Desired work-leisure balance in a partial equilibrium job search model with multiple job holding

Corollary 1(i) and (ii) state that a higher current wage increases the payoff of working at any given working hours. These corollaries ensure that there is a unique solution for ξ₁ in equation (2) at any given w₁, h₁ and h1′ h_1^\prime . Similarly, they ensure the uniqueness of ξ₂ and ξ₀ in (3) and (8). These corollaries simplify the expression of the option values in (1) and (7).

Corollary 1(i) leads to Corollary 1(iii) because a higher current wage implies a higher current expected value, which reduces the potential gain from searching another single job. Corollary 1(iv) is less obvious when workers are searching for a second job. On the one hand, the current wage w₁ has a negative effect on the potential gain V2w1h1,w2′h2′−V1w1h1 {V_2}\left( {{w_1}{h_1},\;w_2^\prime h_2^\prime } \right) - {V_1}\left( {{w_1}{h_1}} \right) , as V₁(w₁h₁) is larger due to a larger w₁. On the other hand, workers keep their current job in a multiple job spell and so a higher current wage in job 1 increases the expected value for a future multiple job spell, i.e., V2w1h1,w2′h2′ {V_2}\left( {{w_1}{h_1},\;w_2^\prime h_2^\prime } \right) is larger due to a larger w₁. Corollary 1(iv) states that the positive effect dominates the negative effect. Corollary 1(iii) and (iv) have the following implications. Workers earning a high wage currently are less likely to quit their current job, as the option value is smaller. However, these workers are more likely to accept a second job, as the option value is larger. It is consistent with Auray et al. (2021) who find that workers having higher wages are more likely to take a second job.

The optimal working time for a single job spell with a given wage w₁ is defined as h1* h_1^* , which is solved from the following optimization problem: (13) maxh1V1w1h1. \mathop {\max }\limits_{{h_1}} {\rm{\;}}{V_1}\left( {{w_1}{h_1}} \right).

By taking the first order condition of V₁(w₁h₁) with respect to h₁, we derive the following corollary regarding the properties of h1* h_1^* . Proofs are found in Appendix.

Interpretation of (14) is straightforward. The marginal gain of increasing working time in job 1 is the wage rate w₁ and the marginal loss is the forgone leisure K′(1 − h₁). Since workers keep job 1 after a second job is accepted, the marginal gain and loss include Eh2′w1−K′1−h1−h2′ {E_{h_2^\prime }}\left[ {{w_1} - K'\left( {1 - {h_1} - h_2^\prime } \right)} \right] , which is the expected gain from wage w₁ and the expected loss of leisure K′1−h1*−h2′ K'\left( {1 - h_1^* - h_2^\prime } \right) upon the acceptance of the second job with offered hours h2′ h_2^\prime . The expected net gain is computed by integrating out the acceptance probability P₂ and is represented by the expectation operator Eh2′ {E_{h_2^\prime }} defined in (12).

The optimal working time for job 1, denoted as h1* h_1^* , is unique. When there is no second job offer (λ₂ = 0), our model is reduced to a static model (Conway and Kimmel, 1998) and the optimal working time denoted as h10 h_1^0 solves w1−K′1−h10=0 {w_1} - K'\left( {1 - h_1^0} \right) = 0 . Optimality requires longer working hours when the second job is unavailable, i.e., h10>h1* h_1^0 > h_1^* . Consider the special case where K(1 − h) = a(1 − h)(1 − log(1 − h)), the optimal working time is given by (15), which is an increasing function of the wage rate w₁. If w₁ is infinity, it is optimal to work for all available time h10=1 \left( {h_1^0 = 1} \right) ; if w₁ is zero, it is optimal not to work h10=0 \left( {h_1^0 = 0} \right) .

Corollary 2(iv) states that the uncompensated wage elasticity of optimal leisure time (ɛ_l,w1) is negative, which is the consequence of using a quasilinear utility function, where wages have no income effect on labor supply decisions but only a substitution effect. This theoretical result parallels the modest empirical consensus that the uncompensated wage elasticity of labor supply is positive. For reviews and meta-analyses, see McCelland and Mok (2012) and Bargain and Peichl (2016).

Using K(1 − h) = a(1 − h)(1 − log(1 − h)), ɛ_l,w1 simplifies to (17). If there is no second job offer (λ₂ = 0), B in (18) equals to one and the elasticity becomes ɛ_l,w1 = − w₁ / a. This elasticity depends on the wage rate w₁ scaling down by a, the minimum dollar value of leisure. a can be conveniently interpreted as the preference for leisure. If a > w₁, leisure is more valuable than work and |ɛ_l,w1| < 1; leisure is inelastic and people prefer to have a fixed amount of leisure time. If a < w₁, leisure is less valuable and |ɛ_l,w1| > 1; leisure is elastic, and people are willing to sacrifice more leisure time in exchange for a higher wage income. By assuming that a is individual specific and depends on a vector of covariates, we can discover which factors and to which extent they determine the desired work-leisure balance. If obtaining a second job is possible (λ₂ > 0), the value of leisure is scaled up by B (> 1) and is equal to aB (> a). Leisure becomes less elastic. In other words, the wage elasticity of leisure time is overstated when the possibility of a second job offer is ignored. We reported in Section 4.2.5 that the estimated elasticity is inflated by more than 50% from −0.26 to −0.41 in a model ignoring multiple job holdings. This theoretical implication is comparable to Ham (1982) and Kahn and Lang (1999), who reported that the wage elasticity of labor supply is biased upward when actual hours instead of desired hours are considered.

We have derived the optimal working time for a single job holder in the last section. Nevertheless, workers might not be able to find a single job that offers h1* h_1^* due to hours constraints. We consider a general case in which the accepted hours h₁ are allowed to be different from the desired hours h1* h_1^* in this Section. After accepting h₁, the workers continue to search for a second job. 98.7% of the workers in our dataset have no change in h₁ by the time they accept the second job. We reason that certain kinds of contractual restrictions forbid them from doing so. Workers can only attain their desired work-leisure balance by choosing their optimal working time h2** h_2^{**} for the second job subject to the given h₁ in the following optimization problem: (19) maxh2V2w1h1,w2h2. \mathop {\max }\limits_{{h_2}} {V_2}\left( {{w_1}{h_1},{w_2}{h_2}} \right).

By taking the first order condition of (19), the properties of h2** h_2^{**} are derived in Corollary 3. Proofs are found in Appendix.

