HC Economist Models

A model of Medicaid’s poverty trap

The design of legislation which regulates Medicaid eligibility creates a poverty trap. In California, generally those who have income below 250% of the federal poverty level and who have limited assets are eligible for Medicaid. (In reality California’s Medicaid eligibility is more complicated than this. For full details of eligibility requirements see “Medi-Cal Facts and Figuresâ€? produced by the California HealthCare Foundation.)

Let me explain how the poverty trap functions. Let us examine an individual which earns 249% of the federal poverty level. The individual may desire to work more hours, but by doing so, he/she will lose the Medicaid benefit. Thus, by working more, the individual is actually worse off.

Here is my model which will describe this phenomenon mathematically. Consumers are utility maximizers:

max(c,h,l) U; U=(1-p)c + p[c-S-a*ln (1+ h +M*)]+g(l)

  • s.t.: c+p(1+j)h+wl=w{L_bar}
  • c>=0, h>=0, 0<=l<={L_bar}
  • c: consumption
  • S: disutility from being sick
  • p: probability of getting sick
  • M*: Medicaid Benefit, M*=0 if wL>E. Otherwise M*=M
  • E: Eligibility limit on Medicare, in California 250% of the poverty line
  • h=health insurance purchased
  • w= wage
  • l=leisure, L=hours worked, {L_bar}=total hours in time period
  • gamma=shadow price of income
  • g’>0; g”<0


The first order conditions are:

    • d c: gamma=1
    • d h: h*=a/(1+j) – 1 – M*
    • d l: g'(l)=w; L={L_bar} – g’_^{-1}(w)

Since h* does not depend on c, L, or w; dc/dL>0 on the support where c, L are continuous. However, if L=(E-e)/w where e is an arbitrarily small positive number, then increasing L by a non-differential amount will decrease consumption. Since h* does not depend on l or c, the individual will choose the same h* with or without the Medicaid benefit (assuming h*>=M). Thus, the poor individual near the Medicare eligibility limit will actually be worse off by working more. This is a poverty trap.

Conservatives often claim that the poor are lazy. While this is certainly true in some cases, and while drugs are an overwhelming problem for the poor, the vast majority of the poor are hard working individuals. They are not stupid, however. As this model shows, when working more hours leads to less disposable income, it is entirely rational to limit hours worked in order to collect the government benefit.

A voucher program could eliminate this problem. The voucher program would have the following model:

max(c,h,l) U; U=(1-p)c + p[c-S-a*ln (1+ h)]+g(l)

  • s.t.: c+p(1+j)h+wl=w{L_bar} + V*
  • c>=0, h>=0, 0<=l<={L_bar}
  • V*={h_bar}p(1+j) if h>={h_bar} and V*=0 if h<{h_bar}

In essence, this would allow the government to pay for an individual’s insurance up to {h_bar}, but workers who decide not to purchase insurance would not receive this subsidy. Since, the government will be giving free health insurance, it is irrational to not choose h*>={h_bar}. V* does not depend on earnings and thus there is no disincentive to work.