Welfare Improvement and the Extension of the Income Gap under Monopoly ()

Yuhki Hosoya^{1}, Soh Kaneko^{2}

^{1}Department of Economics, Kanto-Gakuin University, Yokohama, Japan.

^{2}Faculty of Economics, Keio University, Tokyo, Japan.

**DOI: **10.4236/tel.2015.54069
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This study constructs a model of a monopoly where investors are also actors, and shows that, in contrast to traditional models, this model admits the welfare improvement caused by monopoly. This study also reveals that if a huge income gap exists in the initial stage, then monopoly exacerbates the expansion of the income gap caused by market trades. Moreover, we show that this exacerbation occurs in general situations under some additional (but natural) assumptions.

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Hosoya, Y. and Kaneko, S. (2015) Welfare Improvement and the Extension of the Income Gap under Monopoly. *Theoretical Economics Letters*, **5**, 590-597. doi: 10.4236/tel.2015.54069.

1. Introduction

Economics traditionally considers a monopoly to be bad for an economy. The most famous research indicating that monopolies are bad is the classical partial equilibrium analysis performed by Hicks [1] . This research indicates that a monopoly lowers the total surplus, and thus, the economy with a monopoly is not Pareto efficient. The result of this research is summerized in most of the textbook in microeconomics, e.g. Varian (1992), Okuno (2008) or Mas-Colell, Whinston, and Green (1995) [2] -[4] .

This research focuses on monopoly from a fresh perspective. The traditional monopoly model includes two characters: the monopolistic firm and the consumer. However, a real monopolistic situation necessarily involves a third character, namely, the investor. Under capitalism, investors are also consumers. Therefore, in our model, consumers invest in the monopolistic firm, which distributes its profit into its investors.

We formalize the above circumstance in a model, and analyze its model. We find that the total surplus of an economy may improve under a monopoly, which contradicts the traditional rationale for monopolies being bad. Meanwhile, in such a case the income gap often is expanded by market trade. If the initial income gap is sufficiently large, then a monopoly exacerbates this expansion of the income gap. The reason for this is as follows. Consider there are two consumers, where one is poor and another is rich. Both consumers invest in a firm that sells their own products and transfers its margin to investors in the form of dividends. However, the poor consumer has only limited ability to invest, and thus receives only a small share of the margin on product sales. The bulk of the margin is expropriated by the rich consumer. In this scenario, monopoly exacerbates this expansion of the income gap by enlarging firm’s profit.

This is the case in which the initial income gap is very high. In the case where the initial income gap is not so high, under certain assumptions monopoly also exacerbate the expansion of the income gap. Although these assumptions are not clear in the theoretical sense, we believe that these assumptions are intuitively natural.

In Section 2, we introduce our model and show the results. Section 3 is the conclusion.

2. The Model

We construct two models, named model 1 and model 2, to compare the competitive case with the monopolistic case. Model 1 corresponds with the competitive case, while model 2 corresponds with the monopolistic case. Both models consist of two consumers and one firm. Both consumers have a utility function, where

denotes private consumption and denotes the amount of money. We assume that , and. In the beginning of the model, consumer has units of money and one unit of labor. Without loss of generality, we assume. In the first-stage of the model, consumer determines the amount of investment at same time. Then the stock ratio is defined as and the capital of the firm is defined as. The product function of the firm is denoted as. We assume that is homogeneous of degree one, for all, and and for all.

The second-stage is different from each model. In model 1, each consumer and firm participates in the competitive market and the equilibrium arises. In model 2, the firm determines the price of consumption monopolistically and the wage is determined competitively^{1}.

2.1. The First Model

First, we solve the second-stage. The first-order condition of consumer is,

Hence,

and thus, in equilibrium. Meanwhile, the equilibrium condition of this market is

and,

Hence, the equilibrium price is

Next, the first-order condition of the firm is,

Thus, the equilibrium wage is

Then, the profit of firm is^{2}

where the subscript 1 represents that it is the profit of the first model. Hence, is positive, and the average profit is decreasing.

Therefore, the payoff function of this model is^{3},

In the first-stage, consumer chooses simultaneously and the Nash equilibrium arises.

Define

and as the unique solution of. Then,

(1)

Hence, and thus if and only if Note that K* is the social optimal level of capital, since for any and thus

We show the following proposition:

Proposition 1: There exists a Nash equilibrium. If then is the unique Nash equilibrium. If not, then for any Nash equilibrium, and, and thus,.

Proof: We first suppose. We can easily verify that is not a Nash equilibrium. Note that is always negative since is decreasing. By Equation (1), 1) if and, then and, which implies that is not a Nash equilibrium; 2) if$ and, then and, which implies that is not a Nash equilibrium; 3) if and, then and, which implies that is not a Nash equilibrium. Hence, there is no Nash equilibrium other than.

To show that is in fact a Nash equilibrium, consider the function. By Equation (1), if and if. Thus, is the best response to. Likewise, we can show that is the best response to. This completes the proof of this case.

Next, suppose. It can easily be verified that there is no Nash equilibrium such that. Next, since the function is continuous on, there uniquely exists which attains maximum. If, then and thus, a contradiction. Hence,. Also, if, then and, a contradiction. Hence,. Therefore, and thus $, which implies is a Nash equilibrium. This completes the proof.