The marginal gain of increasing working hours in job 2 is the wage rate w₂ and the marginal loss is the forgone leisure K′(1 − h₁ − h₂) in Corollary 3(i). If separated from job 1 at rate δ₁, workers’ payoff changes to V₁(w₂h₂). Therefore, the marginal gain includes δ₁∇_h2[V₁(w₂h₂)], which is the marginal change in V₁(w₂h₂) when h₂ changes. The optimal time h2** h_2^{**} is unique. The value of leisure in a multiple job spell is scaled up by E in (22). Since E < B, leisure time is less valuable in multiple job spells (aE) than in single job spells (aB). It results in Corollary 3(iv): Job seekers are more willing to sacrifice their leisure time for a second job than for a single job when both jobs have the same wage. Conway and Kimmel (1998) found a similar conclusion, stating that the desired labor supply is more responsive to wage changes when the possibility of multiple jobholding is considered. Renna and Oaxaca (2006) found that the labor supply elasticity for a second job is 1.8, which is three times that for a single job.

The conditional reservation wages defined in (2) and (3) depend on the offered working hours in the following manners:

Corollary 4 implies that job seekers require a higher reservation wage to compensate the loss due to unattractive working hours, In particular, the conditional reservation wage has a U-shaped against the offered hours. Workers require the least reservation wage when the offered hours are equal to the optimal value. This result is consistent with the findings of Altonji and Paxson (1988), and Bloemen (2008).

We describes the dataset and the empirical models we utilize to estimate the structural parameters of the above-presented model in the next section.

The observed wage distribution is truncated at the conditional reservation wage ξjz˜ {\xi _j}\left( {\tilde z} \right) . Instead of solving ξjz˜ {\xi _j}\left( {\tilde z} \right) using (2), (3) and (8), the reservation wage is commonly estimated by the minimum wage observed in the dataset, as no one would accept a job offer if the wage were lower than the reservation wage (Flinn and Heckman, 1982; Lancaster, 1990). This strategy results in one single reservation wage that applies to all individuals, which is inappropriate in our model as ξjz˜ {\xi _j}\left( {\tilde z} \right) is the reservation wage condition on z˜ \tilde z . Although one might stratify the entire sample by different values of z˜ \tilde z and find the corresponding minimum in each sub-sample, its practicability is questionable when z˜ \tilde z contains numerous continuous variables. Similar issues were discussed in Gørgens (2002). Here, we approximate the conditional minimum wages by their q−centile for a sufficiently small q and estimate them with the quantile regression model: (24) ξjz˜=inf{wj′:Gj(wj′|z)≥q}=z˜′βξ,j, {\xi _j}\left( {\tilde z} \right) = {\rm{\;inf\;}}\{ w_j^\prime :{G_j}(w_j^\prime |z) \ge q\} = \tilde z'{\beta _{\xi ,j}}, for j = 0,1,2. We let β_ξ = (β_ξ0, β_ξ1, β _ξ2)′. (24) can be estimated by the least absolute deviation estimators (Koenker and Hallock, 2001).

The estimated conditional minimum wages are consistent estimates of the actual conditional minimum wages when q → 0. If the chosen q is too large, the estimated conditional reservation wages would be biased upward, and a fraction of observed wages would be smaller than the estimated reservation wages. Conversely, if q is too small, the estimates would be sensitive to measurement errors. That the observed minimum hourly wage in our sample is one cent only (see Table 1) is probably driven by measurement errors. Unbiased estimation and robustness require a balance.

Table 2 shows the estimated conditional reservation wages using some selected values of q. Since ξz˜ \xi \left( {\tilde z} \right) depends on z˜ \tilde z , different observations have different values of ξz˜ \xi \left( {\tilde z} \right) . We report the average value ξ¯ \bar \xi in Table 2. Also reported is the fraction of observed accepted wages w′ that are smaller than the estimated ξz˜ \xi \left( {\tilde z} \right) . We denote it as Pr(w′ < ξ) in Table 2. The estimated reservation wages have critical impact on the job offer acceptance probability P_j(z) (j = 1,2,0) defined in (9) – (11), as the smaller the reservation wage is, the more likely a worker accepts an offer. Different observations have different P_j(z)'s as they depend on z. We report the average values denoted as Pj¯ \overline {{P_j}} in Table 2.

We use q = 2 as the benchmark scenario when we present the main results in Section 4.1. Sensitivity checks using other values of q will also be provided there. With q = 2, ξ¯ \bar \xi is 158 cents, around one-third of the average offered wage (445 cents). There are 1.6% of the observed accepted wages w′ smaller than the estimated ξz˜ \xi \left( {\tilde z} \right) , and we replace the latter by the former in such cases. The estimated offer acceptance probabilities range from 60% to 72%. Unemployed job seekers are less selective in the job market, as they accept 72% of the offers in the job market on average, while people holding a job already are more selective, as they consider only 60% of the available offers. Beyond the average values, there is a large variety of individual job acceptance rates (see Table 4). For instance, the estimated P₁(z) ranges from 6% to 99%, meaning that some individuals accept only the highest wages in the wage ladder while some accept almost all offers in the market.

After estimating the conditional reservation wage ξz˜ \xi \left( {\tilde z} \right) , we estimate offered wage distribution G_j(w′|z) (j = 0,1,2), which is truncated by ξz˜ \xi \left( {\tilde z} \right) . We assume that G_j(w′|z) has a normal distribution, with mean denoted as μ_w,j and standard deviation as σ_w,j. The truncated normal density function (Amemiya, 1973; Heckman, 1976) is (25) gjw′w′ξ;z=ϕw′−μw,j/σw,j/σw,j1−Φξ, w>ξ. {g_j}\left( {w'\left| {w'} \right\rangle \xi ;z} \right) = {{\phi \left( {\left( {w' - {\mu _{w,j}}} \right)/{\sigma _{w,j}}} \right)/{\sigma _{w,j}}} \over {1 - \Phi \left( \xi \right)}},\;\;\;w > \xi .

