Price competition between subsidized organizations

Abstract

Many firms and organizations compete for customers while at the same time receiving substantial funding from outside sources, such as government subsidies. In this paper, we study the effects of two commonly observed subsidy systems on the strategic behavior of competing firms. We compare a per unit subsidy to a subsidy allocated according to the firms’ market shares. We show that, holding the total subsidy budget constant, the per unit subsidy results in lower prices, higher output, lower profits and higher overall welfare as compared to the market-share based alternative. However, we also find that a market-share based subsidy makes collusive behavior between firms much harder. Our results suggest a potential trade-off between short-run and long-run objectives: subsidy systems designed to widen participation may favor collusive behavior. The welfare implications of this trade-off are discussed. Our findings have important policy implications for the design of subsidy systems.

Appendices

Appendix 1

In this appendix we consider, for completeness sake, a move from a market-share based to a per unit subsidy system in which the user price (and hence demand) is kept constant. As we focus on symmetric solutions, we delete firm-specific subscripts. Denote the subsidy, the price and demand after the move to the per unit system as $\bar{\gamma },\bar{p},\bar{Q}$, respectively. We then require:

$$\begin{aligned} \overline{p} = p^{*} ->\;\frac{1-\overline{\gamma }}{2-d}=\frac{( {3-2d})-\sqrt{1+( {1+d})Z} }{2( {2-d})( {1-d})}. \end{aligned}$$

(32)

Solving leads to

$$\begin{aligned} \bar{\gamma }=\frac{-1+\sqrt{1+(1+d)Z} }{2(1-d)}. \end{aligned}$$

(33)

This unit subsidy implies the same price and the same demand ($\bar{p}=p^{*} ,\bar{Q}=Q^{*}$, respectively) that we had under the market-share based system. However, it implies a lower budgetary cost to the government. To see this, note that the total cost is$2\bar{\gamma }\bar{Q}$. Using (33) and (9) we find:

$$\begin{aligned} 2\bar{\gamma }\bar{Q}=\frac{(1+d)Z}{2(2-d)(1-d)}. \end{aligned}$$

Using the definition of $Z$, we have

$$\begin{aligned} 2\bar{\gamma }\bar{Q}=\frac{\beta (1+d)}{2}. \end{aligned}$$

Since $d<1$ we have that $2\bar{\gamma }\bar{Q}<\beta$.

Hence, the unit subsidy is more efficient in the sense of Anderson et al. (2001a, b). This is just a collorary of the result we showed above (higher output and hence consumer surplus for given subsidy cost). The intuition is clear. The unit subsidy is more efficient in stimulating demand, so that a smaller subsidy is needed to generate the same demand effect as the subsidy according to market share. Note that, using (17) and (33), we also see that $\hat{\gamma }>\bar{\gamma }$.

Appendix 2

In this appendix, we show that the market-share based system necessarily yields higher profits for firms for all positive subsidies and $0\le d<1$. To see this, differentiate (23) with respect to $\beta$ to find:

$$\begin{aligned} \frac{\partial (\hat{\pi }_{i}^{{o^{*}}} -\pi _{i}^{{c^{*}}} )}{\partial \beta }=\left[ {\frac{1-d}{4(2-d)}} \right]\left[ {\frac{2}{\sqrt{1+2\beta Z} }-\frac{(1+d)}{\sqrt{1+\beta (1+d)Z} }-1} \right]. \end{aligned}$$

The sign of the final term between brackets on the right-hand side is the same as the sign of

$$\begin{aligned} \left[ {2\sqrt{1+\beta (1+d)Z} -(1+d)\sqrt{1+2\beta Z} -(\sqrt{1+\beta (1+d)Z} )(\sqrt{1+2\beta Z} )} \right]. \end{aligned}$$

Reformulation gives

$$\begin{aligned}&\left[ {\sqrt{1+\beta (1+d)Z} -\sqrt{1+2\beta Z} } \right]+\left[ {\sqrt{1+\beta (1+d)Z} ( {1-\sqrt{1+2\beta Z} })} \right]\\&\quad -\left[ {d\sqrt{1+2\beta Z} } \right]. \end{aligned}$$

This is necessarily negative, given $d<1$. Therefore, raising the subsidy necessarily reduces the right-hand side of (23). As the profit difference is zero at zero subsidies this implies that the market-share based subsidy system yields higher profits for firms for all positive subsidies and less than perfect substitution between goods.

Appendix 3

In the main body of the paper, we assumed that the organization maximized profit. In this appendix, we extend the firms’ objective functions. Specifically, we assume that firms not only care about profit, but they also attaches some importance to the surplus generated for consumers.

