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## Taxing a monopoly firm

The model of a monopoly firm I made had a simple demand function of Q = 500 – P and a marginal cost of MC = 150 with no fixed costs.

This firm was producing output of 175 and selling at price 325, bringing in profits of 30265.

Now we can look at what would happen if the government introduced a tax on the monopoly. There are two common types of taxes that are analysed in microeconomics – specific taxes (also known as sales taxes or quantity taxes) and ad valorem taxes.

Specific taxes are charged per unit of output. This means that the price that the supplier (the firm) receives is going to be different from the price the demander (the consumer) pays.

Consumers will not change their view on how they value the product just because there is a tax on it, they will still demand the same amount of product at the same price (to them) as before. So the consumer demand function remains the same, $P_D = 500 - Q$.

However, the price the supplier receives will now be less, as the supplier has to pay $\tau$, the specific tax, on each unit, to the government. So the effective demand function to the supplier is $P_S = 500 - Q - \tau$.

So lets see what happens if the government introduces a specific tax of 50 per unit. The effective demand function to the supplier is $P_S = 500 - Q - 50 \Rightarrow P_S = 450 - Q$.

The total revenue to the supplier is $TR_S = 450Q - Q^2$ so marginal revenue is $MR_S = \frac{d(TR_S)}{dq} = 450 - 2Q$.

Setting MR = MC when MC is 150 means $450 - 2Q = 150 \Rightarrow 150 = Q$.

Now if the supplier produces output of 150, we can look back at the original consumer demand curve to see the price at which consumers would demand 150 – this is 500-150 = 350.

So the supplier produces output of 150 and sells at a price of 350. However of the price of 350 for each item, 50 goes to the government and 300 goes to the supplier, the tax drives a wedge between the price paid by the consumer and the price received by the supplier.

On the graph D1 shows the original consumer demand curve, and D2 shows the effective demand curve faced by the firm once the tax is taken into account. MR2 shows the marginal revenue curve faced by the supplier once the tax is taken into account.

Pm1 and Qm1 give the original price and quantity combinations for the monopoly firm before the tax was introduced. Pm2 and Qm2 give the new price and quantity combinations when the tax is in place. Ps2 is the price received by the supplier when the tax is in place.

The green shaded area represents the profits for the government (the tax revenue). The red shaded area shows the profits of the firm.

Remember that originally without the tax, the firm made profits of 30265. Now, the firm’s profits are $\pi = TR - TC = PQ - (MC)Q = 300(150) - (150)(150) = 22500$. Governments profits are $\tau Q = 50(150) = 7500$. If you add the two together you get total profits of 30000, which is lower than the firm’s profits in the situation without the tax, so there is some deadweight loss here.

Ad valorem taxes are charged as a proportion of the price of the product. VAT is an example of an ad valorem tax, you pay an additional 20% of the price to the producer, in tax.

Again the price that the supplier (the firm) receives is going to be different from the price the demander (the consumer) pays. The supplier has to pay $\alpha$, the ad valorem tax, as a proportion of the price on each unit, to the government. The relationship between the price the demander pays and the price the supplier gets is $P_D = P_S (1+\alpha)$

So here the effective demand function to the supplier is $P_S (1+\alpha) = 500 - Q$.

We can look at what would happen if the government introduced an ad valorem tax of 33.333% of the price received by the supplier. The effective demand function to the supplier is $1.333 P_S = 500 - Q \Rightarrow P_S = 375 - 0.75Q$.

The total revenue to the supplier is $TR_S = 375Q - 0.75Q^2$ so marginal revenue is $MR_S = \frac{d(TR_S)}{dq} = 375 - 1.5Q$.

Setting MR = MC when MC is 150 means $375 - 1.5Q = 150 \Rightarrow 150 = Q$.

Again looking back at the original consumer demand curve, if the supplier produces output of 150, the price at which consumers will demand this amount is 350. The reason I chose the ad valorem rate of 33.333% was to fit with the other example – this level of ad valorem tax has the same impact on consumers as the specific tax of 50 per unit, it means the quantity produced falls to 150 and the price rises to 350.

But this time the wedge between the price paid by consumers and price received by suppliers is different. As $P_D = P_S (1+\alpha)$, $350 = 1.333 P_S \Rightarrow P_S = 262.5$, which means that of the price of 350 for each item, 262.5 goes to the supplier and 87.5 goes to the government. The government is making more revenue from each unit this time.

