Relative Absence Concerns, Positional Consumption Preferences and Working Hours

Abstract

Positional income or consumption preferences can induce individuals to work too much to enhance their relative standing. Empirical evidence suggests that people are also characterised by relative considerations with respect to sickness-related absence from work. We analyse theoretically how relative consumption and absence concerns interact. Although relative absence concerns may mitigate the consequences of relative consumption preferences, the market outcome will never be efficient. Hence, we derive the income tax rate and the level of sick pay which induce efficient behaviour as the market outcome.

Manfred Holler’s list of publications (in Google Scholar) contains more than 400 entries. Few of them explicitly deal with labour issues—although he has written two textbooks in German on labour economics (Goerke & Holler, 1997; Holler, 1986). My academic interests, which developed during the work as Manfred’s chair in Hamburg, attest to the climate of intellectual openness and curiosity, which he created. Braham and Steffen (2008, p. vi) write in their introduction to the Festschrift for Manfred’s 60th birthday: ‘(H)e has always made every effort to free his staff from unnecessary administrative burdens and he never burdened anyone with his own work’. This implied that staff members were free to pursue their own research projects, also if only modestly linked to or even without any relationship to Manfred’s work. While, to the best of my knowledge, Manfred has not worked on positional concerns, the present analysis was certainly inspired by his attitude of openness and tolerance.

Appendices

Appendix

1. Utility Maximum

From an individual’s perspective, the reference levels of consumption and absence, profits, the tax rate and the lump-sum transfer are constant. Therefore, the second-order conditions for a maximum are given by Z_hh, Z_aa < 0 < Z_hhZ_aa − (Z_ah)², where Z_h and Z_a are defined in Eqs. (3a) and (3b) and the net wage equals $w^{{\text{N}}}$ = w[1 − t].

$$Z_{{{\text{hh}}}} = u_{11} \left[ {w^{{\text{N}}} } \right]^{2} - H^{\prime\prime} < 0$$

(15a)

$$Z_{{{\text{aa}}}} = u_{11} \left[ {s - w^{{\text{N}}} } \right]^{2} + v_{11} < 0$$

(15b)

$$Z_{{{\text{ha}}}} = Z_{{{\text{ah}}}} = u_{11} w^{{\text{N}}} \left[ {s - w^{{\text{N}}} } \right] \ge 0$$

(15c)

Hence, we have

$$Z_{{{\text{hh}}}} Z_{{{\text{aa}}}} - \mathop {Z_{{{\text{ah}}}} }\nolimits^{2} = u_{11} \left[ {v_{11} \left[ {w^{{\text{N}}} } \right]^{2} - H^{\prime\prime}\left[ {s - w^{{\text{N}}} } \right]^{2} } \right] - H^{\prime\prime}v_{11} > 0$$

(16)

2. Stability of Market Equilibrium

In equilibrium, lump-sum payments, T, are determined endogenously in order to balance the budget. Thus, T = wt[h − a]. Moreover, profits as defined in (4) are paid out to individuals and affect their consumption. Hence, the equilibrium level of consumption equals production:

$$\begin{aligned} c^{{\text{m}}} = & w\left[ {1 {-} t} \right]\left[ {h^{{\text{m}}} {-}a^{{\text{m}}} } \right] + sa^{{\text{m}}} + wt\left[ {h^{{\text{m}}} {-}a^{m}} \right] + f\left( {h^{{\text{m}}} {-}a^{{\text{m}}} } \right) - w\left[ {h^{{\text{m}}} {-}a^{{\text{m}}} } \right] - sa^{{\text{m}}} \\ = & \,f\left( {h^{{\text{m}}} {-}a^{{\text{m}}} } \right) \\ \end{aligned}$$

(17)

To ascertain whether the market equilibrium is stable, we calculate the Jacobian determinant $\left| J \right|$ of the system defined by Eqs. (3a), (3b) and (5), taking into account (17). Moreover, all individuals behave identically. Hence, changes in consumption, c, and the reference level, $\overline{c}$, are the same. Similarly, the variations in a and $\overline{a}$ coincide. Thus, the derivatives of (3a), (3b) and (5) with respect to contractual hours, h, absence, a, and wages, w, incorporating (17), are given by π_hh = f″ = −π_ha < 0, π_hw = −1 and:

$$Z_{{{\text{hh}}}}^{{\text{e}}} = \left[ {u_{11} + u_{12} } \right]f^{\prime}w^{{\text{N}}} - H^{\prime\prime} < 0$$

