Calculating equilibrium values for perfect competition, monopolies, and considering externalities. A rare earth product example, part 1

This post will be part of a series that considers three different market types, how to work the math to arrive at equilibrium values, and the associated consumer, producer, and economy wide surplus values associated with them.

Suppose that the market for “rare earths” given by the following Demand (inverse demand, marginal benefit or average revenue), supply (marginal costs), and marginal external social cost equations:

Market Quantity Demanded: QD = 120 – 2 P P = 60 – (1/2)QD
Marginal Revenue: MR = 60 - QD
Market Quantity Supplied: QS = (14/5) P P = (5/14) QS
Marginal external social cost: MESC = 15

A. Please graph the supply and demand and marginal external social costs on a single market graph, clearly identifying the quantity and market price associated the
Competitive Market Equilibrium, the Social Equilibrium (i.e. where net economic
surplus is maximized) and the Monopolistic Equilibrium..

B. (all prices are in $) Please fill out the following table (all prices are in $) . 

Equilibrium	Q	P	PS	CS	Ext. Cost	Eco. Surplus
Competitive
Social
Monopoly

The graph of this problem looks like:

Where D represents private demand, S is the supply function of the firm (or marginal cost).  The line above it includes the external cost caused by producing rare earths.  The MR line shows marginal revenue for the monopoly which has twice the slope of the demand curve. 

Point A represents the monopoly outcome, note that the price paid by the consumer is not at point A, but at the point where point A’s quantity intersects the demand curve.  So point A has the highest consumer price.  Point B shows the perfectly competitive outcome, where quantity is the highest of the three, and price is the lowest of the three.  Finally, point C shows the optimal society outcome, where surplus is maximized and the externality is taken into account.

To solve for these equilibrium values we simply need to equate MB (marginal benefit) to MC (marginal cost) in each of the different scenarios.  The trick is to remember what is MB and MC for these scenarios.

In a perfectly competitive market, MB will be price, and MC will be marginal cost or the supply curve.  And we want to find out at what price level this will occur at.  One method is to equate the demand function with the supply function.  The easiest way to do this is to set Qd equal to Qs and then solve:

Qs = (14/5)P = Qd = 120 -2P  or

(14/5)P = 120 – 2P

Add 2P to both sides, then divide both sides by 4.8 to get:

P = 25

So in a perfectly competitive market, equilibrium price will be 25, we can plug this back into our supply or demand function to find out what Q is.

Qd = 120 – 2P = 120 – 2*25 = 120 – 50 = 70  and

Qs = (14/5)P = (14/5)*25 = 70, so we know we did it right.

To find the social equilibrium we have to set MB = MC again, but this time MC will be society’s cost instead of the private firm’s cost.  So we add MESC to our supply function.

P = 15 + (5/14) Qs (social)   or, inverted:

Qs (social) = (14/5)P - 42

And we set this equal to our demand curve function to get:

Qs (social) = Qd = (14/5)P - 42 = 120 – 2P   or

(14/5)P – 42  = 120 – 2P

Now add 2P to both sides, add 42 to both sides, and divide by 4.8 to get:

P = 33.75

Plug this P back into our demand function, and solve for Qd to get:

Qd = 120 – 2* 33.75 = 52.5

Notice that when the negative externality is taken into account, we end up with a lower equilibrium quantity and a higher equilibrium price.  This is because the negative externality poses costs on society, and a lower quantity is beneficial.

Finally let’s consider the monopoly outcome.  With this market structure, our MB will be marginal revenue, while our MC is again the supply function for the firm.  So we can set MR = MC and get:

MR = MC =  60 – Qd = (5/14)Qs

Remember that we are solving for equilibrium so we want Qs=Qd eventually.  To get this, we add Q to both sides, then divide by both sides by (19/14).  This gives us:

Q = 44.2 (rounded)

We can then plug this Q into either our MR or demand function to find the associated marginal revenue and consumer paid price.

The marginal revenue for the monopoly is going to be 15.8 (60 – 44.2).  And the price paid by the consumer will be 37.9 (60 – 0.5*44.2).

The next post will go over how to calculate the associated surpluses for these outcomes.