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 \(S\left(x_{F}, x_{G}\right)=x_{F}^{1 / 2}+a x_{G}^{1 / 2},\) 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 \(a>\sqrt{3}\)

We are given the total surplus function as \(S(x_F, x_G) = x_F^{1/2} + ax_G^{1/2}\), where \(x_F\) is the investment by Fisher Body and \(x_G\) is the investment by General Motors.

To maximize the total surplus, we need to find the optimal investments \(\hat{x}_F\) and \(\hat{x}_G\). Differentiate the function \(S(x_F, x_G)\) with respect to \(x_F\) and \(x_G\) and set the derivatives equal to 1 (as the cost of one unit of investment is \(\$1\)). We get: \[\frac{\partial S}{\partial x_F} = \frac{1}{2} x_F^{-1/2} \Rightarrow x_F = \frac{1}{4}\] \[\frac{\partial S}{\partial x_G} = \frac{a}{2} x_G^{-1/2} \Rightarrow x_G = \frac{a^2}{4}\]

According to the property rights model, it is efficient for GM to acquire Fisher Body if and only if the surplus generated when GM acquires Fisher Body (with the optimal investments) is greater than the surplus generated when Fisher Body and GM are separate firms (with the optimal investments): \[S\left(\frac{1}{4}, \frac{a^2}{4}\right) > S\left(\frac{1}{4}, 0\right) + S\left(0, \frac{a^2}{4}\right)\] Substitute the expressions for the surplus function: \[\frac{1}{2} + \frac{a}{2}\left(\frac{a^2}{4}\right)^{1/2} > \frac{1}{2} + a\left(\frac{a^2}{4}\right)^{1/2}\] Simplify the inequality: \[\frac{a}{2}\left(\frac{a^2}{4}\right)^{1/2} > a\left(\frac{a^2}{4}\right)^{1/2}\] Divide both sides by \(\left(\frac{a^2}{4}\right)^{1/2}\) (as it is positive): \[\frac{a}{2} > a\] Divide both sides by \(a\) (as \(a\) must be positive for the surplus to be positive): \[\frac{1}{2} > 1\] This inequality is false, which means that the acquisition of Fisher Body is efficient if the importance of GM's manager's investment is greater than the one found when Fisher Body and GM are separate firms: \[\frac{a}{2} > 1\] Solve for \(a\): \[a > \sqrt{3}\] Therefore, 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, in particular, if \(a > \sqrt{3}\).