Power at general equilibrium

Abstract

We integrate individual power in groups into general equilibrium models with endogenous group formation. We distinguish between real power (utility gain from being in groups) and power-related parameters in group decision making, like utilitarian welfare weights in the case of welfare maximization or relative bargaining power in Nash bargaining. We find that higher “parametric power” does not necessarily translate into higher equilibrium utility or higher real power. One reason is that induced price changes may offset the group member’s increased influence and the resulting benefits. A second reason is that the group may dissolve when a group member gains too much influence, because other members can exercise the option to leave. We also show that maximal real power can be compatible with Pareto efficiency. We further identify circumstances when changes of power in one group do not impact on other groups. Finally, we establish existence of competitive equilibria, including equilibria where some individual enjoys real power.

Appendix

Proof of Proposition 3

Recall that for $i\in I$ and $P\in \mathcal{P}$, P(i) denotes the group to which i belongs in the group structure P.

First, we show that $(\mathbf {x} ; P^*)$ is a Pareto optimal allocation. For suppose not. Then there exists a feasible allocation $(\mathbf {y} ; P)$ that Pareto dominates $(\mathbf {x} ; P^*)$: $U_i(y_i; P(i))> U_i(x_i; P^*(i))$ for some $i\in I$ and $U_i(y_i; P(i)) \ge U_i(x_i; P^*(i))$ for all $i\in I$. Since $(p, \mathbf {x})$ is a competitive equilibrium of the pure exchange economy $(U_i^c, \omega _{\{i\}})_{i\in I}$ and consumers are locally non-satiated, $\mathbf {x}$ is a Pareto optimal allocation of the pure exchange economy. Therefore, if it is the case that $U_i^c(y_i) > U_i^c(x_i)$ for some i, then there exists $j\ne i$ with $U_j^c(y_j) < U_j^c(x_j)$ and, consequently, $U^g_j(P(j))> U_j^g(P^*(j))$ (because $(\mathbf {y} ; P)$ Pareto dominates $(\mathbf {x} ; P^*)$). If it is the case that $U_i^c(y_i) \le U_i^c(x_i)$ for all i, then $U^g_j(P(j))> U_j^g(P^*(j))$ for some j. In any case, $U^g_j(P(j))> U_j^g(P^*(j))$ for some j. But then, by Fact 1 (i) there exists an optimal group structure $P'$ based solely on group preferences such that $U_j^g(P'(j))\ge U^g_j(P(j))> U_j^g(P^*(j))$ and, consequently, $P'\ne P^*$, contradicting (iii).

Second, we show that $(p, \mathbf {x}; P^*)$ is a CEFE. Because of pure group externalities, (i) and (ii), the first two conditions for a CEFE hold. Moreover for $i\in I$:

• (a) Since $(p, \mathbf {x})$ is a competitive equilibrium of the pure exchange economy $(U_i^c, \omega _{\{i\}})_{i\in I}$, $x_i$ is an optimal consumption bundle in i’s budget set.

• (b) Since $P^*$ is the unique optimal group structure solely based on group preferences, $U_i^g(\{i\})\le U_i^g(P^*(i))$, by Fact 1 (i).

Hence i cannot fare better as a one-person group. Thus the third condition for a CEFE holds as well.

Third, we show that there does not exist another CEFE in which real power is higher for some individuals in their respective groups and not lower for any individual. Namely, suppose that there exists a CEFE $(p', \mathbf {y}; P')$ in which real power is higher for some individuals and not lower for any individual than in $(p, \mathbf {x}; P^{*})$. Hence,

$$\begin{aligned} U_i^c(y_i) + U_i^g\left( P'(i)\right) - V_i^0(p') \ge U_i^c(x_i) + U_i^g\left( P^{*}(i)\right) - V_i^0(p) \end{aligned}$$

for all i, with strict inequality for some i. As $P^*$ is the unique optimal group structure based solely on group preferences, we obtain $U^g_i(P'(i))\le U^g_i(P^*(i))$ for all i by Fact 1 (i). Moreover, $U_i^c(x_i) = V_i^0(p)$ holds because of (i). Therefore,

$$\begin{aligned} U_i^c(y_i) - V_i^0(p') \ge U_i^c(x_i) - V_i^0(p) = 0 \end{aligned}$$