We assume that σ_w,j is a constant, while μw,j=zβμw,j′ {\mu _{w,j}} = z\beta _{{\mu _{w,j}}}^\prime . We let β_μ = (β_μw,0, β_μw,1, β_μw,2)′ and σ = (σ_w,0, σ_w,1, σ_w,2)′. Model (25) is estimated by maximum likelihood method where L(βμ′,σ′|wi,ξi,zi)=∏i=1N∏j=02gj(wi′|wi′,ξi,zi) L(\beta _\mu ^\prime ,\sigma '|{w_i},{\xi _i},{z_i}) = \mathop \prod \limits_{i = 1}^N \mathop \prod \limits_{j = 0}^2 {g_j}(w_i^\prime |w_i^\prime ,{\xi _i},{z_i}) . The estimated G_j(w′|z) is used to construct the empirical P_j(z) in (27). As an alternative to (25), we fit another truncated model by assuming that G_j(w′|z) has log-normal distribution. Since the two competing models are non-nested, we use the model selection test suggested by Vuong (1989). We let L_i,0 and L_i,1 be the ith contributions to the likelihood function under the normal and log-normal specification, respectively. Defining y_i = ln L_i,0 − ln L_i,1 and y¯ \bar y be the sample mean of y_i, the test statistic is V=n−1/2∑i=1nyin−1∑i=1n(yi−y¯)2→DN0,1. V = {{{n^{ - 1/2}}\mathop \sum \nolimits_{i = 1}^n {y_i}} \over {\sqrt {{n^{ - 1}}\mathop \sum \nolimits_{i = 1}^n {{({y_i} - \bar y)}^2}} }}\buildrel D \over \longrightarrow N\left[ {0,1} \right].

The computed value of V is 62.9, which is highly significant. The Vuong's test strongly favors the normal specification against the log-normal specification.

Let h˜ \tilde h be the non-normalized weekly working hours ranging from 0 to 24 × 7 = 168. The distribution of h˜ \tilde h for single job offers is displayed in the left panel of Figure 1. In addition to a distinctive spike at h˜=40 \tilde h = 40 , working hours tend to be concentrated at certain discrete values and are roughly symmetric around the mode at h˜=30 \tilde h = 30 . It is common to use discrete distributions to estimate the discretized working hours (see Blundell and MaCurdy, 1999, for review), for examples, multinominal logit model (Van Soest, 1995; Keane and Moffitt, 1998; Creedy and Kalb, 2005), multinominal Probit model (Bingley and Walker, 1997), Poisson model (Jansen et al., 2003), and nonparametric models (Dickens and Lundberg, 1985; Hoynes, 1996; Bloemen, 2008). We fit a mixture Poisson distribution combined with a spike to the data. We recode the working hours into 14 categories defined by v, where v=minceilh˜/5,14 v = \min \left\{ {{\rm{ceil}}\left( {\tilde h/5} \right),14} \right\} . Hence, v = 1 when the working hours fall into the range of h˜∈0,5 \tilde h \in \left( {0,5} \right) ; v=2 for h˜∈5,10 v = 2\;{\rm{for}}\;\tilde h \in \left( {5,10} \right) ; and v=14 for h˜>65 v = 14\;{\rm{for}}\;\tilde h > 65 . The histogram for the categorized working hours v is displayed in the right panel of Figure 1. We let Poisson_j (v) = Pr (V = v) = γ_j exp (−γ_j) / v! be the Poisson density function and v_j be the residual density at v=8h˜=40 v = 8\left( {\tilde h = 40} \right) for job j = 0,1 and 2. The mixed density function is (26) mj(v|z)=Poissonj(v|z)+vjz𝕀v=8/∑x=114Poissonj(x|z)+vjz {m_j}(v|z) = \left( {Poisso{n_j}(v|z) + {v_j}\left( z \right)\mathbb{I}\left\{ {v = 8} \right\}} \right)/\left( {\sum\limits_{x = 1}^{14} {Poisso{n_j}(x|z) + {v_j}\left( z \right)} } \right) for j = 0,1,2. We assume that γ_j = exp (z′ β_γj) and v_j(z) = exp (z′ β_vj) with β_γ = (β_γ0, β_γ1, β_γ2)′ and β_v = (β_v0, β_v1, β_v2)′. Model (26) is estimated by maximum likelihood method, where L(βγ′,βν′|vi,zi)=∏i=1N∏j=02mjvi|zi L(\beta _\gamma ^\prime ,\beta _\nu ^\prime |{v_i},{z_i}) = \mathop \prod \limits_{i = 1}^N \mathop \prod \limits_{j = 0}^2 {m_j}\left( {{v_i}|{z_i}} \right) . The estimated m_j(v|z) is used to construct the empirical P_j(z) in (27). Remarkably, we use the normalized working time h=h˜/168∈0,1 h = \tilde h/168 \in \left[ {0,1} \right] in the Bellman equations, but we use the non-normalized weekly working hours h˜ \tilde h and the categorized working hours v to elaborate the estimation model, as using them is easier to present than using h.

Figure 1

The job offer and separation rates are λ_j (z) = exp (z′ β_λj) and δ_j (z) = exp (z′ β_δj), respectively, with β_λ = (β_λ0, β_λ1, β_λ2)′ and β_δ = (β_δ1, β_δ2)′. The hazard rate of getting a new job is the product of the job offer rate and the offer acceptance probability P_j(z). Using the discretized density function m_j(v|z), P_j(z) defined in (9) – (11) becomes (27) Pjz= ∑v=014G¯jξj|zmjv|zdv. {P_j}\left( z \right) = \;\sum\limits_{v = 0}^{14} {{{\bar G}_j}\left( {{\xi _j}|z} \right){m_j}\left( {v|z} \right)dv} .

The hazard rate of accepting a new offer for job j at any time t is (28) θjz=λjz∑v=014G¯jξj|zmjv|zdv. {\theta _j}\left( z \right) = {\lambda _j}\left( z \right)\sum\limits_{v = 0}^{14} {{{\bar G}_j}\left( {{\xi _j}|z} \right){m_j}\left( {v|z} \right)dv.}

Single job holders have three possible changes in job status: accepting a new job, accepting a second job, and job separation. Since these three possible changes are competing risks, the probability of these three transitions at time t (sub-density functions) are (29) fλjt;z=θjzexp−θ1z+θ2z+δ1zt for j=1,2, and {f_{{\lambda _j}}}\left( {t;z} \right) = {\theta _j}\left( z \right){\rm{\;exp\;}}\left( { - \left( {{\theta _1}\left( z \right) + {\theta _2}\left( z \right) + {\delta _1}\left( z \right)} \right)t} \right)\;\;\;{\rm{for}}\;\;\;j = 1,2,\;\;{\rm{and}} (30) fδ1t;z=δ1zexp−θ1z+θ2z+δ1zt. {f_{{\delta _1}}}\left( {t;z} \right) = {\delta _1}\left( z \right){\rm{\;exp\;}}\left( { - \left( {{\theta _1}\left( z \right) + {\theta _2}\left( z \right) + {\delta _1}\left( z \right)} \right)t} \right).

