A Welfare Analysis of Location Space Constraints with Vertically Separated Sellers

Abstract

This paper studies the welfare effects of location space constraints when the duopoly sellers are vertically separated. As the downstream firms respond to higher input prices by locating further away from the center of the market, constraining them to locate within the linear city allows the upstream manufacturers better to exploit the downstream industry. This leads to higher final good prices and lower consumer welfare (despite the savings on transportation). This result – which is robust to the inclusion of R&D decisions – is in sharp contrast to the case in which the sellers are vertically integrated. Also different, the incentive to invest in cost-reducing R&D is always insufficient compared with the social optimum. Our results thus suggest the importance of taking into account the vertical market structure in formulating zoning (product standard) and R&D policies.

Appendix

Proof of Lemma 1

We use backward induction to solve this three stage game. Given the costs of the downstream firms (the input prices), the solutions to the pricing and location stages are

$$\begin{aligned} p_a= & {} \frac{1}{3}(2tx_b-2tx_a+2w_1+w_2+tx_b^2-tx_a^2), \quad p_b=\frac{1}{3}(4tx_b-4tx_a+w_1+2w_2-tx_b^2+tx_a^2),\\ x_a^{vc}=\,& {} 0, \qquad x_b^{vc}=1. \end{aligned}$$

The location of the marginal consumer is

$$\begin{aligned} \hat{x}=\frac{1}{2}+\frac{w_2-w_1}{6t}. \end{aligned}$$

In the first stage of input pricing, the first order conditions are

$$\begin{aligned} \frac{\partial \pi _1}{\partial w_1}=\frac{3t+w_2-2w_1+c}{6t}=0, \qquad \frac{\partial \pi _2}{\partial w_2}=\frac{3t-2w_2+w_1+c}{6t}=0. \end{aligned}$$

It is easy to verify that the second-order conditions are satisfied.

Solving the two first order conditions, we have

$$\begin{aligned} w_1^{vc}=w_2^{vc}=3t+c. \end{aligned}$$

Substituting them back into $p_a$ and $p_b$, we have $p_a=p_b=4t+c$.

The profits that are earned by the downstream firms and the upstream manufacturers are respectively $\pi _1^{vc}=\pi _1^{vc}=\frac{3t}{2}$ and $\pi _a^{vc}=\pi _b^{vc}=\frac{t}{2}$, and consumer surplus is

$$\begin{aligned} CS^{vc}=V-\frac{49t}{12}-c. \end{aligned}$$

$\square$

Proof of Lemma 2

Given the input prices and the locations, the pricing stage is the same as that in the location-constrained model. Solving the first order conditions in the location stage, we have

$$\begin{aligned} x_a=\frac{4w_2-4w_1-3t}{12t}, \qquad x_b=\frac{4w_2-4w_1+15t}{12t}. \end{aligned}$$

The location of the marginal consumer is

$$\begin{aligned} \hat{x}=\frac{1}{2}+\frac{2w_2-2w_1}{9t}. \end{aligned}$$

Moving to the first stage of input-price setting, we have

$$\begin{aligned} \frac{\partial \pi _1}{\partial w_1}=\frac{9t+4w_2-8w_1+4c}{18t}=0, \qquad \frac{\partial \pi _2}{\partial w_2}=\frac{9t+4w_1-8w_2+4c}{18t}=0, \end{aligned}$$

from which we solve for

$$\begin{aligned} w_1^{vu}=w_2^{vu}=\frac{9t}{4}+c. \end{aligned}$$

The following equilibrium outcomes are then obtained:

$$\begin{aligned} p_a^{vu}= & {} p_b^{vu}=\frac{15t}{4}+c, \qquad x_a^{vu}=-\frac{1}{4}, \quad x_b^{vu}=\frac{5}{4},\\ \pi _1^{vu}= & {} \pi _2^{vu}=\frac{9t}{8}, \qquad \pi _a^{vu}=\pi _b^{vu}=\frac{3t}{4}, \qquad CS^{vu}=V-\frac{193t}{48}-c. \end{aligned}$$

$\square$

Proof of Result 2

The equilibrium outcomes in the two models can be calculated as follows:

$$\begin{aligned} p_i^{bc}= & {} t+(c-d^{bc}), \quad \pi _i^{bc}=\frac{t}{2}-I(d^{bc}), \quad CS^{bc}=V-\frac{13t}{12}-(c-d^{bc}).\\ p_i^{bu}= & {} \frac{3t}{2}+(c-d^{bu}), \quad \pi _i^{bu}=\frac{3t}{4}-I(d^{bu}), \quad CS^{bu}=V-\frac{85t}{48}-(c-d^{bu}). \end{aligned}$$