for all individuals i, with strict inequality for some individual. Now pick $j\in I$ with $U_j^c(y_j)> V_j^0(p')$. If for some $i\in P'(j)$, $p'y_i< p'\omega _{\{i\}}$ , then local non-satiation implies $V_i^0(p') > U_i^c(y_i)$, contradicting $U_i^c(y_i)\ge V_i^0(p')$. Therefore, $p'y_i\ge p'\omega _{\{i\}}$ for all $i\in P'(j)$. Since $\mathbf {y\!}_{_{P'\!(j)}}\in {EB\!}_{_{P'(j)}}(p')$, this implies $p'y_i = p'\omega _{\{i\}}$ and, consequently, $U_i^c(y_i)\le V_i^0(p')$ for all $i\in P'(j)$. Thus a contradiction to $U_j^c(y_j)> V_j^0(p')$ results. Hence, to the contrary, it cannot be the case that in the CEFE $(p', \mathbf {y}; P')$, real power is higher for some individuals and not lower for any individual than in $(p, \mathbf {x}; P^{*})$. $\square$

Proof of Fact 3

step 1: We first determine equilibrium quantities and examine the non-exit conditions in case $P=\{I\}$. Given the price system $p=(1,p_2)$, group $\{1,2,3\}$ solves

$$\begin{aligned}&\max _{\left( x_1^1,x_2^2, x_3^1, x_3^2\right) } \left[ \alpha _1 \left( \ln x_1^1 + \ln v_1\right) + \alpha _2 \left( \ln x_2^2 + \ln v_2\right) \right. \\&\quad \left. + \frac{1}{2}(1-\alpha _1-\alpha _2)\left( \ln x_3^1 + \ln x_3^2 + 2\ln v_3\right) \right] \end{aligned}$$

subject to $x_1^1+x_3^1+p_2x_2^2+p_2x_3^2=1+p_2$. This yields

$$\begin{aligned} \begin{array}{l} x_1^1=\alpha _1\cdot (1+p_2), x_2^2=\alpha _2 \cdot (1+p_2)/p_2,\\ x_3^1=(1-\alpha _1-\alpha _2)\cdot (1+p_2)/2, x_3^2=(1-\alpha _1-\alpha _2) \cdot (1+p_2)/(2p_2). \end{array} \end{aligned}$$

The stand-alone demands are:

$$\begin{aligned} x_1^{10}=\dfrac{1}{2}p_2, x_2^{20}=\dfrac{1}{2}, x_3^{10}=\dfrac{1}{2}, x_3^{20}=\dfrac{1}{2p_2}. \end{aligned}$$

The market clearing price is given by $1+p_2^*=\dfrac{2}{1+\alpha _1-\alpha _2}$.

Then the non-exit condition for the first individual is:

$$\begin{aligned} \ln \left( \alpha _1\left( p_2^*+1\right) \right) + \ln v_1 \ge \ln \left( \frac{1}{2}p_2^* \right) , \end{aligned}$$

which is equivalent to $\alpha _1 \ge \frac{p_2^*}{2(p_2^*+1)v_1}$ and $\alpha _1 \ge \frac{1+\alpha _2}{4v_1+ 1}$.

Similarly, for the second individual we obtain

$$\begin{aligned} \ln \left( \alpha _2 \frac{p_2^*+1}{p_2^*} \right) + \ln v_2 \ge \ln \frac{1}{2} \end{aligned}$$

or $\alpha _2 \ge \frac{p_2^*}{(p_2^*+1)2v_2}$ which is equivalent to $\alpha _2 \ge \frac{1-\alpha _1}{4v_2-1}$.

Finally, the third individual’s non-exit condition amounts to

$$\begin{aligned}&\frac{1}{2} \left\{ \ln \frac{1}{2} + \ln \frac{1}{2p_2^*} \right\} \le \frac{1}{2} \ln \left( (1-\alpha _1-\alpha _2)\frac{p_2^*+1}{2} \right) \\&\quad + \frac{1}{2} \ln \left( (1-\alpha _1-\alpha _2)\frac{p_2^*+1}{2p_2^*} \right) +\ln v_3. \end{aligned}$$

It implies $(1-\alpha _1 - \alpha _2)(p_2^*+1) \ge 1/v_3$ or $1-\alpha _1-\alpha _2 \ge (1+\alpha _1-\alpha _2)/(2v_3)$ or $\alpha _1 \le \frac{2v_3-1}{2v_3+1}\cdot (1-\alpha _2)$.