The only risk for unemployed individuals is accepting a new offer. Its sub-density function is (31) fλ0t;z=θ0zexp−θ0zt. {f_{{\lambda _0}}}\left( {t;z} \right) = {\theta _0}\left( z \right){\rm{\;exp\;}}\left( { - {\theta _0}\left( z \right)t} \right).

For multiple job holders, the two possible changes are separations from either job. A job separation at time t implies that the employment duration for another job exceeds t. Assuming that the employment duration for jobs 1 and 2 are independent after controlling for all observable z, the sub-density functions are (32) fδjt;z=δjzexp−δjztexp−δkzt, for j≠k∈1,2. {f_{{\delta _j}}}\left( {t;z} \right) = {\delta _j}\left( z \right){\rm{\;exp\;}}\left( { - {\delta _j}\left( z \right)t} \right){\rm{\;exp\;}}\left( { - {\delta _k}\left( z \right)t} \right),\;\;\;{\rm{for}}\;\;\;j \ne k \in \left\{ {1,2} \right\}.

The probability of separation in job j at time t is δ_j(z) exp (−δ_j(z)t) and the probability that the duration of job k exceeds t is exp (−δ_k(z)t). The independence assumption may be violated if there are unobserved factors governing the durations for jobs 1 and 2. For instance, workers having lower preference of leisure may work longer in both jobs j and k. To evaluate the validity of the independence assumption, we apply the Cox proportional hazard model (Cox, 1972) using the ongoing duration for job k as the dependent variable while the duration for job j together with the other covariates are regressors. The Cox model is chosen here as it is the canonical semi-parametric duration model without specifying the distribution of the dependent variable. Result shows that the duration for job j is insignificant with a p-value equals 0.48.

The likelihood for all spells (indexed as i = 1,…, n) is the product of the sub-density functions defined in (29) – (32). Let Δ_λ1, Δ_λ2, and Δ_δ1 be the risk indicators for obtaining a new single job, a second job, and job separation, respectively, in single job spells. Let Δ_δ1 and Δ_δ2 be the risk indicators of separation in jobs 1 and 2, respectively, in multiple job spells. Let Δ_λ0 be the risk indicator of taking a post unemployment offer in unemployment spells. The conditional likelihood function is (33) L(βλ′,βδ′|βξ′,βμ′,σ′,βγ′,βν′)=∏i=1N∏j=02∏m=12fλjti;ziΔλj,ifδmti;ziΔδm,i. L(\beta _\lambda ^\prime ,\beta _\delta ^\prime |\beta _\xi ^\prime ,\beta _\mu ^\prime ,\sigma ',\beta _\gamma ^\prime ,\beta _\nu ^\prime ) = \mathop \prod \limits_{i = 1}^N \mathop \prod \limits_{j = 0}^2 \mathop \prod \limits_{m = 1}^2 {\left( {{f_{{\lambda _j}}}\left( {{t_i};{z_i}} \right)} \right)^{{\Delta _{{\lambda _j}}},i}}{\left( {{f_{{\delta _m}}}\left( {{t_i};{z_i}} \right)} \right)^{{\Delta _{{\delta _m}}},i}}.

After β_ξ, β_μ, σ, β_γ, and β_v are estimated using (24), (25), and (26), β_λ and β_δ are estimated using (33) by the maximum likelihood method.

We consider the case of κ(1 − h) = a(1 − log(1 − h)). The minimum value of leisure a_i is affected by z_i in the form of ai−1=−zi′α a_i^{ - 1} = - z_i^\prime \alpha . The marginal effect of a covariate z_ik on a_i is (34) ∂ai∂zik=ai2αk∝αk. {{\partial {a_i}} \over {\partial {z_{ik}}}} = a_i^2{\alpha _k} \propto {\alpha _k}.

The marginal effect is individual specific and has the same sign as α_k. To estimate α, we rewrite the wage elasticity of optimal leisure time in (22) as semi-elasticity (35) ∂logli**∂w2i=−1aiEi=zi′αEi. {{\partial {\rm{\;log\;}}l_i^{**}} \over {\partial {w_{2i}}}} = - {1 \over {{a_i}{E_i}}} = {{z_i^\prime \alpha } \over {{E_i}}}.

As discussed in the introduction, observed working hours h_1i in the first job could differ from the optimal h1i* h_{1i}^* derived in Corollary 2(i). We assume that workers use the second job to adjust their total working hours to their desired level at any given h_1i. Consequently, the observed working hours h_2i for the second job represent the optimal h2i** h_{2i}^{**} derived in Corollary 3(i) and the observed leisure time l_i = 1 − h_1i − h_2i is the optimal leisure li**=1−h1i−h2i** l_i^{**} = 1 - {h_{1i}} - h_{2i}^{**} at any given h_1i. We can use the information on multiple job spells to identify the structural parameter a_i in (35) using the following reduced form semi-log regression equation: (36) logli=zi′αE^iw2i+ei. {\rm{\;log\;}}{l_i} = {{z_i^\prime \alpha } \over {{{\hat E}_i}}}{w_{2i}} + {e_i}.

Ê_i is estimated from (23) using the estimated parameters obtained in the previous steps. e_i is the residual term and is assumed to be uncorrelated with w_2i and z_i. This assumption is verified in Section 4.2.6.

After α is estimated, we estimate the optimal working time for each single job spell h1i* h_{1i}^* defined in (14) of Corollary 2(i) by solving the following equation (see proof in Appendix): (37) 1−h1i*∏v=0141−h1i*−vλ^2iG¯^2iξ^m^2iv/ρ+δ^1i+δ^2i=exp−w1ia^i1+λ^2iP^2iρ+δ^1i+δ^2i \left( 1-h_{1i}^{*} \right)\underset{v=0}{\overset{14}{\mathop \prod }}\,{{\left( 1-h_{1i}^{*}-v \right)}^{{{{\hat{\lambda }}}_{2i}}{{{\hat{\bar{G}}}}_{2i}}\left( {\hat{\xi }} \right){{{\hat{m}}}_{2i}}\left( v \right)/\left( \rho +{{{\hat{\delta }}}_{1i}}+{{{\hat{\delta }}}_{2i}} \right)}}=\text{ }\!\!~\!\!\text{ exp }\!\!~\!\!\text{ }\left( -\frac{{{w}_{1i}}}{{{{\hat{a}}}_{i}}}\left( 1+\frac{{{{\hat{\lambda }}}_{2i}}{{{\hat{P}}}_{2i}}}{\rho +{{{\hat{\delta }}}_{1i}}+{{{\hat{\delta }}}_{2i}}} \right) \right)

The estimated optimal hours h^1i* \hat h_{1i}^* is then compared with the actual hours h_1i to estimate the extent of working hours mismatches in single job spells.