(18a)

$$Z_{{{\text{aa}}}}^{{\text{e}}} = - \left[ {u_{11} + u_{12} } \right]f^{\prime}\left[ {s - w^{{\text{N}}} } \right] + v_{11} + \gamma v_{12} < 0$$

(18b)

$$Z_{{{\text{ha}}}}^{{\text{e}}} = - \left[ {u_{11} + u_{12} } \right]w^{{\text{N}}} f^{\prime} > 0$$

(18c)

$$Z_{{{\text{ah}}}}^{{\text{e}}} = \left[ {u_{11} + u_{12} } \right]\left[ {s - w^{{\text{N}}} } \right]f^{\prime} > 0$$

(18d)

$$Z_{{{\text{hw}}}}^{{\text{e}}} = u_{1} \left[ {1 - t} \right] = - Z_{{{\text{aw}}}}^{{\text{e}}} > 0$$

(18e)

In (18a) to (18e), we use the superscript e to indicate that equilibrium repercussions via lump-sum payments T and profits are incorporated. The Jacobian determinant $\left| J \right|$ of the system defined by the modified Eqs. (3a), (3b) and (5) is negative.

$$\begin{aligned} \left| J \right| = & - \left[ {Z_{{{\text{hh}}}}^\text{e} Z_{{{\text{aa}}}}^\text{e} - Z_{{{\text{ha}}}}^\text{e} Z_{{{\text{ah}}}}^\text{e} } \right] - f^{\prime\prime}Z_{{{\text{hw}}}}^\text{e} \left[ {Z_{{{\text{ha}}}}^\text{e} + Z_{{{\text{aa}}}}^\text{e} + Z_{{{\text{hh}}}}^\text{e} + Z_{{{\text{ah}}}}^\text{e} } \right] \\ = & \, - H^{\prime\prime}\left[ {\left[ {u_{11} + u_{12} } \right]\left[ {s - w^{{\text{N}}} } \right]f^{\prime} - \left[ {v_{11} + \gamma v_{12} } \right]} \right] \\ & + \,f^{\prime\prime}u_{1} \left[ {1 - t} \right]\left[ {H^{\prime\prime} - \left[ {v_{11} + \gamma v_{12} } \right]} \right] < 0 \\ \end{aligned}$$

(19)

3. Pareto-Efficient Allocation

The second-order conditions for a maximum of Γ are

$$\Gamma_{{{\text{hh}}}} = \left[ {u_{11} + 2u_{12} + u_{22} } \right]\left[ {f^{\prime}} \right]^{2} + \left[ {u_{1} + u_{2} } \right]f^{\prime\prime} - H^{\prime\prime} < 0$$

(20a)

$$\Gamma_{{{\text{aa}}}} = \left[ {u_{11} + 2u_{12} + u_{22} } \right]\left[ {f^{\prime}} \right]^{2} + \left[ {u_{1} + u_{2} } \right]f^{\prime\prime} + v_{11} + 2\gamma v_{12} + \gamma^{2} v_{22} < 0$$

(20b)

and ${\text{Det}} = \Gamma_{{{\text{hh}}}} \Gamma_{{{\text{aa}}}} - \mathop {\Gamma_{{{\text{ha}}}} }\nolimits^{2} > 0$. Using

$$\Gamma_{{{\text{ha}}}} = - \underbrace {{\left[ {u_{11} + 2u_{12} + u_{22} } \right]}}_{\left( - \right)}\left[ {f^{\prime}} \right]^{2} - \underbrace {{\left[ {u_{1} + u_{2} } \right]f^{\prime\prime}}}_{\left( - \right)} = - \Gamma_{{{\text{hh}}}} - H^{\prime\prime} > 0,$$

(21)

the determinant of the system of Eqs. (8a) and (8b) is found to be positive.

