Barefoot and footloose doctors: optimal resource allocation in developing countries with medical migration

Abstract

In light of the shortage of healthcare professionals, many developing countries operate a de facto two-tiered system of healthcare provision, in which Community Health Workers (CHWs) supplement service provision by fully qualified physicians. CHWs are relatively inexpensive to train but can treat only a limited range of medical conditions. This paper explicitly models a two-tiered structure of healthcare provision and characterizes the optimal allocation of resources between training doctors and CHWs, and implications for population health outcomes. We analyze how medical migration alters resource allocation and population health outcomes, shifting resources towards training CHWs. In the model, migration stimulates health care provision at the lower end of the illness severity spectrum, improving health outcomes for those patients; sufferers of relatively severe medical conditions who can only be treated by doctors are made worse off. It is further shown that donor countries must be reimbursed by at least the training cost of emigrating physicians in order to restore aggregate population health to the pre-migration level.

Appendix

Proofs of theorems are provided below.

Proof of Lemma 1

Suppose doctors spend time t(s) treating patients with illnesses in the interval $s\in (s_1 ,s_1 +\updelta )\subset [1,\hat{{s}}]$. The outcome for this group of patients is

$$\begin{aligned} \int \limits _{s_1 }^{s_1 +\updelta } \hbox {log}{\left( {\frac{1+q_1 t(s)}{s}} \right) } dF(s). \end{aligned}$$

(24)

The cost of this treatment is $c_1 \int \nolimits _{s_1 }^{s_1 +\updelta } {t(s)dF(s)=C_1 }$ . Now let $\hat{{t}}(s)=\frac{q_1 t(s)}{q_2 }$, and let these patients be treated instead by CHWs with treatment times $\hat{{t}}\left( s \right)$. The welfare outcome is identical but the cost of treatment is:

$$\begin{aligned} c_2 \int \limits _{s_1 }^{s_1 +\updelta } {\frac{q_1 t(s)}{q_2 }dF(s)=\frac{c_2 q_1 }{q_2 }\frac{C_1 }{c_1 }=\frac{r_2 }{r_1 }C_1 <C_{1} }, \end{aligned}$$

(25)

where the inequality follows from assumption (3). Hence it is not optimal to treat these patients using doctors. $\square$

Proof of Proposition 1

• 1. Observe first that ${Q}'(x)=1-F(x)>0$, so Q is increasing.

• 2. The proof is based on the fact that program (7) is a convex program. Define the Lagrangian function:

  $$\begin{aligned} L(\upvarepsilon )= & {} \int \limits _{1}^{\hat{s}}\hbox {log}\left( {\frac{1+q_2 (t_2 (s)+\upvarepsilon \Delta t_2 (s))}{s}}\right) dF(s)\\&+\,\int \limits _{\hat{{s}}}^\infty {\log } \left( {\frac{1+q_1 (t_1 (s)+\upvarepsilon \Delta t_1 (s))}{s}} \right) dF(s) \\&+\,{\upmu }_2 \left( {m_2 +\upvarepsilon \Delta m_2 -\int \limits _{1}^{\hat{s}} {(t_2 (s)+\upvarepsilon \Delta t_2 (s))dF(s)} } \right) \\&+\,\upmu _1 \left( {m_1 +\upvarepsilon \Delta m_1 -\int \limits _1^{\hat{{s}}} {(t_1 (s)+\upvarepsilon \Delta t_1 (s))dF(s)} } \right) \\&+\,\int \limits _1^{s^{*}} {\upalpha (\hbox {s})(s-(1+q_2 (t_2 (s)+\upvarepsilon \Delta t_2 (s))))} dF(s)\\&+\,\uplambda (\bar{{M}}-c_1 (m_1 +\upvarepsilon \Delta m_1 )-c_2 (m_2 +\upvarepsilon \Delta m_2 )) \end{aligned}$$

  where the functions $t_1 (s),t_2 (s)$ and the numbers $m_1 \hbox { and }m_2$ comprise the candidate for the optimal solution of the program. If we can produce a non-negative function $\upalpha (\cdot )$ on $[1,s^{*}]$ and non-negative constants $({\upmu }_1 ,{\upmu }_2 ,\uplambda )$ such that ${L}'(0)=0$ at the values of these variables stated in the proposition, then the proposition is proved. For this will mean that the concave function L is maximized at $\upvarepsilon =0$. In the Lagrangian function L the functions $\Delta t_i (\cdot ),i=1,2$ are arbitrary feasible variations from the conjectured optimal solution, as are the numbers .$\Delta m_i$.