This time the green shaded area representing the profits for the government (the tax revenue) is bigger, and the red shaded area showing the profits of the firm, is smaller.

Under this ad valorem tax, the firm’s profits are $\pi = TR - TC = PQ - (MC)Q = 262.5(150) - (150)(150) = 16875$. Governments profits are $\alpha P Q = 0.333(262.5)(150) = 13125$. If you add the two together you get total profits again of 30000, but there has been more redistribution of profits away from the firm towards the government.

Generally when a firm has market power, the government will make more revenue by charging an ad valorem tax as it takes advantage of the firm’s power to charge a price higher than marginal cost. Monopolists generally produce less output than the competitive level, and use their market power to charge a higher price for less production. A specific tax will not be as much of a revenue earner when output is cut as it is a constant tax charged per unit sold, but the value of an ad valorem tax increases as the firm increases the price. So ‘pound-for-pound’ as it were, when a specific tax and ad valorem tax on a monopolist have the same effect on the price faced by the consumer, the ad valorem tax will be better for government and worse for the firm.

## A model of a monopoly firm

After looking at the basic theory behind the monopoly firm it is useful to sketch up a simple model to see how to work out basic things like profit maximising price and quantity combinations, total profits and so on.

The monopolist’s total revenue will be given by the price x quantity at the profit maximising amount. The total cost will be the average total cost x quantity at the profit maximising point. The profit will be the total revenue minus the total cost.

The firm’s profit function can be written as $\pi = TR - TC$. In order to maximise profit, we differentiate profit with respect to output, and set it equal to zero. As both total revenue and total cost are functions of Q, we can write this problem as $\pi = TR(Q) - TC(Q) = 0\Rightarrow \frac{d(\pi)}{dQ}=\frac{d(TR)}{dQ}-\frac{d(TC)}{dQ} = 0 \Rightarrow \frac{d(TR)}{dQ}=\frac{d(TC)}{dQ}$

The expression $\frac{d(TR)}{dQ}$ is the change in total revenue when output is increased by one unit – this is marginal revenue.

The expression $\frac{d(TC)}{dQ}$ is the change in total cost when output is increased by one unit – this is marginal cost.

To maximise profit a firm sells where its marginal revenue equals its marginal cost. Unlike the competitive market, marginal revenue does not just equal price.

To make a simple model lets take a firm with a downward sloping market demand curve of $Q=500-P$ and assume the firm has no fixed costs and variable costs of 150Q.

The inverse demand function is $P=500-Q$

So the total revenue function is $TR=PQ=500Q-Q^2$

And the marginal revenue function is $MR=\frac{dTR}{dQ}=500-2Q$

The total cost is $TC = FC + VC = 0 + 150Q$.

Its marginal cost is $MC = \frac{d(TC)}{dq} = 150$. This firm has constant marginal costs.

It sets its production level where MR=MC, so

$500-2Q = 150 \Rightarrow Q = 175$.

At this output, the price will be $P=500-175=325$

So we have a profit maximising price/quantity combination of P=325, Q=175.

The Lerner Index is $\frac {325-150}{325}=0.538$.

What profits does it earn at this point?

Profit is equal to total revenue minus total cost, so

$\pi = TR - TC = PQ - FC - VC = 325(175) - 0 - 150(175) = 30625$.

This diagram shows what is going on:

The shaded area illustrates the profit of 30625. Pm – the monopoly price, is 325, and Qm – the monopoly quantity, is 175.

The amount of market power the monopolist has depends on the elasticity of demand for the good. We can read the elasticity at the profit maximising point from the Lerner Index, $\frac {P-MC}{P}=-\frac{1}{\epsilon}$ so here $\frac {325-150}{325}=-\frac{1}{\epsilon} \Rightarrow {\epsilon}=-\frac{325}{325-150}=-1.857$

When a firm advertises, it is trying to make the demand for its good more inelastic, and/or to shift out the demand curve.

Lets say our monopoly firm runs an advertising campaign which raises its fixed costs by cost 5000. The campaign is successful and it changes the demand function to $Q=600-P$, so the inverse demand function is $P=600-Q$, total revenue is $TR=600Q-Q^2$ and marginal revenue is $MR=600-2Q$ so when MR=MC, $600-2Q=150 \Rightarrow Q=225$. At this output the price will be $P=600-225=375$ so our new profit maximising price is 375 and quantity is 225.

The new Lerner Index is $\frac {375-150}{375}=0.6$ so the amount of market power has risen.