$$\begin{aligned} {\text{Det}} = & \underbrace {{\left[ {\left[ {u_{11} + 2u_{12} + u_{22} } \right]\left[ {f^{\prime}} \right]^{2} + \left[ {u_{1} + u_{2} } \right]f^{\prime\prime}} \right]}}_{\left( - \right)}\underbrace {{\left[ {v_{11} + 2\gamma v_{12} + \gamma^{2} v_{22} - H^{\prime\prime}} \right]}}_{\left( - \right)} \\ & - \,H^{\prime\prime}\underbrace {{\left[ {v_{11} + 2\gamma v_{12} + \gamma^{2} v_{22} } \right]}}_{\left( - \right)} > 0 \\ \end{aligned}$$

(22)

4. Alternative Specification of Preferences

Suppose preferences are given by

$$\tilde{Z}\left( {h,a} \right) = u\left( {c,\overline{c}} \right) - \tilde{H}\left( {h,a} \right) + v\left( {a,\gamma \overline{a}} \right),$$

(23)

where $\tilde{H}_{1}$ < 0 < $\tilde{H}_{2}$. Pareto-efficiency can then be characterised by maximising:

$$\tilde{\Gamma }\left( {h,a} \right) = u\left( {c\left( {h,a} \right),\overline{c}\left( {\overline{h},\overline{a}} \right)} \right) - \tilde{H}\left( {h,a} \right) + v\left( {a,\gamma \overline{a}} \right)$$

(24)

The first-order conditions for individually optimal choices and describing Pareto-efficiency are

$$\tilde{Z}_{h} = u_{1} w\left[ {1 - t} \right] - \tilde{H}_{1} \left( {h,a} \right) = 0$$

(25a)

$$\tilde{Z}_{{\text{a}}} = u_{1} \left[ {s - w^{{\text{N}}}} \right] - \tilde{H}_{2} \left( {h,a} \right) + v_{1} = 0$$

(25b)

$$\tilde{\Gamma }_{{\text{h}}} = \left[ {u_{1} + u_{2} } \right]f^{\prime}\left( {h - a} \right) - \tilde{H}_{1} \left( {h,a} \right) = 0$$

(26a)

$$\tilde{\Gamma }_{{\text{a}}} = - \left[ {u_{1} + u_{2} } \right]f^{\prime}\left( {h - a} \right) - \tilde{H}_{2} \left( {h,a} \right) + v_{1} + \gamma v_{2} = 0$$

(26b)

The properties of the model will be unaffected if (1) $\tilde{H}_{2}$ is not too large in absolute value such that (25b) and (26b) define interior solutions for absence choices, and (2) v₁(a, $\gamma \overline{a}$) − $\tilde{H}_{2} \left( {h,a} \right)$ exhibits the same qualitative features as v₁(a, $\gamma \overline{a}$) with respect to a.

5. Proof of Proposition 2

Part (a): If income is untaxed, working hours will be excessive, there will be too little absence from work and, hence, effort and consumption will also be excessive.

The combination of (11a), (11b), and (12a), (12b) shows that

$$H^{\prime}\left( {h^{{\text{m}}} } \right) = v_{1} \left( {a^{{\text{m}}} } \right)$$

(27a)

and

$$H^{\prime}\left( {h^{*}} \right) = v_{1} \left( {a^{*}} \right).$$

(27b)

Given the assumptions on the derivatives (H′, H″, v₁ > 0 > v₁₁), there are three possible combinations of market outcomes relative to the efficient combination:

Case (1): h^m = h* and a^m = a*,

Case (2): h^m < h* such that H′(h^m) < H′(h*) and v₁(a^m) < v₁(a*), which implies a^m > a*,

Case (3): h^m > h* and a^m < a*, according to the same line of argument as in Case (2).

In Case (1), Eqs. (27a) and (27b) hold, but (11a) and (12a), respectively (11b) and (12b), cannot be satisfied simultaneously. Therefore, h^m = h* and a^m = a* do not guarantee that the conditions which characterise the market equilibrium and the efficient outcome are both fulfilled.