• 2. Evaluating the derivative at zero:

  $$\begin{aligned} {L}'(0)= & {} \int \limits _1^{\hat{{s}}} \frac{q_2 \Delta t_2 (s)}{1+q_2 t_2 (s)}dF(s)+\int \limits _{\hat{{s}}}^\infty \frac{q_1 \Delta t_1 (s)}{1+q_1 t_1 (s)}dF(s) \nonumber \\&+\,\upmu _2 \left( {\Delta m_2 -\int \limits _1^{\hat{{s}}} {\Delta t_2 (s)} dF(s)} \right) \nonumber \\&+\,\upmu _1 \left( \Delta m_1 -\int \limits _{\hat{{s}}}^\infty {\Delta t_1 (s)} dF(s)\right) -\lambda (c_1 \Delta m_1 +c_2 \Delta m_2) \nonumber \\&-\int \limits _1^{s^{*}} {\upalpha (s)q_2 \Delta t_2 (s)dF(s)}. \end{aligned}$$

  (26)

  If we can choose non-negative Lagrangian multipliers $(\upmu _1 ,\upmu _2 ,\uplambda )$ and a non-negative function $\upalpha (s)$ on $[1,s^{*}]$ so that the coefficients of $(\Delta t_1 (s),\Delta t_2 (s),\Delta m_1 ,\Delta m_2 )$ are all annihilated, then the result is proved.

• 3. The coefficients of these variations are:

  $\Delta t_2 (s):\frac{q_2 }{1+q_2 t_2 (s)}-\upalpha (s)q_2 \updelta [1,s^{*}]-\upmu _2 =0, \hbox { for } s\in [1,\hat{{s}}]. \updelta [\hbox {a},\hbox {b}]$ is the function that is 1 on [a,b], and 0 elsewhere;

  

• 4. From the $\Delta t_2$ condition, we have $\upalpha (s)=-\frac{\upmu _2 }{q_2 }+\frac{1}{s}\hbox { for }s\in [1,s^{*}]$. At $s^{*}$, we choose $\upalpha (s^{*})=0$, implying $\upmu _2 =\frac{q_2 }{s^*}$. It follows that $\upalpha (s)>0\hbox { for }s<s^{*}$. Therefore $\uplambda =\frac{q_2 }{c_2 s^{*}}=\frac{1}{r_2 s^{*}}$ and so $\upmu _1 =\frac{c_1 }{r_2 s^{*}}.$ Therefore $(1+q_1 t_1 )\frac{c_1 }{r_2 s^{*}}=q_1$, giving $t_1 =(\frac{r_2 s^{*}}{r_1 }-1)/q_1$. For $t_1$ to be positive we require $s^{*}>\frac{r_1 }{r_2 }$, which we will attend to below. We must also check that we are not treating patients with illness severity $s>\hat{{s}}$ with more treatment than they require: that is, we want $1+q_1 t_1 \le \hat{{s}}$. But this is true because $s^{*}<\hat{{s}}<\frac{r_1 }{r_2 }\hat{{s}}$.

• 5. We now check the budget constraint for the proposed solution, which is:

  $$\begin{aligned} c_2 \int \limits _1^{s^{*}} {\frac{s-1}{q_2 }dF(s)+c_2 \int \limits _{s^{*}}^{\hat{{s}}} {\frac{s^{*}-1}{q_2 }} } dF(s)+c_1 \int \limits _{\hat{{s}}}^\infty {\left( {\left( \frac{r_2 s^{*}}{r_1 }-1\right) /q_1 } \right) } dF(s)=\bar{{M}}. \end{aligned}$$

  This can be written as:

  $$\begin{aligned} r_2 \int \limits _1^{s^{*}} {(s-1)dF(s)+r_2 (s^{*}-1)(F(\hat{{s}})-F(s^{*}))+(r_2 s^{*}-r_1 )(1-F(\hat{{s}}))=M} \end{aligned}$$

  which in turn is equivalent to condition (10) defining $s^{*}$. Finally, our premise (ii) is exactly the condition that tells us a unique solution $s^{*}$ exists to equation such that $\frac{r_1 }{r_2 }<s^{*}<\hat{{s}}$. This is so because the function Q is monotone increasing, and the existence of $s^{*}$ therefore follows from condition (ii) by the intermediate value theorem. $\square$