Capitation taxes and the regulation of professional services

Abstract

We analyze the introduction of capitation taxes as a mean of regulating professional services in a Principal-Agent moral-hazard framework. Although the tax increases cost and lowers participation, we find that it reduces the marginal price associated with quality. The optimal capitation system balances these opposing effects. We use the setup to derive comparative statics results and discuss possible impacts of improvements in ICT on occupational regulation. We find that the results are more ambiguous than often suggested.

Appendix

Proof of Proposition  1

Suppose that $X^{c}\left( \theta ,\phi \right) \le x^{SB}(\alpha ,\theta )$. Geometrically, this case is represented by Fig. 7a. Accordingly, we have $x^{*}\left( \alpha ,\theta ,\phi \right) =x^{SB}(\alpha ,\theta )$. Suppose that $X^{c}\left( \theta ,\phi \right) \ge x^{FB}(\alpha )$ which corresponds to Fig. 7b. Hence, we have $x^{*}\left( \alpha ,\theta ,\phi \right) =x^{FB}(\alpha ).$

Fig. 7

(a, b) The optimal contract

Finally, consider the intermediate case $x^{SB}(\alpha ,\theta )<X^{c}\left( \theta ,\phi \right) <x^{FB}(\alpha )$. By concavity of $V(\cdot )$ with $X^{c}\left( \theta ,\phi \right) <x^{FB}(\alpha )$, we know $\alpha V^{\prime }(X^{c}\left( \theta ,\phi \right) )-1>0$ and, thus, $x^{*}\left( \alpha ,\theta ,\phi \right) \ge X^{c}\left( \theta ,\phi \right)$. Also by concavity, we know $x^{SB}(\alpha ,\theta )<X^{c}\left( \theta ,\phi \right)$ implies $\alpha V^{\prime }(X^{c}\left( \theta ,\phi \right) )-\theta ^{-1}<0$ and therefore $x^{*}\left( \alpha ,\theta ,\phi \right) \le X^{c}\left( \theta ,\phi \right)$. In short, it implies $x^{*}\left( \alpha ,\theta ,\phi \right) =X^{c}\left( \theta ,\phi \right)$. $\square$

Proof of Proposition 3

We proceed in two steps.

• 1. Consider any $\phi$ with $x^{*}\left( \alpha ,\theta ,\phi \right) <X^{c}\left( \theta ,\phi \right)$. We are in a situation as in Fig. 7b so that participating consumers set $x^{*}\left( \alpha ,\theta ,\phi \right) =x^{FB}(\alpha )$. Keeping in mind that (16) is not satisfied, there is a critical transaction cost which is defined by:

  $$\begin{aligned} \tau ^{c}\left( \alpha ,\theta ,\phi \right) =\alpha v(x^{FB}(\alpha ))-x^{FB}(\alpha )-\phi \end{aligned}$$

  (31)

  Now, consider an infinitesimal reduction in $\phi$. By construction, this would not affect the quality implemented, but it would marginally raise $\tau ^{c}$. Hence, it would increase the total number of participating consumers, thereby, raising the total sum of trade gains. Consequently, at $\phi ^{*}\left( \alpha ,\theta ,\mu \right)$, we must have $x^{*}\left( \alpha ,\theta ,\phi ^{*}\right) \ge X^{c}\left( \theta ,\phi ^{*}\right)$.