In Case (2), h^m − a^m < h* − a* results, which implies that c^m = f(h^m − a^m) < c* = f(h* − a*) holds. This, in turn, indicates that f′(h^m − a^m) > f′(h* − a*) and u₁(c^m) > u₁(c*) due to the strict concavity of f and u. Furthermore, deducting (12a) from (11a) yields

$$\begin{aligned} u_{1} \left( {c^{{\text{m}}} ,\overline{c}} \right)f^{\prime}\left( {h^{{\text{m}}} - a^{{\text{m}}} } \right) - & H^{\prime}\left( {h^{{\text{m}}} } \right) - \left[ {\left[ {u_{1} \left( {c^{*},c^{*}} \right) + u_{2} \left( {c^{*},c^{*}} \right)} \right]f^{\prime}\left( {h^{*} - a^{*}} \right) - H^{\prime}\left( {h^{*}} \right)} \right] \\ = & \,\underbrace {{u_{1} \left( {c^{{\text{m}}} ,\overline{c}} \right)f^{\prime}\left( {h^{{\text{m}}} - a^{{\text{m}}} } \right) - u_{1} \left( {c^{*},c^{*}} \right)f^{\prime}\left( {h^{*} - a^{*}} \right)}}_{{ = {\text{A}}1}} \\ & + \,\underbrace {{H^{\prime}\left( {h^{*}} \right) - H^{\prime}\left( {h^{{\text{m}}} } \right)}}_{{ = {\text{A}}2}} - \underbrace {{u_{2} \left( {c^{*},c^{*}} \right)f^{\prime}\left( {h^{*} - a^{*}} \right)}}_{\left( - \right)} = 0 \\ \end{aligned}$$

(28)

In Case (2), the terms A1 and A2 are positive. Therefore, equality (28) cannot hold and h^m < h*, a^m > a* do not describe the market outcome relative to the efficient situation.

In consequence, the only constellation of working hours and absence which simultaneously guarantees the conditions which describe the market equilibrium and the Pareto-efficient allocation is described by Case (3). If h^m > h* and a^m < a*, c^m > c* must also hold. ∎

Part (b): The tax rate $\hat{t}$:  = t(s = γ = 0) which ensures that individuals choose the optimal number of working hours and the optimal duration of absence is given by

$$0 < \hat{t} = - \frac{{u_{2} \left( {c^{*},c^{*}} \right)}}{{u_{1} \left( {c^{*},c^{*}} \right)}} < 1.$$

(29)

This part can be demonstrated by substituting $\hat{t}$:  = t(s = γ = 0) = −u₂(c*, c*)/u₁(c*, c*) in Eqs. (11a) and (11b). Given a unique market equilibrium, it can only be characterised by the values of h and a which fulfil Eqs. (12a) and (12b), i.e. the Pareto-efficient combination. As tax receipts are returned to individuals and they obtain all profit income, consumption will be the same as in the Pareto-efficient allocation, given the same levels of working hours and absence. ∎

6. Sick Pay

In market equilibrium, consumption equals production, c^m = f(h^m − a^m). Therefore, the derivatives of the first-order conditions (3a), (3b) and (5) with regard to sick pay, s, are $Z_{{{\text{hs}}}}^{{\text{e}}}$ = π_hs = 0 and $Z_{{{\text{as}}}}^{{\text{e}}} = u_{1}$. Also taking into account (18a) to (18e), the changes in contractual hours, absence and effort due to a rise in sick pay, s, are found to be

$$\frac{{{\text{d}}h^{{\text{m}}} }}{{{\text{d}}s}} = \frac{{u_{1} }}{\left| J \right|}\left[ {f^{\prime\prime}Z_{{{\text{hw}}}}^{{\text{e}}} - Z_{{{\text{ah}}}}^{{\text{e}}} } \right] > 0$$

(30a)

$$\frac{{{\text{d}}a^{{\text{m}}} }}{{{\text{d}}s}} = \frac{{u_{1} }}{\left| J \right|}\left[ {Z_{{{\text{hh}}}}^{{\text{e}}} + f^{\prime\prime}Z_{{{\text{hw}}}}^{{\text{e}}} } \right] > 0$$

(30b)

$$\frac{{{\text{d}}\left( {h^{{\text{m}}} - a^{{\text{m}}} } \right)}}{{{\text{d}}s}} = - \frac{{u_{1} \left[ {u_{11} + u_{12} } \right]f^{\prime}s}}{\left| J \right|} < 0$$

(30c)

7. Proof of Proposition 3

Notation: Market outcomes in a world with absence externalities are denoted by h^m(a^m, γ), a^m(h^m, γ) and c^m(h^m, a^m) = c^m(h^m(a^m, γ), a^m(h^m, γ)), while the Pareto-efficient allocation is characterised by h*(a*, γ), a*(h*, γ), and c*(γ).

