minimum wage, which creates a higher number of employer–managers in the equilibrium.

1.2 Voluntary solo self-employed

Taking into account (32), the number of voluntary solo self-employed in the equilibrium is equal to

$$ \frac{{e_{2} - e_{1} }}{{e_{M} - e_{m} }} = \left( {\frac{{e_{2} /e_{M} }}{{1 - \left( {e_{m} /e_{M} } \right)}}} \right)\left[ {1 - \left( {\frac{{\beta^{\beta } (1 - \beta )^{1 - \beta } }}{k}} \right)^{{\frac{1}{1 - \beta }}} } \right] $$

(36)

1.2.1 Distribution of skills

Because (e ₂/e _M ) and k are independent of the extreme values of the uniform distribution, the effective sign from the changes in the mean and variance of the distribution of skills on the equilibrium number of voluntary solo self-employed is the same as for the aforementioned employer–managers.

1.2.2 Organizational size diseconomies

The indeterminacy in the sign of the changes in (e ₂/e _M ) to the changes in β implies that the sign of the changes in the number of voluntary solo self-employed from the changes in the organizational size diseconomies is ambiguous. The simulations suggest a positive effect of β on the number of voluntary solo self-employed.

1.2.3 Productivity parameter k and minimum wage

Based on (33), (e ₂/e _M ) increases with k and decreases with m. Accounting these results, the equilibrium number of voluntary solo self-employed in (36) increases when the comparative disadvantage from the lack of specialization in solo self-employed is lower (higher k), and it decreases as the minimum wage becomes lower (higher m).

1.3 Involuntary solo self-employed

The equilibrium number of involuntary solo self-employed is given by the following (remembering that e _s  = e _m ):

$$ \frac{{e_{0} - e_{s} }}{{e_{M} - e_{m} }} = \frac{{\frac{{e_{1} }}{{e_{M} }}\left( {\frac{1}{2m - 1}} \right) - \frac{{e_{s} }}{{e_{M} }}}}{{1 - \left( {e_{m} /e_{M} } \right)}} = \frac{{\frac{{e_{2} }}{{e_{M} }}\left( {\frac{1}{2m - 1}} \right)\left( {\frac{{\beta^{\beta } (1 - \beta )^{1 - \beta } }}{k}} \right)^{{\frac{1}{1 - \beta }}} - \frac{{e_{m} }}{{e_{M} }}}}{{1 - \left( {e_{m} /e_{M} } \right)}} $$

(37)

All the terms in (37) appear in the equilibrium values of the aforementioned employers and voluntary solo self-employed. Therefore, we already know the sign of the changes in these terms from the changes in the model parameters.

1.3.1 Distribution of skills

We compare the value of (37) with the respective value for the new distribution of skills with higher mean and equal variance [e _m  + d, e _M  + d] for d positive. It is straightforward that (37) increases with d positive, so the number of involuntary solo self-employed decreases as the average skills increase (for a given variance). We then repeat the exercise with the new distribution with equal mean and higher variance [e _m  − d, e _M  + d]. Now, the comparison of the respective values of (37) for d equal and greater than zero provides the opposite results: the number of involuntary solo self-employed increases with the dispersion of the distribution of skills for the equal mean.

1.3.2 Organizational size diseconomies

The indeterminacy in the sign of the changes in (e ₂/e _M ) to the changes in β implies that the sign of the changes in the number of voluntary solo self-employed from the changes in the organizational diseconomies is ambiguous. However, the numerical analysis shows that the number of involuntary solo self-employed will be lower in economies with higher organizational size diseconomies.

1.3.3 Productivity parameter k and minimum wage

Based on the numerator of (37), the equilibrium number of the involuntary solo self-employed decreases with the parameter k and with a lower minimum wage (higher m). These findings imply a lower number of involuntary solo self-employed.

1.4 Employees

The equilibrium number of employees is given by

$$ \frac{{e_{1} - e_{0} }}{{e_{M} - e_{m} }} = \frac{2(m - 1)}{2m - 1}\left( {\frac{{\beta^{\beta } (1 - \beta )^{1 - \beta } }}{k}} \right)^{{\frac{1}{1 - \beta }}} \frac{{e_{2} /e_{M} }}{{1 - \left( {e_{m} /e_{M} } \right)}} $$

(38)

Because (e ₂/e _M ) is independent of the extreme values of the uniform distribution, the sign of the effect of the changes in the mean and the variance of the distribution of skills on the equilibrium number of voluntary solo self-employed and employees is the same as for the aforementioned employers.

The number of employees decreases with k and increases with m (the term $\frac{2(m - 1)}{2m - 1}$ in (38) increases with m).

1.5 Unemployment

Unemployment occurs when e _s  > e _m . The number of individuals with a lower skills level that are unwilling to become solo self-employed in the economy is given by

$$ \frac{{e_{s} - e_{m} }}{{e_{M} - e_{m} }} = \frac{{\left( {\frac{U}{\theta k}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} - e_{m} }}{{e_{M} - e_{m} }} $$

(39)

Shifts to the right of the distribution of skills (with a constant variance) lower unemployment, but a higher variance for a given mean increases unemployment.

Unemployment increases with the reservation rent U and decreases with positive productivity shocks (higher θ) and higher k.

1.6 Average span of control

The endogenous variable of interest now is

$$ \frac{{e_{1} - e_{0} }}{{e_{M} - e_{2} }} = \frac{{e_{1} }}{{e_{M} - e_{2} }}\frac{2(m - 1)}{2m - 1} = \frac{2(m - 1)}{2m - 1}\left( {\frac{{\beta^{\beta } (1 - \beta )^{1 - \beta } }}{k}} \right)^{{\frac{1}{1 - \beta }}} \frac{{e_{2} /e_{M} }}{{1 - \left( {e_{2} /e_{M} } \right)}} $$

(40)

The average span of control is independent of the bounds of the skills distribution and therefore independent of the mean and the dispersion of skills. The sign of the changes in the average span of control to changes in k an m is undetermined. The calculations indicate that the average span of control increases with parameter k and with a lower relative minimum wage (higher m). Furthermore, the higher organizational size diseconomies imply a lower average span of control, so long as (e ₂/e _M) also decreases with β.

1.7 Produced output

The endogenous variable is now the total produced output from (34). The output increases with the mean of the skills distribution for a given variance and with the variance in the skills for a given mean. The positive effect on the total output from the higher upper bound, e _M  + d, is higher than the negative effect from the change in the lower bound of the distribution, e _M  − d.

The total output decreases with β in the range of reasonable values. It increases with k and m (lower relative minimum wage).