First reconsider the case of a per unit subsidy. Given a subsidy $\gamma$, we write the objective function as

$$\begin{aligned} (p_i +\gamma )q_i +\lambda \left[ {\int \limits _0^{q_i } {p_i (q_i ,q_j )dq_i -p_i q_i } } \right]. \end{aligned}$$

The first term is profit, the second term is net consumer surplus weighted by a factor $\lambda$. The larger $\lambda$, the more important is net consumer surplus relative to profit.

Using demand specification (2), it easily follows that the optimization problem facing firm $i$ can be rewritten as

$$\begin{aligned} \mathop {\max }\nolimits _{p_i } \quad (p_i +\gamma )q_i +0.5\lambda (q_i )^2. \end{aligned}$$

Following the same steps as in Sect. 3.1, and assuming both firms serve a positive share of the market, the reaction function for firm $i$ can be shown to read

$$\begin{aligned} p_i^o =\frac{(1-d^2)( {1-d-\gamma +dp_j })-\lambda ( {1-d+dp_j })}{2(1-d^2)-\lambda }. \end{aligned}$$

Solving the two reaction functions and assuming symmetry yields the following Nash equilibrium prices

$$\begin{aligned} p_{i}^{{o^{*}}} =p_{j}^{{o^{*}}} =\frac{(1+d)(1-d-\gamma )-\lambda }{(2-d)(1+d)-\lambda }. \end{aligned}$$

(34)

For $\lambda =0$, this result just reproduces (8). Moreover, simple differentiation shows that Nash equilibrium prices are declining in $\lambda$. For future reference we note that, using (2) in the main body of the paper, Nash equilibrium quantities are given by

$$\begin{aligned} q_{i}^{{o^{*}}} =q_{j}^{{o^{*}}} =\frac{1+\gamma }{Z-\lambda } \end{aligned}$$

(35)

where, as always, $Z=(2-d)(1+d)$.

Next take the case of market-share based funding. Given a subsidy budget $\beta$, the objective function for the firm can be written as

$$\begin{aligned} \mathop {\max }\limits _{p_i } \quad p_i q_i +\beta \frac{q_i }{q_i +q_j }+0.5\lambda (q_i )^2. \end{aligned}$$

Following the same steps as in Sect. 3.2, and assuming a symmetric Nash equilibrium, straightforward algebra produces the extended version of (14); the Nash equilibrium prices are given by

$$\begin{aligned} p_{i}^{{c^{*}}} =\frac{(3-2d)-\lambda -\sqrt{\lambda (2+\lambda )+1+\beta (1+d)Z} }{2(2-d)}. \end{aligned}$$

(36)

This reduces to (14) when the weight of consumer surplus is zero; moreover, Nash equilibrium prices are again declining in $\lambda$.

Reconsider, then, the comparison of the two subsidy systems. Suppose the total available subsidy budget is constant and given by $\beta$. The Nash equilibrium prices are then given by (34). How do these prices compare to those under a system of per unit subsidies, holding the total cost of the subsidies constant at $\beta$? To find out, we first derive the Nash equilibrium prices that will be observed under a per unit subsidy system with a total cost of $\beta$. Denote the per unit subsidy that generates the same total subsidy cost $\beta$ in the per unit system by $\hat{\gamma }$; in other words, $\beta =2\hat{\gamma }\hat{q}_i^o$, where $\hat{q}_i^o$ is the quantity demanded in the per unit system if the subsidy is $\hat{\gamma }$. Using (35), it immediately follows that

$$\begin{aligned} \beta =2\hat{\gamma }\left( {\frac{1+\hat{\gamma }}{Z-\lambda }}\right). \end{aligned}$$

Solving this expression for $\hat{\gamma }$ leads to

$$\begin{aligned} \hat{\gamma }=\frac{-1+\sqrt{1+2\beta (Z-\lambda )} }{2}. \end{aligned}$$

Observe that the per unit subsidy is a declining function of the weight $\lambda$ associated with net consumer surplus. Finally, using (34) we find that this subsidy $\hat{\gamma }$ gives Nash equilibrium prices

$$\begin{aligned} \hat{p}_i^{o*} =\frac{(1+d)\left[ {3-2d-\sqrt{1+2\beta (Z-\lambda )} } \right]-2\lambda }{2(Z-\lambda )}. \end{aligned}$$

As should be the case, this boils down to (18) when $\lambda =0$.

