The Cost of Unemployment from a Social Welfare Approach: The Case of Spain and Its Regions

Abstract

This paper proposes a protocol to measure the cost of unemployment by taking into account three different aspects: incidence, severity and hysteresis. Incidence refers to the conventional unemployment rate; severity takes into account both unemployment duration and the associated income loss; and hysteresis refers to the probability of remaining unemployed. The cost of unemployment is regarded as a welfare loss, which is measured by a utilitarian social welfare function whose arguments are the individual disutilities of unemployed workers. Each individual disutility is modelled as a function of income loss, unemployment duration and hysteresis. The resulting formula is simple and easy to understand and implement. We apply this assessment protocol to the Spanish labour market, focusing on the regional differences and using the official register of unemployed workers compiled by the Public Employment Service.

Appendices

Appendix 1: The Formal Model

Consider the following extremely simple model of a worker whose utility depends on income and leisure according to the symmetric Cobb–Douglas function given by:

$$u\left( {y,L} \right) = ay^{1/2} L^{1/2}$$

where y stands for income, L for leisure, and a is a coefficient that defines the units in which utility is measured. Let T stand for the total amount of time available in a given period to be allocated between labour and leisure and let w denote the corresponding wage rate. Then, y = w(T − L), which results in:

$$u\left( {y,L} \right) = a\left[ {w(T - L)} \right]^{1/2} L^{1/2}$$

The consumer’s optimal choice consists of working for half of the available time and devoting the remaining half to leisure.⁹ That is, y* = wT/2, L* = T/2, so that: u* = u(y*, L*) = aw^1/2(T/2). By letting a = 2/T the following emerges:

$$u^{*} \, = u\left( {y^{*},L^{*}} \right) \, = w^{1/2}$$

That is, utility in equilibrium can be approximated by the square root of the market wage. When a worker h is unemployed he/she may receive: (a) an unemployment benefit s_h per period (typically for a maximum of q* periods provided he/she has earned the pertinent rights); or (b) a social subsidy z_h > 0, or nothing, z_h = 0, per period. Therefore, we will have:

$$u_{h}^{0} = \left\{ {\begin{array}{*{20}l} {\left( {s_{h} } \right)^{1/2} } \hfill & {ifunemployment\;benefit} \hfill \\ {\left( {z_{h} } \right)^{1/2} } \hfill & {otherwise} \hfill \\ \end{array} } \right.$$

We can now define the average utility loss of a worker h unemployed for q_h periods as the following cost function:

$$c_{h} (.) = \left\{ {\begin{array}{*{20}l} {\left. {\begin{array}{*{20}l} {\left( {w_{h} } \right)^{1/2} - \left( {s_{h} } \right)^{1/2} ,} \hfill & {if\;q_{h} \le q*} \hfill \\ {\frac{{\left( {w_{h} } \right)^{1/2} q_{h} - \left( {s_{h} } \right)^{1/2} q* - \left( {z_{h} } \right)^{1/2} \left( {q_{h} - q*} \right)}}{{q_{h} }}} \hfill & {if\;q_{h} > q*} \hfill \\ \end{array} } \right\}} \hfill & {with\;unemp.\;benefits} \hfill \\ {\left( {w_{h} } \right)^{1/2} - \left( {z_{h} } \right)^{1/2} } \hfill & {with\;no\;unemployment\;benefits} \hfill \\ \end{array} } \right.$$

That is, the average disutility per period is measured by a cost function that reflects the impact on the agent’s utility of the corresponding income loss with respect to being employed. When the unemployed worker is entitled to unemployment benefits, the average utility loss is simply the square root of the lost wage minus the square root of the unemployment benefit. The worker may not be receiving those benefits either because was never entitled to them or because the unemployment spell exceeds the critical value q* (2 years in the case of Spain). In the first case the average utility corresponds to the difference of the square root of the lost wage and the square root of the social subsidy, if any. In the second case the average is slightly more involved because one has to compute the utility loss when receiving unemployment benefits for part of the time and social subsidies, if any, for the remaining time.¹⁰

The simplest way of obtaining an overall measure of disutility for a worker h unemployed for q_h months would be multiplying the average disutility per period by the number of periods. Yet it is sensible to think of duration entering disutility as an increasing and convex function, f(q_h), as an extra month of unemployment is worse the longer the unemployment spell. Recall on this point that the degree of convexity of a function is related to its curvature, which is controlled by its second derivative, usually expressed in terms of the elasticity of the first derivative. In our case that elasticity measures the relative change in the marginal impact of duration on the relative change of individual unemployment length. The simplest constraint to control for the degree of convexity in this context is to assume constant elasticity. This makes it possible to parameterize the impact of unemployment duration by a single number: the value of the elasticity of marginal impact of duration, ν. The function that performs this task is well known and can be expressed as $f\left( {q_{h} } \right) = q_{h}^{{1 + \nu_{h} }}$, where $\nu_{h}$ stands for the elasticity of the marginal impact of duration for agent h.¹¹ This gives the following formula for the aggregate disutiluty of an agent:

$$d_{h} = c_{h} (.)q_{h}^{{1 + \nu_{h} }}$$

The overall per capita disutility can thus be expressed as:

$$D_{N} = \frac{1}{n}\sum\limits_{{h \in U_{N} }} {c_{h} (.)q_{h}^{{1 + \nu_{h} }} = \frac{{n^{U} }}{n} \times \frac{{\sum\nolimits_{{h \in U_{n} }} {c_{h} (.)q_{h}^{{1 + \nu_{h} }} } }}{{n^{U} }}}$$

This equation can be decomposed multiplicatively into three different components, along the lines in Shorrocks (2009b). To so let:

$$C^{U} = \frac{{\sum\nolimits_{{h \in U_{N} }} {c_{h} ( \cdot )} }}{{n^{U} }}, \ldots q_{N} = \frac{{\sum\nolimits_{{h \in U_{N} }} {q_{h} } }}{{n^{U} }}$$

denote the average income loss of the unemployed and the average duration of unemployment in society, respectively, and let $\nu_{N}$ the average probability or remaining unemployed. Now consider the following elementary rewriting of D_N:

$$D_{N} = \frac{{n^{U} }}{n} \times C^{U} q_{N}^{{1 + \nu_{N} }} \times \frac{1}{{n^{U} }}\sum\limits_{{h \in U_{N} }} {\frac{{c_{h} (.)q_{h}^{{1 + \nu_{h} }} }}{{C^{U} q_{N}^{{1 + \nu_{N} }} }}}$$

The first component of this expression corresponds to the incidence of unemployment (the head count ratio, H_N). The second term, $C^{U} q_{N}^{{1 + \nu_{N} }}$, is a measure of the intensity of unemployment, S_N, given by the average disutility of the unemployed. Finally, the third term is a measure of inequality in disutility among the unemployed, I_N, which is given by the sum of the shares of individual disutility in average disutility: To get an inequality measure that yields a zero value when all disutilities are equal, we take the following:

$$I_{N} = \frac{1}{{n^{U} }}\sum\limits_{{h \in U_{N} }} {\frac{{c_{h} ( \cdot )q_{h}^{{1 + \nu_{h} }} }}{{C^{U} q_{N}^{{1 + \nu_{N} }} }}} - 1$$

Appendix 2: Estimation Results

See Table 4.

Table 4 Probability of finding a job (1 − v)