Table 3 shows the summary statistics for the estimated conditional reservation wages. To facilitate comparisons, we report the observed offered wages as well. The estimated conditional reservation wages are highest in single job offers and lowest in post-unemployment offers, whereas the second job offers are somewhere in between. The observed offered wages have the same patterns. We reason that unemployed people have zero wage income and have a lower opportunity cost to accept new offers relative to working people who perform on-the-job search. Conditional reservation wages for second job offers are lower than single job offers as the former are a device to adjust the working hours to the optimal level.

Figure 2 provides the histograms of the estimated reservation wages and the observed offered wages. The observed offered wages are skewed to the right, arguably due to truncations by the reservation wages from below. Since different observations have different reservation wages, the truncation points are also different for different observations. We plot the fitted truncated models for using three truncation points: the mean, 10th percentile, and 90th percentile of the reservation wages. The models fit the data quite well except that the estimated densities are underestimated at the mode for the post-unemployment offers. To evaluate whether G₁, and have identical distributions, we apply the cross-model comparison test suggested by Clogg et al. (1995). The chi-square test statistics with 13 degrees of freedom range from 55 to 78 and are rejected at 1% level, indicating that the three offered wage distributions are different.

Figure 2

Figure 3 reports the histograms for the categorized weekly working hours and the estimated mixed Poisson models. Compared with single jobs, the proportion of second jobs offering the standard 40 hours (v = 8) is much smaller, and there is a remarkable proportion having working hours shorter than 25 (v = 5). The working hours for post-unemployment offers are somewhere in between. The fitted models match the actual data closely.

Figure 3

Table 4 summarizes the estimated job offer rates λ_j, separation rates δ_j, and job acceptance rates P_j. Second jobs have lower offer rates (inflow) and higher separation rates (outflow) than single jobs. These explain why total number of multiple job spells is only one-sixth that of single job spells. Higher separation rates also explain why the average duration for second jobs (34 weeks) is shorter than single jobs (106 weeks). Post-unemployment offer has higher acceptance rates than single and second jobs, implying that unemployed people are less selective in accepting a job offer than job seekers holding job(s).

Results are reported in Table 5. The estimated average minimum value of leisure ($16.8) is around 3.8 times the observed average offered real wage ($4.45). We compare this figure with a few previous studies that estimated similar objects. Bockstael et al. (1987) and Larson and Shaikh (2001) estimated the shadow values of time using static structural models with hours constraints, which are 3.5 to 7 times the wage. Using a reduced form Tobit model with hours constraints, Feather and Shaw (1999) found that the reservation wages are around 106% to 112% of the accepted wages. Allowing for intertemporal substitution in a structural model, Phaneuf et al. (2000) and Lloyd-Smith et al. (2019) found that the values of time are around 90 percent of the wages. It is reminded that the above studies used different theoretical and empirical approaches, which make comparisons not straightforward. In particular, the Tobit model concerns the extensive margin (work or not work) rather than the intensive margin (number of working hours) of labor supply. It is well-known that the extensive margin is more elastic than the intensive margin (see, e.g., Keane and Rogerson, 2012). A higher elasticity of labor supply is accompanied by a lower value of leisure according to Corollary 2(iii), which plausibly explains why the estimated value in Feather and Shaw (1999) is relatively small. For similar reasons, models allowing for intertemporal substitutions tend to provide smaller values of leisure, as the intertemporal elasticities of labor supply are larger than the non-intertemporal elasticities. For instance, Chetty et al. (2011) found that the former are about 3 times larger than the latter. The implied value of leisure would then be 3 times smaller than our estimated values.

The estimated values of B in Table 5 are greater than or equal to one, which is consistent with Corollary 2(iii), but the sizes are relatively small (with mean 1.02). The values of E are smaller than B as predicted by Corollary (3)(iv), and the difference is relatively substantial (with mean 0.67). These suggest that the possibility of multiple job searches adds only a little to the values of leisure in single job spells (measured by aB), but it reduces the values of leisure for multiple job spells (measured by aE) by around one-third.

Agreeing with Corollary (2)(iv) and Corollary (3)(iii), the estimated wage elasticities of leisure time, ɛ_l,w1 and ɛ_l,w2, are negative. The average values are −0.26 for single job spells and −0.40 for multiple job spells. Leisure times are around 50% more elastic when multiple jobs are held. Since most of the previous studies estimated the elasticities of labor supply, we transformed them into elasticities of leisure so that direct comparisons are possible. Transformations as such were discussed in Mankiw et al. (1985) and Aguiar et al. (2021): (38) εh,w=wh∂h∂w=−wl∂l∂wlh=−lhεl,w {\varepsilon _{h,w}} = {w \over h}{{\partial h} \over {\partial w}} = - {w \over l}{{\partial l} \over {\partial w}}{l \over h} = - {l \over h}{\varepsilon _{l,w}}

Leisure l is about four times labor supply h in the dataset of Mankiw et al (1985). In our data sample, the average leisure for a single job spell is 127 hours a week while the average working hours is 39 hours a week. Hence, we use l/h = 3.3 to do similar calculations. In Chetty et al. (2011) the estimated wage elasticity in a model without considering hours constraints is 0.33. The implied leisure elasticity is then −0.10. For models considering hours constraints: the implied leisure elasticities for Zabel (1993) range from −0.02 to −0.17; for Conway and Kimmel (1998) it is −0.32; for Feather and Shaw (2000) they range from −0.15 to −0.21; and for Renna and Oaxaca (2006) they range from −0.12 to −0.15. Our estimated average leisure elasticity for a single job (−0.26) is closed to the lowest end of these previous findings. The above results concern single job spells. For multiple job spells, the implied elasticities of leisure for Shishko and Rostker (1976) range from −0.55 to −0.78; for Conway and Kimmel (1998) they range from −0.01 to −0.14; for Renna and Oaxaca (2006) it is −0.54. Our estimated average elasticity for multiple job spells is −0.40, which is somewhere in between the previous findings.