The differences of equilibrium prices, profits, and consumer and social welfare in the two models are

$$\begin{aligned} p_i^{bc}-p_i^{bu}=d^{bu}-d^{bc}-\frac{t}{2},\quad \pi _i^{bc}-\pi _i^{bu}=I(d^{bu})-I(d^{bc})-\frac{t}{4},\quad CS^{bc}-CS^{bu}=\frac{11t}{16}+d^{bc}-d^{bu}. \end{aligned}$$

With $d^{bu}>d^{bc}$ and $I(d^{bu})>I(d^{bc})$, the signs are all ambiguous. $\square$

Proof of Lemma 3

The solutions to the pricing and location stages (and the location of the marginal consumer) are the same as those in Lemma 1. In the input pricing stage, the first-order conditions are

$$\begin{aligned} \frac{\partial \pi _1}{\partial w_1}=\frac{3t+w_2-2w_1+c-d_1}{6t}=0, \qquad \frac{\partial \pi _2}{\partial w_2}=\frac{3t+w_1-2w_2+c-d_2}{6t}=0. \end{aligned}$$

With the second-order conditions being satisfied ($\frac{\partial ^2 \pi _1}{\partial w_1^2}=\frac{\partial ^2 \pi _2}{\partial w_2^2}=-\frac{1}{3t}.$), we have

$$\begin{aligned} w_1=3t+c-\frac{d_2+2d_1}{3}, \qquad w_2=3t+c-\frac{2d_2+d_1}{3}. \end{aligned}$$

Substituting in the values of p, x and w, we can write manufacturers 1 and 2’s profit as:

$$\begin{aligned} \pi _1=\frac{3t}{2}-\frac{d_2-d_1}{3}+\frac{(d_1-d_2)^2}{54t}-I_1, \qquad \pi _2=\frac{3t}{2}+\frac{d_2-d_1}{3}+\frac{(d_1-d_2)^2}{54t}-I_2. \end{aligned}$$

The first-order conditions for R&D choices are

$$\begin{aligned} \frac{\partial \pi _1}{\partial d_1}=\frac{1}{3}-\frac{(d_2-d_1)}{27t}-I_1', \qquad \frac{\partial \pi _2}{\partial d_2}=\frac{1}{3}+\frac{(d_2-d_1)}{27t}-I_2'. \end{aligned}$$

The second-order conditions ($\frac{1}{27t}-I''<0$) are satisfied when $I''$ is sufficiently large.

Impose symmetry, and we obtain

$$\begin{aligned} d_1=d_2=d^{vc}, \qquad such \ that \ \qquad \frac{1}{3}-I'(d^{vc})=0. \end{aligned}$$

$\square$

Proof of Lemma 4

The equilibrium prices and locations (and the location of the marginal consumer) given input prices are the same as those in Lemma 2. The first-order conditions in the input-pricing stage are

$$\begin{aligned} \frac{\partial \pi _1}{\partial w_1}=\frac{4w_2-8w_1+9t+4c-4d_1}{18t}=0, \qquad \frac{\partial \pi _2}{\partial w_2}=\frac{9t+4w_1-8w_2+4c-4d_2}{18t}=0, \end{aligned}$$

from which we obtain

$$\begin{aligned} w_1=\frac{9t}{4}+c-\frac{2d_1+d_2}{3}, \qquad w_2=\frac{9t}{4}+c-\frac{2d_2+d_1}{3}. \end{aligned}$$

Thus manufacturers 1 and 2’s profit could be written as

$$\begin{aligned} \pi _1=\frac{9t}{8}-\frac{d_2-d_1}{3}+\frac{2(d_1-d_2)^2}{81t}-I_1, \qquad \pi _2=\frac{9t}{8}+\frac{d_2-d_1}{3}+\frac{2(d_1-d_2)^2}{81t}-I_2. \end{aligned}$$

The first-order conditions in the R&D stage are

$$\begin{aligned} \frac{\partial \pi _1}{\partial d_1}=\frac{1}{3}-\frac{4(d_2-d_1)}{81t}-I_1', \qquad \frac{\partial \pi _2}{\partial d_2}=\frac{1}{3}+\frac{4(d_2-d_1)}{81t}-I_2'. \end{aligned}$$

The second-order conditions are satisfied when $I''$ is sufficiently large ($\frac{4}{81t}-I''<0$).

At the symmetric equilibrium we have

$$\begin{aligned} d_1=d_2=d^{vu}\quad such \ that \ \quad \frac{1}{3}-I'(d^{vu})=0. \end{aligned}$$

$\square$