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

Short answer

Answer: It is efficient for GM to acquire Fisher Body when the importance of GM's manager's investment (parameter \(a\)) is greater than \(\sqrt{3}\).

Step-by-step solution

Identify the total surplus function

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.

Find the investments maximizing the total surplus

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: \[\frac{\partial S}{\partial x_F} = \frac{1}{2} x_F^{-1/2} \Rightarrow x_F = \frac{1}{4}\] \[\frac{\partial S}{\partial x_G} = \frac{a}{2} x_G^{-1/2} \Rightarrow x_G = \frac{a^2}{4}\]

Calculate the efficiency condition

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 \(a > \sqrt{3}\).

Key concepts

Total Surplus Maximization

Understanding the concept of total surplus maximization is crucial for grasping the dynamics of business decisions, particularly when it comes to mergers and acquisitions, as illustrated by the GM and Fisher Body case. Total surplus represents the combined benefit to all parties involved in an economic transaction.

The ultimate goal is to maximize this total surplus, which can be thought of as the sum of consumer and producer surplus in a market context. In the context of investments by firms, maximizing total surplus means finding the levels of investment that yield the highest total return minus the costs of those investments.

In mathematical terms, this involves solving for the investment levels that maximize a surplus function. The exercise provided an example using a specific surplus function, where the optimal investments were calculated by setting the derivative of the surplus function with respect to each firm's investment equal to the cost per unit of investment, thus ensuring that the marginal benefit of investment equals its marginal cost, a condition for maximization in economics.

Investment Optimization

The concept of investment optimization plays a considerable role in achieving total surplus maximization. Optimization ensures that every dollar invested is yielding the highest possible return. Applying the principles of calculus, as seen in the textbook example, investments are optimized by finding the point where the additional cost of investing one more unit will equal the additional revenue generated by that investment.

In the given function, investments by Fisher Body and GM were represented as variables whose values were to be determined. By finding the derivatives of the total surplus function and equating them to the unit cost, we were able to calculate the investments that would produce the maximum surplus. When the parameter a, representing the significance of GM's investment, is large enough, the optimized investments dictate a scenario where an acquisition becomes more efficient than operating separately, highlighting the link between investment decisions and corporate strategy.

Fisher Body and General Motors Case Study

The Fisher Body and General Motors case study is a historical example of corporate integration that has become pivotal in understanding property rights and transaction cost economics. In the textbook exercise, we explored a simplified model that reflects the essence of the actual situation. We analyzed the efficiency of the acquisition of Fisher Body by GM, given a certain level of importance of GM's managerial investment.

This model breaks down the benefits and costs associated with each firm's investments, showing that it only makes sense for GM to acquire Fisher Body if the parameter a is greater than \(\sqrt{3}\), that is, if the manager's investment in GM has a significant impact on the total surplus.

This case highlights how property rights and the allocation of control rights can affect operational efficiency and long-term strategic decisions. In reality, when GM acquired Fisher Body, it was, in part, to internalize the benefits of investments and reduce transaction costs that arise from market exchanges, portraying intricacies of real-life decision-making that models like the one provided help to elucidate.