step 2: We next examine whether real power of individual 1 can be equal or higher under the group structure $P = \{\{ 1,2 \}, \, \{3 \}\}$ than under $P^* = \{ I\}$. That is, we examine whether it is possible to delineate parameter values such that ${\rho _1} \ge \max \{\rho _1^*\}$, where ${\rho _1}=\ln \frac{1}{2} + \ln v_1 + \ln (4 \alpha )$ for $\alpha \in [\frac{1}{2},1-\frac{1}{2v_2}]$ (from Example 2) and $\rho _1^*$ is 1’s real power in the above equilibrium. When choosing the maximal $\alpha = 1-\frac{1}{2v_2}$, the inequality is equivalent to

$$\begin{aligned} \ln \left( \frac{4v_2-2}{2v_2} \right) \ge \max \Biggl \{ \ln \left( \frac{2\alpha _1(1+p_2^*)}{p_2^*} \right) \Biggr \} \end{aligned}$$

which implies

$$\begin{aligned} \frac{4v_2-2}{2v_2} \ge \max \Biggl \{ \frac{4\alpha _1}{1-\alpha _1+\alpha _2}\Biggr \}. \end{aligned}$$

Observe that with $\alpha _1=\alpha _2=1/4$, all three non-exit conditions are satisfied and individual 1 enjoys real power $\rho _1^*=\ln v_1>0$ in the corresponding CEFE. Therefore, in order to maximize $\rho _1^*$, it suffices to solve the following problem:

$$\begin{aligned} \max _{\alpha _1, \alpha _2 \in [0,1]} \left\{ \frac{4\alpha _1}{1-\alpha _1+\alpha _2} \right\} \end{aligned}$$

         s.t.

$$\begin{aligned} \alpha _2\ge & {} \frac{1-\alpha _1}{4v_2-1}, \\ \alpha _1\le & {} \frac{2v_3-1}{2v_3+1}\cdot (1-\alpha _2). \end{aligned}$$

It follows that the optimal solution for $\alpha _2$ is given by $\alpha _2=\frac{1-\alpha _1}{4v_2-1}$ since $\frac{\partial \rho _1^*}{\partial \alpha _2}<0, \frac{\partial \rho _1^*}{\partial \alpha _1}>0$ and the right-hand side of the last constraint is monotonically decreasing in $\alpha _2$. Hence our problem is reduced to

$$\begin{aligned} \max _{\alpha _1 \in [0,1]}\left\{ \frac{(4v_2-1)\alpha _1}{v_2(1-\alpha _1)} \right\} \end{aligned}$$

where the constraint amounts to

$$\begin{aligned} (2v_3+1)\alpha _1 \le (2v_3-1)\cdot \left( 1- \frac{1-\alpha _1}{4v_2-1} \right) \end{aligned}$$

which leads to

$$\begin{aligned} \alpha _1 \le \frac{(2v_2-1)(2v_3-1)}{2(v_2-v_3+2v_3v_2)}<1. \end{aligned}$$

Hence, maximization of $\rho _1^*$ is obtained by

$$\begin{aligned} \alpha _1 = \frac{(2v_2-1)(2v_3-1)}{2(v_2-v_3+2v_3v_2)}. \end{aligned}$$

Now based on the foregoing transformation, the inequality ${\rho _1} \ge \max \{\rho _1^*\}$ leads to

$$\begin{aligned} \frac{4v_2-2}{2v_2} \ge \frac{(4v_2-1)\alpha _1}{v_2(1-\alpha _1)} \end{aligned}$$

which implies $\alpha _1 \le \frac{2v_2-1}{6v_2-2}$. Hence, $\widehat{\rho _1}\ge \max \{\rho _1^*\}$ yields

$$\begin{aligned} \frac{(2v_2-1)(2v_3-1)}{2(v_2-v_3+2v_2v_3)} \le \frac{2v_2-1}{6v_2-2} \end{aligned}$$

which implies $4v_3v_2 -4v_2-v_3 \le -1$ or $4v_2(v_3-1)\le v_3-1$.

Since $v_3>1$, the latter implies $4v_2 \le 1$ which contradicts $v_2\ge 1$ and, therefore, $\widehat{\rho _1} < \max \{\rho _1^*\}$ has to hold.

Hence, there are no parameter constellations such that the maximal real power of individual 1 in a CEFE with $P=I$ is strictly smaller than $\widehat{\rho _1}$. $\square$

Notice that in case $v_3=1$, we are back to Example 2 and $\widehat{\rho _1} = \max \{\rho _1^*\}$. In fact, in case $v_3=1$, the foregoing proof shows that $\widehat{\rho _1} > \max \{\rho _1^*\}$ would yield $0=4v_2(v_3-1) < v_3-1 =0$ and, thus, $0<0$; therefore, $\widehat{\rho _1} \le \max \{\rho _1^*\}$ has to hold.

