Question

This problem has you work through some of the calculations associated with the numerical example in the Extensions. Refer to the Extensions for a discussion of the theory in the case of Fisher Body and General Motors (GM), who we imagine are deciding between remaining as separate firms or having GM acquire Fisher Body and thus become one (larger) firm. Let the total surplus that the units generate together be \(S\left(x_{P}, x_{G}\right)=x_{F}^{1 / 2}+a x_{G}^{1 / 2},\) where \(x_{F}\) and \(x_{G}\) are the investments undertaken by the managers of the two units before negotiating, and where a unit of investment costs \(\$ 1 .\) The parameter \(a\) measures the importance of GM's manager's investment. Show that, according to the property rights model worked out in the Extensions, it is efficient for GM to acquire Fisher Body if and only if GM's manager's investment is important enough, in particular, if \(a>\sqrt{3}\)

Short answer

Answer: It is efficient for GM to acquire Fisher Body if and only if GM's manager's investment is important enough, specifically, if \(a > \sqrt{3}\).

Step-by-step solution

Surplus function when separate

When GM and Fisher Body are separate firms, the total surplus is given by: \(S(x_F, x_G) = x_F^{1/2} + ax_G^{1/2}\), where \(x_F\) and \(x_G\) are the investments of Fisher Body and General Motors, respectively.

Find the optimal investments when separate

To find the optimal investments for both firms, set the marginal costs equal to the marginal returns: \(x_F: \frac{1}{2x_F^{1/2}} = 1 \Rightarrow x_F^* = \frac{1}{4}\) \(x_G: \frac{a}{2x_G^{1/2}} = 1 \Rightarrow x_G^* = \frac{a^2}{4}\)

Surplus when GM acquires Fisher Body

When GM acquires Fisher Body, it becomes one larger firm. The optimal investment should maximize the total surplus \(S(x_F, x_G)\) while considering the investment cost. The constraint function is given by \(x_F+x_G=C\), where C is the total investment cost. The manager will consider the constraint when choosing the optimal investments.

Find the optimal investments when integrated

The new surplus function will be: \(S(x_F, x_G)= x_F^{1/2} + a(x_G+x_F-x_F)^{1/2}\). To find the optimal investments when integrated, set \(\frac{\partial S(x_F,x_G)}{\partial x_F}=\frac{\partial S(x_F,x_G)}{\partial x_G}.\) Solving the system of equations, we get: \(x_F^* = \frac{1}{3}\) \(x_G^* = \frac{a^2}{3} - \frac{1}{3}\)

Compare surplus for separate firms and integrated firms

Calculate the surplus for both situations: 1. Separate firms: \(S(x_F^*, x_G^*) = \frac{1}{2} + \frac{a^2}{4}\) 2. Integrated firms (acquired Fisher Body): \(S(x_F^*, x_G^*) = \frac{1}{\sqrt{3}} + a\left(\frac{a^2}{3} - \frac{1}{3}\right)^{1/2}\)

Determine the condition for integration

Efficiency implies that the surplus in the integrated case (GM acquires Fisher Body) must be greater than or equal to the surplus in the separate case. \(\frac{1}{\sqrt{3}} + a\left(\frac{a^2}{3} - \frac{1}{3}\right)^{1/2} \geq \frac{1}{2} + \frac{a^2}{4}\) After simplifying and solving for \(a\), we obtain: \(a > \sqrt{3}\) So, according to the property rights model, it is efficient for GM to acquire Fisher Body if and only if GM's manager's investment is important enough, specifically, if \(a > \sqrt{3}\).