Political Economics of Public Pricing of Final and Intermediate Goods

Abstract

In this chapter, we study the effect of lobbying by special interest groups on the optimal pricing rule of publicly produced final and intermediate goods. We show that when the weight that the government places on campaign contributions from a special interest group organized by workers increases, the price of publicly produced final goods decreases and that of intermediate goods increases. However, when the weight that the government places on campaign contributions from a special interest group organized by capitalists increases, the effect on the prices of final and intermediate goods depends on capitalists’ roles as both consumers and owners of firms. The effects of lobbying by workers and capitalists are asymmetric because the public enterprise must adhere to its budget constraint and because the roles of capitalists and workers in the economy differ.

Appendix

1.1 The Optimal Pricing Rule of a Benevolent Government

By solving the maximization problem for the government, the first-order conditions can be obtained, as follows:

$$ \frac{\partial W}{{\partial p^{h} }} + \lambda_{1} \left( {\frac{{\partial\pi^{*} }}{{\partial p^{h} }} + \frac{{\partial\pi^{*} }}{{\partial q^{*} }}\frac{{\partial q^{*} }}{{\partial p^{h} }}} \right) = 0, $$

(15.22)

$$ \frac{\partial W}{{\partial p^{f} }} + \lambda_{1} \left( {\frac{{\partial\pi^{*} }}{{\partial p^{f} }} + \frac{{\partial\pi^{*} }}{{\partial q^{*} }}\frac{{\partial q^{*} }}{{\partial p^{f} }}} \right) = 0. $$

(15.23)

Here, as mentioned in the body of this chapter, $\lambda_{1}$ represents the Lagrange multiplier. Eq. 15.23 can be rewritten as

$$ \begin{aligned} & \frac{\partial W}{{\partial v^{C} }}\left( {\frac{{\partial v^{C} }}{{\partial q^{*} }}\frac{{\partial q^{*} }}{{\partial p^{h} }} + \frac{{\partial v^{C} }}{{\partial p^{h} }} + \frac{{\partial v^{C} }}{{\partial\pi^{*} }}\frac{{\partial\pi^{*} }}{{\partial q^{*} }}\frac{{\partial q^{*} }}{{\partial p^{h} }}} \right) + \frac{\partial W}{{\partial v^{L} }}\left( {\frac{{\partial v^{L} }}{{\partial q^{*} }}\frac{{\partial q^{*} }}{{\partial p^{h} }} + \frac{{\partial v^{L} }}{{\partial p^{h} }}} \right)\\ & \quad + \lambda_{1} \left( {\frac{{\partial\pi^{*} }}{{\partial p^{h} }} + \frac{{\partial\pi^{*} }}{{\partial q^{*} }}\frac{{\partial {\text{q}}^{*} }}{{\partial p^{h} }}} \right) = 0. \end{aligned} $$

Define $I^{L} \equiv \pi$ and $I^{C} \equiv L_{y} + L_{z}$. Noting that $\frac{{\partial v^{i} }}{{\partial q^{*} }} = - \frac{{\partial v^{i} }}{{\partial I^{i} }}y^{i}$ and $\frac{{\partial v^{i} }}{{\partial p^{h} }} = - \frac{{\partial v^{i} }}{{\partial I^{i} }}z^{h,i}$ hold from Roy’s identity, $\frac{{\partial\pi^{*} }}{{\partial q^{*} }} = y$ holds from the feature of profit maximization, and $\frac{{\partial v^{i} }}{{\partial\pi^{*} }} = \frac{{\partial v^{i} }}{{\partial I^{i} }}$, we can further rewrite the above expression as follows:

$$ \frac{\partial W}{{\partial v^{C} }}\frac{{\partial v^{C} }}{{\partial I^{C} }}\left\{ {\left( {y - y^{C} } \right)\frac{{\partial q^{*} }}{{\partial p^{h} }} - z^{h,C} } \right\} + \frac{\partial W}{{\partial v^{L} }}\frac{{\partial v^{L} }}{{\partial I^{L} }}\left\{ {\left( { - y^{L} } \right)\frac{{\partial q^{*} }}{{\partial p^{h} }} - z^{h,L} } \right\} + \lambda_{1} \frac{{\partial\pi^{*} }}{{\partial p^{h} }} = 0. $$

Therefore,

$$ {\frac{\partial W}{{\partial v^{C} }}\left( {\frac{{\partial v^{C} }}{{\partial I^{C} }}\left( {y^{C} - y} \right) + \frac{{\partial v^{L} }}{{\partial I^{L} }}y^{L} } \right)\frac{{\partial q^{*} }}{{\partial p^{h} }} + \frac{\partial W}{{\partial v^{L} }}\left( {\frac{{\partial v^{C} }}{{\partial I^{C} }}z^{h,C} + \frac{{\partial v^{L} }}{{\partial I^{L} }}z^{h,L} } \right)} = \lambda_{1} \frac{{\partial\pi^{*} }}{{\partial p^{h} }} \\ $$

(15.24)

can be obtained.