• 2. Consider any $\phi$ with $x^{*}\left( \alpha ,\theta ,\phi \right) >X^{c}\left( \alpha ,\theta ,\phi \right)$. This corresponds to a situation similar to the one represented in Fig. 6a. By (13) it implies that $x^{*}\left( \alpha ,\theta ,\phi ^{*}\right) =x^{SB}(\alpha ,\theta )$ and, thus, $x^{SB}(\alpha ,\theta )>X^{c}\left( \alpha ,\theta ,\phi \right)$. Consider an infinitesimal increase in the capitation to $\phi +\varepsilon$ with $\varepsilon >0$. It has no effect on the quality implemented – which remains at $x^{SB}(\alpha ,\theta )$ – and no impact on participation. Hence, we have $W\left( \phi ;\alpha ,\theta ,\mu \right) =W\left( \phi +\varepsilon ;\alpha ,\theta ,\mu \right)$. The same process will continue until we reach capitation tax $\phi =\phi ^{SB}\left( \alpha ,\theta \right)$ which is implicitly defined so that the kink of the cost function $C(x;\theta ,\phi ^{SB})$ occurs exactly at $x^{SB}(\alpha ,\theta )$. Accordingly, we have $W\left( \phi ;\alpha ,\theta ,\mu \right) =W\left( \phi ^{SB};\alpha ,\theta ,\mu \right)$. We now argue that $W_{\phi }\left( \phi ^{SB};\alpha ,\theta ,\mu \right) >0$. Consider an additional infinitesimal increase in capitation at $\phi ^{SB}\left( \alpha ,\theta \right)$. By (9) it forces participating consumers to marginally raise the quality they induce. This generates a first-order effect in quality. However, it has only a second-order effect on participation . Hence, $x^{*}\left( \alpha ,\theta ,\phi \right) >X^{c}\left( \theta ,\phi \right)$ can never be optimal with the result that $x^{*}\left( \alpha ,\theta ,\phi ^{*}\right) \le X^{c}\left( \theta ,\phi ^{*}\right)$.

Taking points 1 and 2 together confirms (i). Using (i), point 1 implies $X^{c}\left( \theta ,\phi ^{*}\right)\, \le x^{FB}(\alpha )$ and point 2 validates $x^{SB}(\alpha ,\theta )\le X^{c}\left( \theta ,\phi ^{*}\right)$. $\square$

Proof of Proposition 5

Slightly abusing notation, the first-order condition from (24) becomes:

$$\begin{aligned} \left( \alpha v^{\prime }\left( X^{c}\right) -1\right) X_{\phi }^{c}G(\tau ^{c}|\mu )+\left( \alpha v\left( X^{c}\right) -X^{c}\right) g(\tau ^{c}|\mu )\tau _{\phi }^{c}-\tau ^{c}g(\tau ^{c}|\mu )\tau _{\phi }^{c}=0 \end{aligned}$$

(32)

Taking (20) into account and factorizing allows us to rewrite (32) as follows:

$$\begin{aligned} \left[ \left( \alpha v^{\prime }\left( X^{c}\right) -1\right) X_{\phi }^{c}+\phi \frac{g(\tau ^{c}|\mu )}{G(\tau ^{c}|\mu )}\tau _{\phi }^{c}\right] G(\tau ^{c}|\mu )=0 \end{aligned}$$

(33)

From the definition of $X^{c}$ in (9), we have $X_{\phi }^{c}>0$. Moreover, by Proposition 4 we know $\tau _{\phi }^{c}\le 0$. Hence, we have $\alpha v^{\prime }\left( X^{c}\right) \ge 1$ or equivalently $\phi ^{*}\left( \alpha ,\theta \right) \le \phi ^{FB}\left( \alpha ,\theta \right)$. Note that the equality is not feasible since at $\phi =\phi ^{FB}\left( \alpha ,\theta \right)$, we have $\tau _{\phi }^{c}<0$. Using (22) and (9), the equality (32) can be equivalently rewritten as

$$\begin{aligned} \left( \alpha v^{\prime }\left( X^{c}\right) -1\right) \frac{\theta }{1-\theta }=\frac{\phi g(\tau ^{c}|\mu )}{G(\tau ^{c}|\mu )+\phi g(\tau ^{c}|\mu )}\ , \end{aligned}$$

(34)

which implies $\alpha v^{\prime }\left( X^{c}\right) <\theta ^{-1}$. Accordingly, by (23) we have $\phi ^{*}\left( \alpha ,\theta \right) >\phi ^{SB}\left( \alpha ,\theta \right)$. $\square$