Welfare Improvement and the Extension of the Income Gap under Monopoly

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.


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 suffi-ciently 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.

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 i c denotes private consumption and   denotes the amount of money.We assume that ( ) In the beginning of the model, consumer i has i e units of money and one unit of labor.Without loss of generality, we assume 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 p monopolistically and the wage w is determined competitively1 .

The First Model
First, we solve the second-stage.The first-order condition of consumer i is, ( ) , , , c c F K L + = and, 2. L = Hence, the equilibrium price is Next, the first-order condition of the firm is, ( ) , .

L pF K L w =
Thus, the equilibrium wage is where the subscript 1 represents that it is the profit of the first model.Hence, ( ) π is positive, and the average profit ( ) Therefore, the payoff function of this model simultaneously and the Nash equilibrium arises.

Define
( ) ( ) and * K as the unique solution of ( ) .
Hence, ( ) . a a K + = Note that K* is the social optimal level of capital, since ( ) , a a and thus ( ) ( ) ( ) We show the following proposition: Proposition 1: There exists a Nash equilibrium ( ) is the unique Nash equilibrium.If not, then for any Nash equilibrium ( ) .We can easily verify that ( ) 0, 0 is not a Nash equilibrium.Note that ( ) ( ) , a a is not a Nash equilibrium; 2) if 2 Use the Euler equation .
3 If 0 K = , then no production arises and 1 i i U e = .But we can easily verify that such situation is not a Nash equilibrium, since ( ) ( ) is in fact a Nash equilibrium, consider the function ( ) . By Equation (1), ( ) ( ) is the best response to a K e < − .Therefore, ( ) , e a is a Nash equilibrium.This completes the proof.

The Second Model
The demand function of consumer i on private consumption is simply ( ) ( ) Hence, the total demand is ( ) ( ) Now, we introduce an assumption.ASSUMPTION 1: For any 0 K > , there exists By first-order condition, we have is the unique value such that , 2 c c F K + = .Thus, in equilibrium, the profit of the firm is Then, the payoff function of this model 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: Here, we provide a sufficient condition of ASSUMPTION 2 to show this assumption is not too strong. .
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 strong5 .Define and K + as the unique solution of ( ) 0 h K = .If such K + does not exist, then let .K + = +∞ Then, 4 For example, to differentiate ( ) We will show the following proposition: Proposition 3: Under ASSUMPTIONS 1-2, there exists a Nash equilibrium ( ) , , 2 2 is the unique Nash equilibrium.If not, then for any Nash equilibrium ( )

Example: Improvement of Total Welfare
Suppose ( ) ( ) . By easy calculation, we have in model 1, ( ) This example demonstrates that the existence of the case where monopoly improves the total surplus.

Comparative Statics
First, we argue the following result.Proposition 4: Suppose that 1 e is sufficiently low.Define ( ) ( ) ( ) ( ) 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, ( ) ( ) Proposition 5: Suppose ASSUMPTIONS 1-2 hold, and choose any Nash equilibria ( ) , a a of model 1 and , a a Meanwhile, the equilibrium condition of this market is It can be verified in the same way as Proposition 1.