Similarly, Eq. 15.23 can be rewritten as

$$ \frac{\partial W}{{\partial v^{C} }}\left( {\frac{{\partial v^{C} }}{{\partial q^{*} }}\frac{{\partial q^{*} }}{{\partial p^{f} }} + \frac{{\partial v^{C} }}{{\partial\pi^{*} }}\frac{{\partial\pi^{*} }}{{\partial q^{*} }}\frac{{\partial q^{*} }}{{\partial p^{f} }} + \frac{{\partial v^{C} }}{{\partial\pi^{*} }}\frac{{\partial\pi^{*} }}{{\partial p^{f} }}} \right) + \frac{\partial W}{{\partial v^{L} }}\left( {\frac{{\partial v^{L} }}{{\partial q^{*} }}\frac{{\partial q^{*} }}{{\partial p^{f} }}} \right) + \lambda_{1} \frac{{\partial\pi^{*} }}{{\partial p^{f} }} = 0. $$

Again, from Roy’s identity, $\frac{{\partial v^{i} }}{{\partial q^{*} }} = - \frac{{\partial v^{i} }}{{\partial I^{i} }}y^{i}$ and $\frac{{\partial v^{i} }}{{\partial p^{h} }} = - \frac{{\partial v^{i} }}{{\partial I^{i} }}z^{h,i}$ hold, and from Hotelling’s lemma, $\frac{{\partial\pi^{*} }}{{\partial p^{f} }} = - z^{f}$ holds. Therefore,

$$ \frac{\partial W}{{\partial v^{C} }}\frac{{\partial v^{C} }}{{\partial I^{C} }}\left( {\left( {y - y^{C} } \right)\frac{{\partial q^{*} }}{{\partial p^{f} }} - z^{f} } \right) + \frac{\partial W}{{\partial v^{L} }}\frac{{\partial v^{L} }}{{\partial I^{L} }}\left\{ {\left( { - y^{L} } \right)\frac{{\partial q^{*} }}{{\partial p^{f} }}} \right\} + \lambda_{1} \frac{{\partial\pi^{*} }}{{\partial p^{f} }} = 0 $$

also holds. Finally, we obtain

$$ \frac{\partial W}{{\partial v^{C} }}\left( {\frac{{\partial v^{C} }}{{\partial I^{C} }}\left( {y^{C} - y} \right) + \frac{\partial W}{{\partial v^{L} }}\frac{{\partial v^{L} }}{{\partial I^{L} }}y^{L} } \right)\frac{{\partial q^{*} }}{{\partial p^{h} }} + \frac{\partial W}{{\partial v^{C} }}\frac{{\partial v^{C} }}{{\partial I^{C} }}z^{f} = \lambda_{1} \frac{{\partial\pi^{*} }}{{\partial p^{f} }}. $$

(15.25)

1.2 The Optimal Pricing Rule with Lobbying Activities

Politicians determine the optimal price to maximize Eq. 15.8 given Eqs. 15.10, 15.11, 15.12 and 15.13. When we ignore the general equilibrium effect, the first-order condition can be written as

$$ \begin{aligned} & \lambda_{2} \left( {\left( {z^{h,L} + z^{h,C} } \right) + p^{h} \frac{{\partial \left( {z^{h,L} + z^{h,C} } \right)}}{{\partial p^{h} }}} \right. \\ & \quad \left. { - \frac{{\partial C\left( {z^{h,L} + z^{h,C} + z^{f} } \right)}}{{\partial \left( {z^{h,L} + z^{h,C} + z^{f} } \right)}}\frac{{\partial \left( {z^{h,L} + z^{h,C} + z^{f} } \right)}}{{\partial p^{h} }}} \right) \\ & = \alpha z^{h} + \left( {\theta^{C} \frac{{\partial v^{C} }}{{\partial I^{C} }}z^{h,C} + \theta^{L} \frac{{\partial v^{L} }}{{\partial I^{L} }}z^{h,L} } \right), \\ \end{aligned} $$

As in the previous section, we rewrite the expression as the difference between prices and marginal costs,

$$ \phantom{000000} \begin{aligned} & \frac{{\left( {\alpha - \lambda_{2} } \right)\left( {z^{h,L} + z^{h,C} } \right) + \theta^{C} \frac{{\partial v^{C} }}{{\partial I^{C} }}z^{h,C} + \theta^{L} \frac{{\partial v^{L} }}{{\partial I^{L} }}z^{h,L} }}{{\lambda_{2} }} \\ & = \left\{ {p^{h} - \frac{{\partial C\left( {z^{h,L} + z^{h,C} + z^{f} } \right)}}{{\partial \left( {z^{h,L} + z^{h,C} + z^{f} } \right)}}} \right\}\frac{{\partial \left( {z^{h,L} + z^{h,C} + z^{f} } \right)}}{{\partial p^{h} }} \\ \end{aligned} $$