Part (a): If the tax rate is zero (t = 0), sick pay is non-negative (s ≥ 0), and higher absence by the reference group does not raise utility from absence (v₂ ≤ 0), working hours in market equilibrium will be excessive, while the differences between Pareto-efficient and market outcomes with respect to absence and consumption are indeterminate.

The comparison of the first-order conditions characterising the market equilibrium and the Pareto-efficient outcome or of a combination of them does not provide insights with respect to the relative levels of working hours and absence. However, it can be shown that only a number of combinations of h, a and h − a are feasible. Basically, the differences [h^m(a^m, γ) − a^m(h^m, γ)] − [h*(a*, γ) − a*(h*, γ)], h^m(a^m, γ) − h*(a*, γ) and a^m(h^m, γ) − a*(h*, γ) could be positive, zero or negative. Hence, the theoretically maximal number of outcomes is 27. To simplify the subsequent argument, note that imposing a sign on the term Diff 1: = h^m(a^m, γ) − a^m(h^m, γ) − [h*(a*, γ) − a*(h*, γ)] = h^m(a^m, γ) − h*(a*, γ) − [a^m(h^m, γ) − a*(h*, γ)] implies that the same sign applies to the difference Diff 2: = c^m(h^m, a^m) − c*(γ) because c = f(h − a).

Some of the 27 feasible combinations are logically impossible. If Diff 2 > (<) 0 holds, h^m(a^m, γ) − h*(a*, γ) ≤ (≥) 0 and a^m(h^m, γ) − a*(h*, γ) ≥ (≤) 0 cannot occur simultaneously. This argument rules out 4 (and another 4) of the 27 combinations. Additionally, if Diff 2 = 0 holds, h^m(a^m, γ) − h*(a*, γ) and a^m(h^m, γ) − a*(h*, γ) must have the same signs. Hence, another six combinations cannot describe the market outcome relative to the Pareto-efficient allocation (cf. Table 1).

Table 1 Feasible and impossible combinations of working hours, absence and effort

We next consider the case of Diff 2 ≤ 0 again. This implies that u₁(c^m) ≥ u₁(c*), given u₁₁ + u₁₂ < 0 and f′(h^m − a^m) = w ≥ f′(h* − a*). As a result, u₁(c^m)w > [u₁(c*) + u₂(c*)]f′(h* − a*), since u₂ < 0. The comparison of (3a) and (8a), assuming t = 0, clarifies that H′(h^m) > H′(h*) < 0 must hold, because otherwise the equations cannot be fulfilled simultaneously. Given the convexity of H in h, H′(h^m) > H′(h*) implies that h^m(a^m, γ) > h*(a*, γ) holds. Accordingly, all theoretically feasible cases for which Diff 2 ≤ 0 is assumed are only compatible with h^m(a^m, γ) > h*(a*, γ), ruling out a further 6 of the remaining 13 (27 − 4 − 4 − 6) combinations as incompatible with h^m(a^m, γ), a^m(h^m, γ) characterising the market equilibrium and h*(a*, γ), a*(h*, γ) the Pareto-efficient allocation (argument A).

Note that thus far the proof has required no restrictions with respect to sick pay and the sign of v₂. Suppose, next, that h^m(a^m, γ) ≤ h*(a*, γ) holds. In accordance with the above line of argument, this implies that H′(h^m) < H′(h*) is true. Combining (3a), (3b) and (8a), (8b) yields

$$u_{1} \left( {c^{{\text{m}}} } \right)s - H^{\prime}\left( {h^{{\text{m}}} } \right) + v_{1} \left( {a^{{\text{m}}} } \right) = 0$$

(31a)

$$\gamma v_{2} - H^{\prime}\left( {h^{*}} \right) + v_{1} \left( {a^{*}} \right) = 0$$

(31b)

For γv₂ < 0 and s ≥ 0 or γv₂ = 0 and s > 0, Eqs. (31a) and (31b) can only hold at the same time if v₁(a^m) < v₁(a*), that is for a^m(h^m, γ) > a*(h*, γ), and given v₁₁ < 0 (argument B). Hence, two further combinations have been ruled out. Because no further incompatibilities of the first-order conditions, or combinations thereof, can be discerned, the above considerations leave 5 of the 27 permutations (see Table 1). All of them are characterised by h^m(a^m, γ) > h*(a*, γ). ∎

The proof that h^m(a^m, γ) > h*(a*, γ) is the only feasible outcome, assumes either a positive level of sick pay (s > 0) and γv₂ ≥ 0, or non-negative sick pay (s ≥ 0) and envy with respect to absence (γv₂ < 0); cf. argument B. Therefore, it also covers the case of positive sick pay and no absence externality. Hence, the above argument constitutes an alternative to the proof provided in Appendix 6 establishing that working hours in market equilibrium will be excessive if sick pay is positive.