The new profits are $\pi = 375(225) - 5000 - 150(225) = 45625$. So the advertising campaign has paid off.

Here is a new graph in the case where there is some advertising:

Now I have distinguished between the original demand curve (D1) and profit maximising price/quantity combinations (Pm1, Qm2) and the new demand curve and price/quantity combinations after advertising (D2, Pm2, Qm2). The effect of the advertising was to shift the demand curve out.

This time I’ve had to include an average total cost curve to be able to draw on the profits at the profit maximising quantity. In the original case we had no fixed costs so the average total cost were just equal to the marginal cost, but here we have fixed costs.

The average total cost is $ATC = \frac{TC}{Q} = \frac{FC + VC}{Q}$ so in this case is $ATC = \frac{500 + 150Q}{Q} = \frac{500}{Q}+150$.

We can see the effect it has had on changing the elasticity again through the Lerner Index, $\frac {375-150}{375}=-\frac{1}{\epsilon} \Rightarrow {\epsilon}=-{375}{375-150}=-1.667$, so the demand has now become less elastic.

## Price discrimination strategies

Sometimes you can divide a firm’s potential customers into different groups, with different demand elasticities. A clothes shop for instance might find that its customers who have jobs, have a relatively inelastic demand whilst students (who are keen on clothes but short on cash) have a much more elastic demand, ie they are much more willing to buy clothes in greater quantities when the price drops.

Consider now a firm that has two customer groups, call them ’employed’ and ‘students’. It can produce at a constant marginal cost of 10.

The demand function for ’employed’ is $Q_{emp}=63-0.5P$
The demand function for ‘students’ is $Q_{stu}=200-2P$

If it set a uniform price based on the demand function for ’employed’ then the inverse demand function for ’employed’ would be $P=126-2Q_{emp}$ so $TR=PQ=126Q-2Q_{emp}^2 \Rightarrow MR = \frac{d(TR)}{dQ}=126 - 4Q_{emp}$.

When MR = MC, $10 = 126 - 4Q_{emp} \Rightarrow Q=29$

Subbing that back into the inverse demand function, $P=126-2(29)=68$

So the firm would sell at a price of 68. It would sell a quantity of 29 to the ’employed’ market and at that price it would sell $Q_{stu}=200-2(68) = 64$ a quantity of 64 to the ‘student’ market.

Total sales would be 93 and at a price of 68 each that gives total revenue of 6324. With a cost of 10 per unit, total cost (assuming no fixed cost here) would be 930 so total profit would be 5394.

However, its possible to increase that profit, if you can set a separate price for students. If we want to find the profit maximising price for students, we would have an inverse demand function of $P=100-0.5Q_{stu}$ so $TR=100Q-0.5Q_{stu}^2 \Rightarrow MR = 100-Q_{stu}$.

When MR = MC, $10 = 100-Q_{stu} \Rightarrow Q=90$
Subbing that back into the inverse demand function, $P=100-0.5(90) = 55$

So for the ‘student’ market the profit-maximising price would be 55.

If we sell to the ’employed’ market at a price of 68, and offer a student discount to sell to the ‘student’ market at a price of 55, then our total revenues will be (68 x 29)+(55 x 90) = 6922. Total cost would be (10 x 29)+(10 x 90) = 1190. So total profit will be 5732.

We have increased total profit by being able to sell at a lower price to the student market, simply because their demand was more elastic, so by reducing the price for students from 68 to 55, we profited by the large increase in output of sales to students from 64 to 90.

Of course this strategy does not work if we can’t identify who the students are. If people from the ’employed’ market start being able to pass themselves off as students then our strategy falls down!

So in order to use a price discrimination strategy, you generally have to have three things:
1. You need to have some form of market power.
2. You need to have consumer groups with different sensitivities to price (different elasticities of demand) and you need to be able to identify them.
3. You need to be able to prevent or limit resales (its no good if students can buy up a large stock of your goods at the lower price, then sell them on to ’employed’ customers!)

Price discrimination strategies are all around us. Different ticket pricing on trains for instance, where you pay a higher price for travelling on ‘peak hours’ than you do at other times of the day, is basically a way of identifying which consumers have relatively inelastic demand (commuters who need to get the train to go to work in the morning) and charging them higher prices, while charging a lower price to customers who are travelling for other reasons, and may have other alternative forms of transport (car, bus etc) and so need to be enticed on to the train with a lower price.