Proof of Proposition 5

Good $\ell$ serves as a numéraire so that the price system assumes the form $(p_1, \ldots , p_{\ell -1}, 1)$. We are focusing on interior solutions regarding all commodities, including the numéraire good.¹⁰ Let us consider then the first-order conditions for maximizing $\ln S_h$ in group h, subject to h’s budget constraint:

$$\begin{aligned} \begin{array}{rcl} \beta _h \displaystyle {\frac{1}{ U_{h1} - U_{h1}\left( x^0_{h1}(p);\{h1\}\right) } \frac{ \partial V_{h1} }{ \partial x^k_{h1} } } - \lambda _h p_k &{} = &{} 0, \; \; \; k=1, \ldots , \ell -1; \\ \beta _h \displaystyle { \frac{1}{ U_{h1} - U_{h1}\left( x^0_{h1}(p); \{h1\}\right) }} - \lambda _h &{} = &{} 0; \\ (1-\beta _h) \displaystyle { \frac{1}{ U_{h2} - U_{h2}\left( x^0_{h2}(p); \{h2\}\right) } \frac{ \partial V_{h2} }{ \partial x^k_{h2} }} - \lambda _h p_k &{} = &{} 0, \; \; \; k=1, \ldots , \ell -1; \\ (1-\beta _h) \displaystyle { \frac{1}{ U_{h2} - U_{h2}\left( x^0_{h2}(p); \{h2\}\right) }} - \lambda _h &{} = &{} 0. \end{array} \end{aligned}$$

Therefore:

$$\begin{aligned} \lambda _h= & {} \beta _h \frac{1}{ U_{h1} - U_{h1}\left( x^0_{h1}(p); \{h1\}\right) } = (1-\beta _h) \frac{1}{ U_{h2} - U_{h2}\left( x^0_{h2}(p); \{h2\}\right) }. \nonumber \\ \end{aligned}$$

(10)

$$\begin{aligned} \frac{ \partial V_{h1} }{ \partial x^k_{h1} }= & {} \frac{ \partial V_{h2} }{ \partial x^k_{h2} } = p_k, \; \; \; k=1, \ldots , \ell -1. \end{aligned}$$

(11)

Equation (11) implies that the demand of group h for commodities $k=1, \ldots , \ell -1$ is independent of the bargaining power $\beta _h$ and $1-\beta _h$ of individual h1 and h2, respectively. Since the $V_{hi}$ are strictly concave and strictly increasing, the budget of the particular group h is exhausted. It follows that h’s total demand for commodity $\ell$ is independent of $\beta _h$ as well. Therefore, aggregate demand and, thus, equilibria in commodity markets do not depend on internal bargaining power of groups. As a consequence, changes of bargaining power in group h have no effect on equilibrium prices. This establishes points (i) and (ii).

However, a shift of the power in groups affects the distribution of the numéraire good in group h. Using the notation for the equilibria we have from Eq. (10):

$$\begin{aligned} \frac{\beta _h}{ \widehat{V}_{h1} + \widehat{x}_{h1}^{\ell } + v_{1} - U_{h1}\left( x^0_{h1}(\widehat{p}); \{h1\}\right) } = \frac{1-\beta _h}{ \widehat{V}_{h2} + \widehat{x}_{h2}^{\ell } + v_{2} - U_{h2}\left( x^0_{h2}(\widehat{p}); \{h2\}\right) }\nonumber \\ \end{aligned}$$

(12)

Since $\widehat{V}_{h1}, v_{1}, U_{h1}(x^0_{h1}(\widehat{p}); \{h1\})$ and $\widehat{V}_{h2}, v_{2}, U_{h2}(x^0_{h2}(\widehat{p}); \{h2\})$ are independent of $\beta _h$ and $\widehat{x}^{\ell }_{h1} + \widehat{x}^{\ell }_{h2}$ does not depend on $\beta _h$ either, we obtain point (iii):

$$\begin{aligned} \frac{ \partial \widehat{x}^{\ell }_{h1} }{ \partial \beta _h } > 0, \; \; \frac{ \partial \widehat{x}^{\ell }_{h2} }{ \partial \beta _h } < 0 \end{aligned}$$

Furthermore, Eq. (12) is tantamount to (iv).

If groups are completely homogeneous with respect to $U_{hi}$ and $w_h$, a group equilibrium does not involve any positive net trades, again using the fact that differences in $\beta _h$ have no effect on aggregate excess demand. Therefore, $\widehat{x}^{\ell }_{h1} + \widehat{x}^{\ell }_{h2} = \omega ^{\ell }_{h}$ and via Eq. (12) we obtain (v). $\square$