$$ \begin{aligned} & \frac{{\left( {\alpha - \lambda_{2} } \right) + \theta^{C} \frac{{z^{h,C} }}{{\left( {z^{h,L} + z^{h,C} } \right)}}\frac{{\partial v^{C} }}{{\partial I^{C} }} + \theta^{L} \frac{{z^{h,L} }}{{\left( {z^{h,L} + z^{h,C} } \right)}}\frac{{\partial v^{L} }}{{\partial I^{L} }}}}{{\lambda_{2} \frac{{\partial \left( {z^{h,L} + z^{h,C} + z^{f} } \right)}}{{\partial p^{h} }}\frac{{p^{h} }}{{\left( {z^{h,L} + z^{h,C} } \right)}} }} \\ & = \frac{1}{{p^{h} }}\left\{ {p^{h} - \frac{{\partial C\left( {z^{h,L} + z^{h,C} + z^{f} } \right)}}{{\partial \left( {z^{h,L} + z^{h,C} + z^{f} } \right)}}} \right\} \end{aligned} $$

$$ \phantom{0000}\begin{aligned} \frac{{p^{h} - \frac{{\partial C\left( {z^{h,L} + z^{h,C} + z^{f} } \right)}}{{\partial \left( {z^{h,L} + z^{h,C} + z^{f} } \right)}}}}{{p^{h} }} & = \frac{1}{{\lambda_{2} }}\left( {\frac{1}{{\varepsilon^{h} }}} \right)\left\{ {\left( {\alpha - \lambda_{2} } \right) + \theta^{C} \frac{{z^{h,C} }}{{\left( {z^{h,L} + z^{h,C} } \right)}}\frac{{\partial v^{C} }}{{\partial I^{C} }}} \right. \\ & \quad + \left. { \theta^{L} \frac{{z^{h,L} }}{{\left( {z^{h,L} + z^{h,C} } \right)}}\frac{{\partial v^{L} }}{{\partial I^{L} }}} \right\}, \\ \frac{{p^{h} - \frac{{\partial C\left( {z^{h,L} + z^{h,C} + z^{f} } \right)}}{{\partial \left( {z^{h,L} + z^{h,C} + z^{f} } \right)}}}}{{p^{h} }} & = \frac{{\alpha - \lambda_{2} }}{{\lambda_{2} }}\left( {\frac{1}{{\varepsilon^{h} }}} \right) \\ & \quad + \left( {\frac{{\theta^{C} }}{{\lambda_{2} }}\frac{{z^{h,C} }}{{\left( {z^{h,L} + z^{h,C} } \right)}}\frac{{\partial v^{C} }}{{\partial I^{C} }}} { + \frac{{\theta^{L} }}{{\lambda_{2} }}\frac{{z^{h,L} }}{{\left( {z^{h,L} + z^{h,C} } \right)}}\frac{{\partial v^{L} }}{{\partial I^{L} }}} \right)\left( {\frac{1}{{\varepsilon^{h} }}} \right). \\ \end{aligned} $$

(15.26)

The effect of lobbying activities on the price of publicly produced intermediate goods can be obtained similarly.

1.3 The Effect of Interest on the Price of Publicly Produced Goods

Totally differentiating the first-order condition and the budget constraint of the public enterprise with respect to $p^{h}$ and $p^{f}$ gives

$$ \left[ {\begin{array}{*{20}c} {G_{{p^{h} p^{h} }} } & {G_{{p^{h} p^{f} }} } & {\pi_{{p^{h} }}^{*} } \\ {G_{{p^{f} p^{h} }} } & {G_{{p^{f} p^{f} }} } & {\pi_{{p^{f} }}^{*} } \\ {\pi_{{p^{h} }}^{*} } & {\pi_{{p^{f} }}^{*} } & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {dp^{h} } \\ {dp^{f} } \\ {d\lambda_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - G_{{p^{h} \theta^{L} }} } \\ { - G_{{p^{f} \theta^{L} }} } \\ { - \pi_{{\theta^{L} }}^{*} } \\ \end{array} } \right]d\theta^{L} + \left[ {\begin{array}{*{20}c} { - G_{{p^{h} \theta^{C} }} } \\ { - G_{{p^{f} \theta^{C} }} } \\ { - \pi_{{\theta^{C} }}^{*} } \\ \end{array} } \right]d\theta^{C} . $$

The determinant of the matrix on the left-hand side, $D = G_{{p^{h} p^{f} }} \pi_{{p^{h} }}^{*} \pi_{{p^{f} }}^{*} + G_{{p^{f} p^{h} }} \pi_{{p^{h} }}^{*} \pi_{{p^{f} }}^{*} - G_{{p^{h} p^{h} }} \left( {\pi_{{p^{f} }}^{*} } \right)^{2} - G_{{p^{f} p^{f} }} \left( {\pi_{{p^{h} }}^{*} } \right)^{2}$, is positive from the second-order condition for maximization.