## The welfare loss of monopoly

The welfare losses of monopoly (or any form of market power) can be shown quite easily by illustrating the consumer and producer surplus on a graph.

Consider the effect of a firm with linear demand and supply curves (the supply curve would really be the marginal cost). The diagram below considers the case where the firm is competing in a perfectly competitive market with an infinite number of identical firms, or has a monopoly on the market.

In the case of perfect competition, then the firm will simply produce at the competitive price, Pc, where the supply and demand curves interact. All firms are identical so will face identical supply curves – if this firm’s supply curve (marginal cost curve) was higher and it was unable to profitably produce at Pc then it would have gone out of business, and if its supply curve was lower and it was able to make profits then other firms would enter the market until all firms were making zero profits. When the firm produces at Pc it will supply quantity Qc.

When it has a monopoly, it instead produces at the point where MR = MC, ie where the marginal revenue curve cuts the supply curve. This is quantity Qm which will sell for price Pm.

Now first consider the consumer and producer surplus in the case of perfect competition.

The yellow area shows consumer surplus and orange area shows producer surplus. I have split the graph into five areas, area a, b, c, d and e. Ignore the purple MR line cutting through areas a, b and d, the areas are just bounded by the blue supply and demand curves and the red dotted lines linking price and quantity combinations.

In the competitive case:

Consumer surplus = a + b + c
Producer surplus = d + e

Now consider the consumer and producer surplus in the case of monopoly.

Again yellow is consumer surplus, orange is producer surplus, and I have added a third colour, grey, to show ‘deadweight loss’ – the area that was surplus to consumers or producers in the competitive case but has now been lost.

In the monopoly case:

Consumer surplus = a
Producer surplus = b + d
Deadweight loss = c + e

The effect of going from perfect competition to monopoly is bad for consumers. Consumer surplus has been reduced by (b + c). Area b has gone from consumers to producers, so this is not an overall welfare loss, just a distributional change from consumers to producers.

However the monopoly is good for producers. Producer surplus has increased by (b – e) and as b is a larger area than e this is a net gain.

Areas c and e are deadweight loss. Consumers have lost c and producers have lost e, this is because there is now less output being produced due to the quantity decreasing from Qc to Qm.

So overall society loses out – there is a net welfare loss when the aggregate welfare of consumers and producers is taken into account, although producers are likely to be happy as they have gained at the expense of consumers. From an economic point of view, here there is an efficiency loss caused by going from perfect competition to monopoly.

## The degree of monopoly

Market power is the ability to charge a price above marginal cost. A firm in a competitive market produces where P=MC. Any time a firm is able to charge a price such that P>MC, it has a degree of market power. There’s a nice tool to use which gives us an indication of amount of market power a firm has, called the Lerner Index.

The Lerner Index is $\frac{P-MC}{P}$

This shows us how much of a ‘mark-up’ the firm is charging above its marginal cost, as a proportion of its price. It will be a value between 0 and 1. A competitive firm would have a value of 0, the closer you get to 1, the more market power a firm has.

Lets look at an example. Consider a firm has a monopoly in an industry which faces a market demand curve $Q = 400 - 10P$. It has a constant marginal cost of production of 10. What price and quantity combination will it choose? The monopoly firm always produces where MR = MC. So we will need to use a bit of differentiation to find the marginal revenue.

First write the inverse demand function: $P = 40 - 0.1Q$ and now express it in terms of total revenue: $TR = PQ = 40Q - 0.1Q^2$.

$MR = \frac{d(TR)}{dQ}=40 - 0.2Q$ so when MR=MC, $10 =40 - 0.2Q \Rightarrow Q = 150$

Our inverse demand function now gives us the price: $P = 40 - 0.1(150) = 25$

So at the profit maximising point where it produces, this firm has got a marginal cost of production of 10 and is selling at a price of 25. The Lerner Index is then $\frac{25-10}{25}=0.6$

There is another useful property of the Lerner Index that you need to know, which relates to the elasticity of demand at the profit maximising point.

Here you have to return to the concept that the firm produces where MR = MC. You can express MR in terms of the elasticity of demand as $MR = P [1 + \frac{1}{\epsilon}]$.

As we are producing at the profit maximising point where MR=MC, this means that $MC=P(1+\frac{1}{\epsilon}) \Rightarrow MC = P + \frac{P}{\epsilon} \Rightarrow \frac{MC - P}{P} = \frac{1}{\epsilon} \Rightarrow \frac{P - MC}{P} = -\frac{1}{\epsilon}$

The Lerner Index is equal to the negative of the inverse of the elasticity of demand at the profit maximising point.