We want to compare the prices $\hat{p}_{i}^{{o^{*}}}$ and $p_{i}^{{c^{*}}}$. We have

$$\begin{aligned} \hat{p}_{i}^{{o^{*}}} -p_{i}^{{c^{*}}}&= \left[ {\frac{(1+d)\left[ {3-2d-\sqrt{1+2\beta (Z-\lambda )} } \right]-2\lambda }{2(Z-\lambda )}} \right] \\&-\left[ {\frac{(3-2d)-\lambda -\sqrt{\lambda (2+\lambda )+1+\beta (1+d)Z} }{2(2-d)}} \right]. \\ \end{aligned}$$

This can be reformulated after simple algebra as

$$\begin{aligned}&\hat{p}_{i}^{{o^{*}}} -p_{i}^{{c^{*}}} =\left[ {\frac{1}{2(Z-\lambda )(2-d)}} \right] \\&\quad *\left\{ {(Z-\lambda )\left[ {\lambda +\sqrt{\lambda (2+\lambda )+1+\beta (1+d)Z} } \right]-\left[ {\lambda +Z\sqrt{1+2\beta (Z-\lambda )} } \right]} \right\} . \end{aligned}$$

Of course, this expression boils down to (21) if the weight given to consumer surplus is zero, in which case it was shown to be negative. However, numerical simulations suggested that, depending on the parameter values, it can be negative or positive. It is easy to show that the price difference is increasing in $\lambda$ and in $d$, and it is declining in $\beta$. In other words, for given substitutability and a given subsidy budget, the price difference will become positive at sufficiently high values of the weight the firm attaches to consumer surplus. As a consequence, we found the welfare difference between the per unit and the market-share based systems to be increasing in $\beta$, and decreasing in both $d$ and in the weight of consumer surplus in the firm’s objective function. This implies that the market-share based system will perform better than the per unit subsidy if the weight of consumer surplus is sufficiently large.

Appendix 4

Can anything be said about the optimal second-best subsidy for each of the two subsidy systems? To study this question, we return to the case of profit-maximizing firms for simplicity. In determining the optimal subsidy, we assume the government maximizes a standard welfare function, to be described below.

First, consider a per unit subsidy system. Write the government’s objective function as

$$\begin{aligned} (p+\gamma )q+\left[ {\int \limits _0^q {p(q)dq-pq} } \right]-(1+\mu )\gamma q. \end{aligned}$$

The first term is profit of the firm, the second term is surplus for the consumer, the third term is the cost of the subsidy to the government. A weight ($1+\mu )$ larger than one ($\mu \ge 0)$ is associated with the cost of the subsidy, reflecting the absence of lump sum taxation. It can be interpreted as the cost of funds. Assuming symmetry, we have deleted firm subscripts; everything is expressed on a per firm basis.

The government is interested in maximizing welfare with respect to the subsidy per unit $\gamma$. Using the linear demand functions (2), straightforward algebra shows that the optimal subsidy satisfies

$$\begin{aligned} \gamma =\frac{1-d-\mu (2-d)}{1+2\mu (2-d)}. \end{aligned}$$

As expected, it follows by differentiation of this expression that an increase in the cost of funds $\mu$ reduces the optimal subsidy. Similarly, more substitutability (a higher $d)$ reduces the optimal subsidy. Note that, if the cost of funds equals one ($\mu =0)$ the optimal subsidy $\gamma =1-d$ will push down Nash equilibrium prices to zero, see (8). This is the first-best outcome: as our model assumed zero marginal production cost for the firm and there are no externalities, the optimal price is zero.

Optimal prices and quantities at the optimal subsidy are given by, respectively

$$\begin{aligned} \begin{aligned}&p=\frac{\mu (3-2d)}{1+2\mu (2-d)} \\&q=\frac{1+\mu }{(1+d)\left[ {1+2\mu (2-d)} \right]}. \\ \end{aligned} \end{aligned}$$

Using these results, the total subsidy cost is

$$\begin{aligned} \gamma q=\left( {\frac{1+\mu }{1+d}}\right)\frac{\left[ {1-d-\mu (2-d)} \right]}{\left[ {1+2\mu (2-d)} \right]^2}. \end{aligned}$$

Second, we turn to the optimal market-share based subsidy. The government maximizes

$$\begin{aligned} (pq+\beta )+\left[ {\int \limits _0^q {p(q)dq-pq} } \right]-(1+\mu )\beta . \end{aligned}$$

with respect to $\beta$. Manipulating the first-order condition yields, after simple but substantial algebra, the optimal subsidy as

$$\begin{aligned} \beta =\left[ {\frac{\left[ {(1+d)^2(3-2d)} \right]^2-\left[ {8Z\mu +(1+d)^2} \right]^2}{\left[ {8Z\mu +(1+d)^2} \right]^2(1+d)Z}} \right]. \end{aligned}$$

Again, differentiation shows that the optimal subsidy is a downward sloping function of the cost of funds.

In principle, prices, quantities and the various welfare components (profit, consumer surplus, etc.) can be determined for both subsidy systems. This would allow a welfare comparison of the subsidy systems, assuming the government sets the subsidy levels at their second-best optimal values. However, although calculating welfare is not difficult for the per unit subsidy, doing the same for the market-share based system turned out to be highly complicated.²⁶ Moreover, both the optimal subsidies and optimal welfare are highly nonlinear functions of the shadow cost of funds and other model parameters. A formal comparison of the relative welfare performance of the two systems at their respective second-best optimum is therefore outside the scope of this paper.