Worker’s Problem

Consider an infinite horizon model where individuals choose their level of effort on the job (i.e., whether to shirk, or whether to not shirk), and whether to purchase health insurance if it is not provided to them by their employer or if they are unemployed. An individual can be of either good (g) or bad (b) health. Let y denote her income, T(y) denote her after-tax income, and xdenote her health insurance (“HI”) status, where:

x= 0 ⇒ worker is uninsured (2.2.1)

x= 1 ⇒ worker receives HI from her employer (2.2.2)

Then, each period, the individual’s expected flow utilityvh(y, x) is given by: vh(y, x) =                      Em˜hu(T(y)−m˜h), ifx= 0 u(T(y)), ifx= 1 u(T(y)−Rh), ifx= 2 (2.2.4)

Where the subscripthdenotes the individual’s health status (it can take on the values

borg). Notice that in this simple model, health status only affects the individual through her realization of the medical expenditure shock ˜mh and, as a consequence, health insur-

ance premium Rh. Letting ¯mh =Em˜h, and assuming that ˜mb first order stochastically

dominates ˜mg, I have that ¯mb > m¯g and that Em˜gu(T(y)−m˜g) > Em˜bu(T(y)−m˜b). In other words, an unhealthy worker gets, on average, larger medical expenditure shocks than a healthy worker, and workers prefer to be healthy.

Each period, the individual can be either employed or unemployed. While employed, the worker can either exert effort (not shirk) or be lazy (shirk). If the worker does not shirk, she will incur a cost of effort e and get laid off at the exogenous rate of δ. If the worker shirks, on the other hand, she will incur no cost of effort, but may be “discovered” shirking and consequently fired. This happens with probabilityq, thus increasing her rate of separation to δ +q. Note that q is exogenous in this simple model but can later be made endogenous.

LetVH

h (x) denote the value function of an employed individual with health status h

and health insurance statusx, who decides not to shirk (or exertHigh effort, H). Similarly, Let V_hL(x) denote the value function of an employed individual with health statush and

health insurance status x, who decides to shirk (or exert Low effort, L). I can write the value functions as:

rV_hH(x) = vh(y, x)−e+δ(Uh(ˆxh)−max{VhH(x), VhL(x)}) (2.2.5) rV_hL(x) = vh(y, x) + (δ+q)(Uh(ˆxh)−max{VhH(x), VhL(x)}) (2.2.6)

Now, Uh(x) is the value function of an unemployed individual with health status h

and health insurance status x (to be defined explicitly below), and ˆxh is the individ-

ual’s health insurance status immediately after being laid off (also to be defined explicitly below). In the US, under the Consolidated Omnibus Budget Reconciliation Act of 1985 (“COBRA”), employers are strongly encouraged (via tax deductions) to extend employer- provided health insurance for a period of time after an involuntary separation (provided “gross misconduct” did not occur). Since separations in my model are involuntary, I extend employer-provided health insurance for a single period after an exogenous sepa- ration. Notice, however, that if the worker chose to self-insure herself while employed, she is not required to continue with that coverage. Indeed, it may be optimal for her to terminate her coverage (or begin covering herself, if she previously had not), given that her income will have changed from y tob.64

In other words, an individual’s health insurance status immediately after being laid off is given by:

ˆ xh =            x, ifx= 1 x∗_h, otherwise (2.2.7) 64

Possible extension: if discovered shirking, consider this a “gross misconduct” and prevent COBRA continuation payments in that case.

Wherex∗_h = arg maxx0_∈{0,2}U_h(x0) = arg max_x0_∈{0,2}v_h(b, x0). Next, let us write the value function of an unemployed individual with health status h and health insurance status

x∈ {0,2}: (1 +r)Uh(x) =vh(b, x) +αmax{VhH(y,xˆˆh), VhL(y,xˆˆh)}+ (1−α)Uh(x∗h) (2.2.8) Where ˆ ˆ xh=            1, if HI offered x∗∗_h , otherwise (2.2.9)

And x∗∗_h , the optimal choice of whether to purchase HI while employed when the employer does not offer HI, satisfies: x∗∗_h = arg maxx0_∈{0,2}v_h(y, x0).65

Notice that the individual will always accept the (y, x) offer, providedy > b, since she can always shirk on the job and earn the wage of y at no additional cost to herself. In other words, being employed and shirking is strictly better than being unemployed. The workers only choices, then, are:

1)x∗_h : whether to purchase HI when employed, if employers do not offer HI.

2)x∗∗_h : whether to purchase HI when unemployed, after COBRA continuation ends, or if employers did not provide HI.

Since the individual’s problem is stationary (health status does not change), if the worker shirked today, she will necessarily shirk again tomorrow (the same goes for exerting effort). In other words, “once a shirker always a shirker”. I can get rid of the max operator

65

This follows from equations (2.2.12) and (2.2.13). Note that x∗ may not equalx∗∗, since the flow utilityvh(y, x) atx= 2 depends on income, which differs when employed (y) or unemployed (b).

in the value functions while employed: rV_hH(x) = vh(y, x)−e+δ(Uh(ˆxh)−VhH(x)) (2.2.10) rV_hL(x) = vh(y, x) + (δ+q)(Uh(ˆxh)−VhL(x)) (2.2.11) Solving, I get: V_hH(x) = vh(y, x)−e+δUh(ˆxh) r+δ (2.2.12) V_hL(x) = vh(y, x) + (δ+q)Uh(ˆxh) r+δ+q (2.2.13)

Then, I have two No-Shirk-Conditions (“NSC’s”): V_hH(ˆxˆh) > VhL(ˆxˆh) for h = b, g.

From the previous value function expressions, I can re-write the NSC’s as:

vh(y,xˆˆh)> rUh(ˆxh) + (r+δ+q)e/q≡ˆvh(ˆxh) (2.2.14)

Notice immediately that offering health insurance is a “double-edged sword” (recall that if the firm offers HI, ˆxˆh = ˆxh = 1). On one hand, it raises vh(y, x) and adds to the

efficiency wage compensation required to get workers not to shirk, but on the other hand it increases the value of unemployment for workers, since after termination they get to continue with their health insurance coverage for a few months,66 raising ˆvh(ˆxh).

66

In my model, this continues until the next period, which is defined as the average length of COBRA continuation.