Next, we discuss the estimated working hours mismatch, defined as Δh=h1−h1* \Delta h = {h_1} - h_1^* . The estimated optimal weekly working hours h1* h_1^* have a mean value of 46.5 hours, which are longer than the observed working hours h₁ averaging at 39.0. Workers underwork by 7.5 hours on average. Figure 4 shows the histogram of the estimated values for Δh. A thicker tail on the left represents the underworking cases.

Figure 4

Table 6 reports the disaggregated figures in detail. For our benchmark model using q = 2, 37% of the observations work longer than optimal, and 63% work shorter. The average working hours for the overworked employees are 41.5, which are longer than their desired 33.1 hours by 8.5 hours. The underworked employees work an average of 37.5 hours, which are shorter than their desired 54.4 hours by 16.9 hours. These suggest that working hours mismatches are common phenomena and the discrepancies are quite substantial. If a week has five working days, the average overworked hours are 1.7 (= 8.5/5) per day, whereas the average underworked hours would be 3.4 (= 16.9/5) per day. As robustness checks, Table 6 shows the estimated results using other values of q, which differ from the benchmark case only marginally. According to Bloemen (2008), workers declared dissatisfied with their working hours only when the difference is greater than certain threshold values, averaged at 16 hours. Based on this estimated value, the proportion of workers in our sample would have declared overworked reduces from 37% to 5%, while that declared underworked reduces from 63% to 26%, leaving 69% declared satisfied with their working hours. Survey data would have understated the degree of work-leisure mismatch substantially.

Table 7 shows the estimated α_k and the estimated partial effect of covariates z_ik on the value of leisure a_i defined as ai2αk a_i^2{\alpha _k} in (34). Since ai2αk a_i^2{\alpha _k} is individual specific, its mean value, minimum and maximum values are reported. To make interpretations of the above results easier, we establish the relationship between our estimates and the wage elasticity of labor supply. As mentioned in equation (38), the wage elasticity of working hours ɛ_h,w, has an opposite sign as the elasticity of leisure ɛ_l,w. The more elastic (more positive) the labor supply is, the more elastic (more negative) the demand for leisure is. Note that ɛ_l,w = −w/aB from (17) and the partial effect of z_k on a has the same sign as α_k from (34). Put them together, a positive α_k increases the value of leisure a and leads to a less negative ɛ_l,w, and a less positive ɛ_l,h, so that labor supply is less elastic. The rationale is as follows: a higher value of leisure increases the opportunity cost of work, which makes an individual less willing to substitute work for leisure when wage increases. This rationale was applied in Connolly (2008) who used weather condition to measure the utility of leisure, while González-Chapela (2007) used the price for various leisure-related goods and services to measure the disutility for work. Using this rationale one can compare our results with the estimated elasticities of labor supply in the literature directly.

Table 7 shows that age and net worth have important roles in the value of leisure. When age increases from 25 to 35, the value of leisure increases by $3.20 (=$0.318 × 10) on average, which is approximately 20% of the average value of a ($16.8). Workers with one more thousand net worth value leisure $0.92 (5.4%) more. These results are consistent with Attanasio et al. (2018) who found that younger workers are more elastic in labor supply, as young workers have fewer financial responsibilities and more alternative usages of their time. Also, Domeij and Floden (2006) found that workers having financial constraints are two times more elastic in labor supply.

Education has the strongest effect among the dummy variables: workers without high school and degree qualifications have a lower value of leisure by $4.40 (26%). Types of industry also matter: workers in the public sector and professional services have a higher value of leisure by $3.48 (20.7%) and $1.91 (11.3%), respectively, than other industries. These results agree with Bils et al. (2012) as workers having different kinds of human capital value their leisure differently. Also, Kudoh and Sasaki (2011) found that industries requiring different levels of professional training have different search frictions that affect the costs of working. Generally speaking, workers who have better education and/or receive professional training have a higher opportunity cost for leisure, which makes them less elastic in labor supply.

Females value leisure $0.83 (4.9 %) less than males meaning that females are more elastic in labor supply. It is intuitive as female labors are usually regarded as more substitutable for domestic work than male labors (e.g., Blundell and MaCurdy, 1999), although female elasticities dropped by half over the two decades (Heim, 2007). Workers having at least one child value their leisure $0.52 (3.1%) less. It is in some way in line with Blundell et al. (1998) who found that the elasticity of labor supply for parents is the largest when their child is aged below four years old, as the young parents in our sample arguably have relatively small kids. The remaining results in Table 7 show that Black workers have lower value of leisure by $2.10 (12.5%) than white worker and Hispanics. The effect of previous job turnovers is weak, as the value of leisure drops by $0.27 (1.6%) only when workers have one more previous job. The effects of previous employment duration and marital status are insignificant.

We introduce some new variables in this section. The weekly time endowment is denoted as e˜i {\tilde e_i} , which is allocated to work h˜i {\tilde h_i} and leisure li˜ \widetilde {{l_i}} , such that e˜i=h˜i+li˜ {\tilde e_i} = {\tilde h_i} + \widetilde {{l_i}} . The discussions so far assumed that e˜i {\tilde e_i} is 168 hours a week and we did not distinguish the different usages of leisure. Particularly, we assumed that the marginal utility derived from sleeping is the same as the marginal utility derived from home production, transportation, and recreational activities. Suppose now that people are committed to spend ñ_i hours on certain activities (e.g., sleeping) that cannot be substituted for work h˜i {\tilde h_i} or leisure li˜ \widetilde {{l_i}} , the total allocatable time would be shorter. The question is how far our results are affected by introducing ñ_i.

We consider a simple case, in which ñ_i is not a decision variable in the optimization problem. ñ_i is exogenously given and is a fixed value for each individual i independent of the job status. Workers’ problem is to maximize their payoff by allocating h˜i {\tilde h_i} and li˜ \widetilde {{l_i}} subject to the new time endowment e˜i=168−n˜i {\tilde e_i} = 168 - {\tilde n_i} . The model can easily incorporate such a change by defining the new normalized working hours as hi=h˜i/168−n˜i {h_i} = {\tilde h_i}/\left( {168 - {{\tilde n}_i}} \right) and the new normalized leisure hours as li=li˜/168−n˜i {l_i} = \widetilde {{l_i}}/\left( {168 - {{\tilde n}_i}} \right) . The time constraint implies h_i + l_i = 1 as before. The theoretical results established in Corollaries 1–4 apply equally to this extension.