2.2. The Second Model

The demand function of consumer on private consumption is simply

Hence, the total demand is. Thus, to sell, the firm must choose. Then, the profit function is

Now, we introduce an assumption.

ASSUMPTION 1: For any, there exists such that is a maximum point of.

By first-order condition, we have

Recall that is the unique value such that. Thus, in equilibrium, the profit of the firm is

Then, the payoff function of this model is

We want to focus on the case where the equilibrium of the first stage is well-defined. Therefore, we introduce an additional assumption:

ASSUMPTION 2: is decreasing in.

Here, we provide a sufficient condition of ASSUMPTION 2 to show this assumption is not too strong.

Proposition 2: Suppose that ASSUMPTION 1 holds. Then, ASSUMPTION 2 holds if is decreasing in.

Proof: By ASSUMPTION 1 and the second-order necessary condition, we have

Meanwhile, since is decreasing, we have

By homogeneity of degree one on,

Further, both and are homogeneous of degree zero^{4}. Therefore,

and thus,

Hence,

and thus, ASSUMPTION 2 holds. This completes the proof.

It can be easily verified that is decreasing for any u that has constant or decreasing relative risk aversion. Hence, ASSUMPTION 2 is not too strong^{5}.

Define

and as the unique solution of. If such does not exist, then let Then, and thus if and only if. Note that is well-defined under ASSUMPTION 2. Since for all, we have.

We will show the following proposition:

Proposition 3: Under ASSUMPTIONS 1-2, there exists a Nash equilibrium. If, then is the unique Nash equilibrium. If not, then for any Nash equilibrium, and.

Proof: It can be verified in the same way as Proposition 1.

2.3. Example: Improvement of Total Welfare

Suppose and. By easy calculation, we have in model 1,

and thus,

Therefore, we have

In model 2, we have

and thus,

Therefore,

This example demonstrates that the existence of the case where monopoly improves the total surplus.

2.4. Comparative Statics

First, we argue the following result.

Proposition 4: Suppose that is sufficiently low. Define

.

Then, we have for any Nash equilibria of model 1 and of model 2 with.

Proof: It suffices to show that our claim holds if, because this model is continuous on parameter. Thus, we assume. By calculation in subsection 2.2, we have, and thus for any. Hence, we can easily verify that, and thus

which completes the proof.

Later, 2) of Proposition 5 says that under ASSUMPTION 3, the restriction of proposition 4 is removed. Here we introduce additional assumptions.

ASSUMPTION 3: is increasing in.

ASSUMPTION 4: is increasing in.

Remark: ASSUMPTIONS 3-4 are not clear in the theoretical view. However, we think both conditions are natural in the real world. Usually, the bigger the capital obtained, the richer the firm becomes. Also, if the monopolistic power of the firm becomes strong, then we can expect wages to decrease. Note that by definition, is always positive.

Proposition 5: Suppose ASSUMPTIONS 1-2 hold, and choose any Nash equilibria of model 1 and of model 2. Then,

1) if;

2) Under ASSUMPTION 3, if, and if;

3) Under ASSUMPTIONS 3-4, and. Further, if and.

Proof: If, then and 1) holds.

Suppose ASSUMPTION 3 holds. By easy calculation,

and thus,

Recall that. By ASSUMPTION 2, is a decreasing function, and thus for any. Thus, we have

Since and is decreasing, we have, and thus,

Note that if and if. Hence, for any such that and,.

If, then. Hence, and if, then the inequality is strict. If not, then, and thus since for any. Thus, (2) holds.

Lastly, suppose ASSUMPTIONS 3-4 hold. If, then (1) and (2) imply (3). Otherwise, and. Now,

whenever. Hence, for any,

and thus,. If, then

and thus,. Thus, 3) holds.

3. Conclusion

We constructed a model of a monopoly with investors, and showed that monopoly did not necessarily decrease total welfare. Meanwhile, under mild assumptions monopoly exacerbated the expansion of the income gap. Therefore, we revealed a new aspect of the negative influence of monopoly.

Acknowledgements

We are grateful for Eisei Ohtaki for his comments and suggestions.

Notes

^{1}In the second-stage, we assume that the consumption space of each consumer is . This assumption is made for tha sake of simplicity and is not essential. We note that this setup is introduced in the explanation of the quasi-linear preference in Mas-Colell, Whinston and Green (1995).

^{2}Use the Euler equation

^{3}If, then no production arises and. But we can easily verify that such situation is not a Nash equilibrium, since is positive.

^{4}For example, to differentiate with respect to, we have and thus.

^{5}Actually, we think that there may exist a weaker condition than ASSUMPTION 2 ensuring the following Propositions. However, since ASSUMPTION 2 itself is not too strong, we satiate this assumption, at least in this paper.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | Hicks, J.R. (1938) Value and Capital. Oxford University Press, Oxford. |

[2] | Mas-Colell, A., Whinston, M.D. and Green, J.R. (1995) Microeconomic Theory. Oxford University Press, Oxford. |

[3] | Okuno, M. (2008) Microeconomics. Tokyo University Press, Tokyo. |

[4] | Varian, H.R. (1992) Microeconomic Analysis. W. W. Norton and Company, New York. |

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