Part (b): If a higher absence level by the reference group increases the level of absence chosen individually (v₁₂ > 0), a greater strength of relative absence concerns, as captured by an increase in the parameter γ, raises the number of working hours and the duration of absence in market equilibrium, while effort declines.

Since $Z_{{{\text{h}}\gamma }}^{e} = \pi_{{{\text{h}}\gamma }} = 0$ and $Z_{{{\text{a}}\gamma }}^{e} = v_{12} \overline{a}$, the changes in working hours, absence and effort are

$$\frac{{{\text{d}}h^{{\text{m}}} }}{{{\text{d}}\gamma }} = - Z_{{{\text{a}}\gamma }}^{{\text{e}}} \underbrace {{\frac{{Z_{{{\text{ah}}}}^{{\text{e}}} + f^{\prime\prime}Z_{{{\text{aw}}}}^{{\text{e}}} }}{\left| J \right|}}}_{\left( - \right)}$$

(32a)

$$\frac{{{\text{d}}a^{{\text{m}}} }}{{{\text{d}}\gamma }} = Z_{{{\text{a}}\gamma }}^{{\text{e}}} \underbrace {{\frac{{Z_{{{\text{hh}}}}^{{\text{e}}} + Z_{{{\text{hw}}}}^{{\text{e}}} f^{\prime\prime}}}{\left| J \right|}}}_{\left( + \right)}$$

(32b)

$$\frac{{{\text{d}}\left( {h^{{\text{m}}} - a^{{\text{m}}} } \right)}}{{{\text{d}}\gamma }} = - Z_{{{\text{a}}\gamma }}^{{\text{e}}} \frac{{Z_{{{\text{hh}}}}^{{\text{e}}} + Z_{{{\text{ha}}}}^{{\text{e}}} }}{\left| J \right|} = Z_{{{\text{a}}\gamma }}^{{\text{e}}} \underbrace {{\frac{{H^{\prime\prime}}}{\left| J \right|}}}_{\left( - \right)}$$

(32c)

Part (c): A greater strength of relative absence concerns has ambiguous consequences for the Pareto-efficient allocation.

The partial derivatives of Eqs. (8a) and (8b) with respect to γ are given by Γ_hγ = 0 and Γ_aγ = v₂ + (v₁₂ + γv₂₂)$\overline{a}$. Since Γ_aγ cannot be signed without specifying the utility function v, the changes in working hours, absence and effort in the Pareto-efficient allocation are ambiguous. ∎

Part (d): The tax rate and level of sick pay which induce a Pareto-efficient allocation as market outcomes are given by $0 < t^{*}\left( {c^{*},\gamma } \right) = - \frac{{u_{2} \left( {c^{*},c^{*},\gamma } \right)}}{{u_{1} \left( {c^{*},c^{*},\gamma } \right)}} < 1\;{\text{and}}\;s^{*}\left( {c^{*},a^{*},\gamma } \right) = \frac{{\gamma v_{2} \left( {a^{*},\gamma } \right)}}{{u_{1} \left( {c^{*},c^{*},\gamma } \right)}}.$

Replacing t and s in Eqs. (3a) and (3b) by −u₂(c*, c*, γ)/u₁(c*, c*, γ) and γv₂(a*)/u₁(c*, c*, γ) and using w = f′(h − a) from (5) shows that Eqs. (3a) and (3b) will hold for those values of working hours and absence which characterise the Pareto-efficient allocation described by Eqs. (8a) and (8b). All tax payments are returned to individuals via lump-sum payments. Moreover, individuals obtain the entire profit income. Consequently, income and consumption will be the same as in the Pareto-efficient allocation, given h^m(a^m, γ) = h*(a*, γ) and a^m = a*. ∎