This extension would affect the empirical results. The reason is explained in the following. Our dataset provides us with the individuals’ working hours only, but not their leisure time. Leisure time is computed as the residual hours out of working, i.e., li˜=e˜i−h˜i \widetilde {{l_i}} = {\tilde e_i} - {\tilde h_i} in the empirical application. If we reduce the value of e˜i {\tilde e_i} from 168 to 168 − ñ_i, the normalized working hours h_i would be divided by a smaller denominator which makes h_i larger. Likewise, the normalized leisure time l_i = 1 − h_i would become smaller. For any given w_i, Δw_i, and Δl_i, the elasticity of leisure defined as ɛ_l,w = w_i / l_i × Δl_i / Δw_i would have larger absolute value when a smaller l_i is used. As can be seen from ɛ_l,w = −w_i / a_iE_i in (22), a larger absolute value of elasticity would lead to a smaller estimated value of leisure a_i. To sum up, leisure is more elastic and the value of leisure is smaller when the value of ñ_i increases. Effects on the optimal working hours hi* h_i^* defined in (14) and the working hours mismatches computed by Δh=hi−hi* \Delta h = {h_i} - h_i^* are however unclear.

We have to fix ñ_i to perform the empirical analysis. Since the actual leisure time and the usages of leisure are not provided in the dataset, such a choice is inevitably arbitrary. As robustness checks, we set two values of ñ_i. If people need 4 hours a day for the non-substitutable activities, ñ_i = 4 × 7 = 28, and the new time endowment becomes e˜i=168−28=140 {\tilde e_i} = 168 - 28 = 140 . If people need 6 hours for these activities, ñ_i = 6 × 7 = 42, and the new time endowment becomes e˜i=168−42=126 {\tilde e_i} = 168 - 42 = 126 . Since the results in Table 2 to 4 are not affected by this extension, we provide the new results of some key figures in Table 5 and 6 below. To ease comparisons, the original results using e˜i=168 {\tilde e_i} = 168 are also reported.

Consistent with our predictions, leisure is more elastic (changing from −0.26 to −0.32 for single job holdings and from −0.40 to −0.59 for multiple job holdings) while the value of leisure drops (from $16.8 to $14.9), when people have a smaller time endowment e˜i {\tilde e_i} due to their commitments in non-substitutable activities. Regarding the working hours mismatches, the estimated optimal working hours h1* h_1^* drop from 46.5 hours a week to 38.3 hours a week, which makes the probability of overwork increases from 37% to 53%. The average mismatch Δh changes from underworking for 7.5 hours a week to overworking 0.6 hours a week. For the overworked group, the average number of overworked hours increases from 8.5 to 9.9 a week. For the underworked group, the average number of underworked hours drops from 16.9 to 10.1. The main impact of shortening the endowment e˜i {\tilde e_i} is mitigating the extent of underwork. To conclude, when workers are committed to more non-substitutable activities, their value for leisure would be smaller and their demand for leisure would be more elastic due to a smaller endowment for allocatable time. Consequently, the optimal working hours would be smaller and the chance of underwork would be smaller.

We examine to what extent a policy enhancing working hours flexibility would alleviate working hours mismatches in this section. Policies promoting flexible hours have become popular in the last decades in European countries (see Lewis, 2003; Plantenga, 2009; Messenger, 2018, for reviews). These policies give employees the right to bargain with their employers so that they can have more flexibility in determining their working schedules. It was found that a quarter of workers had access to flexible schedules across 30 European countries by 2015 (Chung and Van der Lippe, 2020). Females with family responsibility were most benefited by these policies (Song and Gao, 2020). There are evidence that these policies increase workers’ leisure satisfaction and reduce their turnover intention (Kröll and Nüesch, 2019). A survey regarding the increasing practices of flexible work arrangements in the U.S. is found in Katz and Krueger (2019).

We start with a closer look on the nature of mismatches by studying the differences between actual and optimal hours in Figure 5. In our model hours distribution is exogenously determined by the employees and is featured by spikes at certain fixed value of hours (see the left panel of Figure 5). Most noticeably, the spike at the standard 40 hours occupies around 35% of the entire distribution. The desired hours in contrast are smoothly distributed. The middle panel of Figure 5 shows the distributions for the overworked group. It is apparent that most of the overworked workers prefer to work shorter than 40 hours while a lot of them have to work for 40 hours or longer. For the underworked group in the right panel of Figure 5, most of them prefer to work longer than 40 hours while a lot of them have to work for 40 hours or less. Mismatches occur as workers do not have enough choices to match their desired hours.

Figure 5

It is natural to ask to what extent the problem of hours mismatches is alleviated when the offered hours have more variety. We introduce flexibility in the offered hours distribution by adding random noises to the working hours. The random noises have a normal distribution with zero mean and we consider different values of the standard deviation defined as SD (measured in weekly hour). If such flexibility were available, the underworked group would choose longer hours, while the overworked group would choose shorter hours. Since shorter hours would never be considered by the underworked group, adding flexibility to reduce the hours is irrelevant to the underworked workers. We add only the positive random noises to the underworked group. Similarly, longer hours would never be considered by the overworked group, we add only the negative random noises to the overworked group. The simulated results using SD = 4 are provided in Figure 6. The simulated working hours in the left panel of Figure 6 do not have a distinct spike at 40 hours anymore, and they are more smoothly distributed than the original data. For the overworked group in the middle panel, the size of the overlapping region between the actual and the preferred hours increases substantially, meaning that the differences between the actual and the preferred hours are smaller. Similar observations are found for the underworked group in the right panel.

Figure 6

We report the numerical results in Table 9 using different values of SD, and SD = 0 refers to original data. Although the probability of overwork Pr(h1>h1*) {\rm{Pr}}({h_1} > h_1^*) does not change much with different SD, the average hours mismatch Δh drops obviously with a larger value of SD. For instance, when SD = 8 hours a week, the average mismatch drops by one-fourth from underworking 7.5 hours to 5.8 hours. For the overworked group, the overworked hours drop from 8.5 to 6.6. Taking 5 working days a week, the average daily overworked hours drop from 1.7 hours to 1.3 hours. For the underworked group, the underworked hours drop from 16.9 to 13.5, which is a drop from 3.4 hours per day to 2.7 hours.

From the policy's point of view, one would want to know the average size of the noises added to the working hours distribution. It is found that the random noises have an average absolute size of 6.5 hours when SD = 8. In other words, a policy allowing employees to adjust their working hours by ± 6.5 hours per week (or ± 1.3 hours per day), the average mismatch would be reduced by almost one-fourth from 7.5 hours to 5.8 hours.

