How to solve dominant firm problems, a question and answer

This post is going over the economics of a dominant firm problem using algebra and math.  For more information on the economic theory of a dominant firm or price leader model go here.  The actual question being solved is:

In the model of a dominant firm, assume that the fringe supply curve is given by Q= -1+0.2P, where P is market price and Q is output. Demand is given by Q=11 –P. What will price and output be if there is no dominant firm? Now assume that there is a dominant firm, whose marginal cost is constant at $6. Derive the residual demand curve that it faces and calculate its profit-maximizing output and price.

In order to solve this you need to follow these steps:

 1)  First you need to find equilibrium price (P*) by setting the Q's (from the fringe supply and industry demand) equal to each other and solving for P.  At this P*, the demand curve for the dominant firm begins.

2)  Now find the P where fringe supply equals zero.  You can do this by using your MCcf equation and plugging in 0 for Q, and solving for P.  At this P, the demand curve for the dominant firm will cross the industry demand line.

3)  Now that we have these two points, we can sketch a demand curve, and since we know the assoicated P's and Q's, we can calculate the demand curve for the dominant firm's slope.

4)  If we double the slope of the dominant firm's demand curve we get its marginal revenue curve.

5)  We can now set marginal revenue (derived from 4) equal to marginal cost, and solve for Q which will give us the profit maximizing quantity.

Now let's go through it step by step:

First, we know two equations for the competitive fringe's supply, and the total industry demand.  Where these two line's cross show us where the perfectly competitive equilibrium would be.   So we get this by setting the Qs equal to each other and solving for P.  For more information on this process, check out the post on solving for equilibrium mathematically.

Qfs = -1 +.2P = Qd = 11 - P

-1+ .2P = 11 - P, add 1 and P to both sides to get:

1.2P = 12, then divide both sides by 1.2 to get:

P = 10, so the perfectly competitive equilibrium price occurs at a price of $10, and this is also where the demand curve for the dominant firm begins.

You can see this interaction in the graph to the right, where the green line shows the demand curve and the blue line shows the marginal cost, or fringe supply curve.  Where they cross gives us the price level, which is marked on the graph as Ppc which stands for the price associated with the perfectly competitive equilibrium, and in this particular case is equal to 10.

The next step is to find the price at which the competitive fringe's supply will be 0.  To do this we need to take the competitive fringe's supply function and figure out at what price will quantity be equal to 0.  Given the function:

Qcf = -1 +.2P, and we can see that Qcf will only be 0 if P is 5, so at a P of 5, we know that the dominant firm's demand curve will cross the industry's demand curve.  We can find the quantity associated with this point by plugging this price into the industry demand function.  We get:

Qd = 11 - P => 11 - 5 = 6, so the quantity will be 6 at a price of 5.  This gives us two points for the dominant firm's demand curve (0,10) and (6,5).   Using the rise/run equation to find the slope we get a slope of -5/6.  If we double the steepness of the slope, we get -5/3 as the slope of the marginal revenue curve for the dominant firm.

We can now draw the dominant firm's demand curve (residual demand) and marginal revenue curve given these points and slopes:

Now we set MR = MC to find the profit maximizing quantity.  We know that MC = 6, so we can use our MR equation set equal to 6 to solve for quantity.  Our marginal revenue function is identical to our demand function with double the slope.  So the intercept is 10, and the slope is -5/3.  So:

MC = MR => 6 = 10 -(5/3)Q

Multiply both sides by 3 to get:

18 = 30 - 5Q, now add 5Q and subtract 18 from both sides to get:

5Q = 12, or Q = 2.4

So the optimal quantity of output for this firm is 2.4 which corresponds pretty well to our graph.  Now to get the price, we have to draw the line up to where it intersects the fringe demand curve, or we can plug in 2.4 into our fringe demand equation to get:

P = 10 - (5/6)2.4 = 10 - 2 = 8

So the price charged by the dominant firm in this example will be $8.

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