Inefficiency from Strong Labor Unions or Strong School District: The Calculus of Bargaining Power and Competitive Markets

The economic efficiency of labor union contracts with a monopsonistic labor market can be analyzed with some simple tools from multivariate calculus and a theorem called the Envelop Theorem of functions.  This mathematical treatment is not the only method of analyzing societies overall welfare between these two agents, Game Theory is another modeling tool used to critically think about strategic situation and bargaining situations where there are threats, domination, and probabilistic calculations being implied or explicitly communicated between two or more parties. Another treatment of this problem using Mathematical Game Theory will be presented in a subsequent post.

This entry will examine the labor market without the existence of a union and then with the existence of a union, finally a model will be examined to explore the bargaining power and techniques of a labor union once it is created.  The welfare of the workers will be measured it the amount of workers employed and their respective wages.  The assumption imposed is that a labor union prefers more members to less and that workers prefer higher wages to lower wages.

Analytical Calculations/Derivations:

Suppose that all of the students that a teacher instructs, no matter what grade level, go on to graduate high school.  Also assume that a teacher can teach “t” kids a year and that the economic value of education for a high school graduate and a high school drop out is , “g” and “d” respectively. Therefore the marginal revenue product of one teacher is given by this general form:

The MR above is equal to a number and suppose that this number is 20 so 20 is the marginal revenue product of one teacher.

The school district is the only hirer of teachers in a local area and faces a labor supply curve of the form

Which merely states that for every dollar increase in wages (w) you will encourage 50 more people to supply (l) their services in the teaching sector.  The number 50 is just an arbitrary number and the actual number for a particular labor market can be calculated using econometrics.  Fifty is a nice round number so that will be arbitrary constant, and most importantly,  this number will not have an impact on our fundamental analysis of this market.

The total wage bill for the school district is equal to the wage (w) multiplied by the labor supplied (l).  When the school district thinks about hiring more teachers it must do so my analyzing at the margin, in other words they must think of the expense incurred from hiring another teacher and weigh those expenses against the economic benefits of hiring that teacher to the students.  This is why the formula for the total wage bill must be differentiated with respect to labor to get the marginal expense:

The quantity and wages paid to teachers is less than in a perfectly competitive labor market when the school district has all hiring power and there is no union.

To study the impact that unionization we must assume that we have a downward-sloping marginal revenue product for a labor curve of the form.

There are a few options for the equilibrium solution of this problem:

1) The Union is totally powerless and the School District imposes its hiring and wage decisions on the Union

The wage and hiring is determined above as 500 teachers get hired at a wage of $10.00

2) The Union is completely powerful and imposes its wages and hiring decisions on the School District

3)  The Union and the School District operate in perfectly competitive market where wages and hiring decisions are made by bargaining by individual principals and individual teachers, then we would have something akin to a perfectly competitive market:

The perfectly competitive price, that is when supply equals demand in the labor market implies that, l=583 and w=$11.66

4)  The Union and the School District each have enough power to make it so the ultimate outcome of hiring and wages are determined through bi-lateral bargaining

The fourth situation is the most realistic situation, where the labor union and the school district bargain for wages and in by the duality principle of commodity pricing, also bargain for the quantity of teachers hired.

Here is a summary of the different labor market outcomes on quantity of teachers hired and their wages,  based on the power of the union and school district in bargaining and non-bargaining and situations:

Conclusion:  A competitive labor market for education would cause the increase in the number of teachers and the wages of teachers, if either the Teacher’s Union or the School District has too much power then there will be inefficiencies in the market.  If the teacher Union has too much power then they will increases wages until the point where the less teachers are hired than in the competitive market.  If the School district has too much power then they will hire less teachers and pay them less than if they were in a perfectly competitive market.  In the competitive market there are the maximum number of teachers hired, but the wages are not as high as they would have been under a strong Union, but they are not as low as they would be under a strong District.  The competitive outcome is preferred by society because they receive the maximum amount of services at a low cost, it is efficient and their is no waste in the system, but this would entail having a large number of competing public and private schools as well as a dismantling of the Teacher’s Union, both of which are unlikely to occur, despite the large welfare gains to society and the ultimate stakeholders, the students.


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