We consider an extension where workers can hold more than two jobs in multiple job spells. We fix J = 3 for illustration. The term ‘multiple job spell’ is reserved for those holding two jobs in a working spell, while we introduce the term ‘triple job spell’ for those holding three jobs. In our dataset, single job holders never accept two new jobs at the same time and therefore it is not possible to switch from a single job spell to a triple job spell. Also, a switch from unemployment to a triple job spell is not found in the data. The only way a worker holds the third job (job 3) is a switch from a multiple job spell to a triple job spell. For this reason, we only need to revise the Bellman equations for the multiple job holders in (6). Let λ₃ be the job offer rate and δ₃ be the job separation rate for job 3. Let w3′ w_3^\prime be the offered wage for job 3 with conditional distribution G₃ and h3′ h_3^\prime be the offered hours for job 3 with distribution M₃. The expected present value for a triple job spell is defined as V₃(w₁h₁, w₂h₂, w₃h₃).

Equation (6) is revised as (39) ρ+δ1+δ2V2w1h1,w2h2=w1h1+w2h2+K1−h1−h2+δ2V1w1h1+δ1V1w2h2 + λ3∫01−h1−h2∫ξ3∞V3w1h1,w2h2,w3′h3′−V2w1h1,w2h2dG3w3′dM3h3′. \matrix{ {\;\;\;\;\left( {\rho + {\delta _1} + {\delta _2}} \right){V_2}\left( {{w_1}{h_1},{w_2}{h_2}} \right)} \hfill \cr { = {w_1}{h_1} + {w_2}{h_2} + K\left( {1 - {h_1} - {h_2}} \right) + {\delta _2}{V_1}\left( {{w_1}{h_1}} \right) + {\delta _1}{V_1}\left( {{w_2}{h_2}} \right)} \hfill \cr {\;\;\;\; +\; {\lambda _3}\mathop \smallint \nolimits_0^{1 - {h_1} - {h_2}} \int_{{\xi _3}}^\infty {\left[ {{V_3}\left( {{w_1}{h_1},{w_2}{h_2},w_3^\prime h_3^\prime } \right) - {V_2}\left( {{w_1}{h_1},{w_2}{h_2}} \right)} \right]d{G_3}\left( {w_3^\prime } \right)d{M_3}\left( {h_3^\prime } \right).} } \hfill \cr }

The conditional reservation wage ξ₃ is the solution of (40) V3w1h1,w2h2,ξ3h3′=V2w1h1,w2h2 {V_3}\left( {{w_1}{h_1},{w_2}{h_2},{\xi _3}h_3^\prime } \right) = {V_2}\left( {{w_1}{h_1},{w_2}{h_2}} \right)

Comparing (6) with (39), the added item is the option value created by the search for job 3. Workers holding three jobs have three possible changes in their job status, i.e., separate from job 1 and hold jobs 2 and 3, or separate from job 2 and hold jobs 1 and 3, or separate from job 3 and hold jobs 1 and 2. The Bellman equation for a triple job spell is (41) (ρ+δ1+δ2+δ3)V3w1h1,w2h2,w3h3=w1h1+w2h2+w3h3+K1−h1−h2−h3 + δ1V2w2h2,w3h3+δ2V2w1h1,w3h3+δ3V2w1h1,w2h2. \matrix{ {\;\;\;\;(\rho + {\delta _1} + {\delta _2} + {\delta _3}){V_3}\left( {{w_1}{h_1},{w_2}{h_2},{w_3}{h_3}} \right)} \hfill \cr { = {w_1}{h_1} + {w_2}{h_2} + {w_3}{h_3} + K\left( {1 - {h_1} - {h_2} - {h_3}} \right)} \hfill \cr {\;\;\;\; + \;{\delta _1}{V_2}\left( {{w_2}{h_2},{w_3}{h_3}} \right) + {\delta _2}{V_2}\left( {{w_1}{h_1},{w_3}{h_3}} \right) + {\delta _3}{V_2}\left( {{w_1}{h_1},{w_2}{h_2}} \right).} \hfill \cr }

Since V₂(w₁h₁, w₂h₂) in (39) involves the option value, the derivations for Corollary 1–4 would be analytically complicated, although it does not provide any additional useful insight. Moreover, triple job spells represent only 1.5% of our dataset, we do not think it would create a significant difference in our estimates. Setting J = 2 is a reasonable tradeoff.

It is possible that multiple job holders take a second job by the same time they quit one of the two existing jobs. The Bellman equation for multiple job holders in (6) would be extended to (42) ρ+δ1+δ2V2w1h1,w2h2=w1h1+w2h2+K1−h1−h2+δ2V1w1h1+δ1V1w2h2 + λ4∫01−h1∫ξ14∞V2w1h1,w4′h4′−V2w1h1,w2h2dG4w4′dM4h4′ + λ4∫01−h2∫ξ24∞V2w2h2,w4′h4′−V2w1h1,w2h2dG4w4′dM4h4′. \matrix{ {\;\;\;\;\left( {\rho + {\delta _1} + {\delta _2}} \right){V_2}\left( {{w_1}{h_1},{w_2}{h_2}} \right)} \hfill \cr { = {w_1}{h_1} + {w_2}{h_2} + K\left( {1 - {h_1} - {h_2}} \right) + {\delta _2}{V_1}\left( {{w_1}{h_1}} \right) + {\delta _1}{V_1}\left( {{w_2}{h_2}} \right)} \hfill \cr {\;\;\;\; +\; {\lambda _4}\mathop \smallint \nolimits_0^{1 - {h_1}} \int_{{\xi _{14}}}^\infty {\left[ {{V_2}\left( {{w_1}{h_1},\;w_4^\prime h_4^\prime } \right) - {V_2}\left( {{w_1}{h_1},{w_2}{h_2}} \right)} \right]d{G_4}\left( {w_4^\prime } \right)d{M_4}\left( {h_4^\prime } \right)} } \hfill \cr {\;\;\;\; +\; {\lambda _4}\mathop \smallint \nolimits_0^{1 - {h_2}} \int_{{\xi _{24}}}^\infty {\left[ {{V_2}\left( {{w_2}{h_2},\;w_4^\prime h_4^\prime } \right) - {V_2}\left( {{w_1}{h_1},{w_2}{h_2}} \right)} \right]d{G_4}\left( {w_4^\prime } \right)d{M_4}\left( {h_4^\prime } \right).} } \hfill \cr }