## Why does a monopoly never produce in the inelastic part of its demand curve?

This is a pretty standard question and it’s a good bet at some point when you start studying microeconomics you will get given this question as an exercise. It is also a question that is good to understand because if you get this you are on the way to getting some of the key concepts about elasticity and marginal revenue.

The first thing to understand is that, apart from the special case of constant elasticity where the demand curve is of the form $Q=aP^{-b}$, the elasticity will vary along different points of the demand curve. This is true even when the gradient of the demand curve is constant (ie the demand curve is linear). This is a point that sometimes confuses students about elasticity, they think “constant gradient = constant elasticity”…no it doesn’t.

Here is an example, this is a simple demand function Q = 20-0.5P.

We can calculate the elasticity at different points, a, b, c, d and e.

Remember the definition of elasticity:

Elasticity of demand is the proportional change in quantity demanded divided by the proportional change in price: $\frac{\triangle Q}{Q}\div\frac{\triangle P}{P} = \frac{\triangle Q}{Q}\frac{P}{\triangle P}$ which is generally expressed as $\frac{P}{Q}\frac{\triangle Q}{\triangle P}$. This will generally be a negative value, because when you increase the price, you decrease the quantity, so $\triangle Q$ will be negative.

As these changes tend to zero (ie at the margin) we can express the elasticity as $\epsilon=\frac{P}{Q}\frac{dQ}{dP}$.
If $-1<\epsilon<0$ then we say the demand is inelastic. If $\epsilon=1$ then it is 'unit elastic'. If $-\infty<\epsilon<-1$ then it is elastic.

With this demand function, $\frac{dQ}{dP} = -0.5$, so the elasticity at different points will be $\epsilon=\frac{P}{Q}\times -0.5$

So at point a, the elasticity is 36/2 x -0.5 = -9
At point b, the elasticity is 24/8 x -0.5 = -1.5
At point c, the elasticity is 20/10 x -0.5 = -1
At point d, the elasticity is 18/11 x -0.5 = -0.818
At point e, the elasticity is 4/18 x -0.5 = -0.111

Notice that at point c, the mid point of the curve, the elasticity is -1, this is where the curve is unit elastic. Above point c, the curve is elastic, it gets more elastic the higher the price and lower the quantity. At the point where the price is 40 and the quantity is 0, the elasticity will be 40/0 x -0.5 which will be infinity. Below point c, the curve is inelastic and gets less elastic the lower the price and higher the quantity. At the point where the price is 0 and the quantity is 20, the elasticity will be 0/20 x -0.5 which will be 0.

We can now think of this with marginal revenue. $MR = \frac{d(TR)}{dQ}$ and $TR = PQ$ so $MR = \frac{d(PQ)}{dQ}$.

Here the inverse demand function is $P=40-2Q$ so $PQ=40Q-2Q^2$ and $\frac{d(PQ)}{dQ}=40-4Q$. So we can draw in the marginal revenue curve MR = 40-4Q:

Notice how the marginal revenue is positive when the demand curve is elastic, it is zero when the demand curve is unit elastic and it becomes negative when the demand curve is inelastic.

This is the answer to the question. Given that the marginal revenue is the amount of revenue gained by selling an extra unit, nobody is going to sell an extra unit if the marginal revenue is negative (ie they lose money by selling it).

You can also think of this in an algebraic way. Given that $MR = \frac{d(PQ)}{dQ}$, we can use the product rule to say $\frac{d(PQ)}{dQ} = P\frac{dQ}{dQ} + Q\frac{dP}{dQ}$ so $MR = P + Q\frac{dP}{dQ}$.

Now multiply both top and bottom parts of the right hand side of that equation by P so you get $MR = P + PQ\frac{dP}{PdQ}$. We can factorise the P out of this to get $MR = P [1 + Q\frac{dP}{PdQ}]$ which can be rewritten slightly differently as $MR = P [1 + \frac{Q}{P}\frac{dP}{dQ}]$.

The right hand side of that equation is the inverse of the elasticity, $\frac{1}{\epsilon}$, so $MR = P [1 + \frac{1}{\epsilon}]$. This is a useful equation to remember.

Elastic demand is where $\epsilon < -1$ and inelastic demand is where $-1 < \epsilon < 0$. So now we can think of why a monopolist won't produce in the inelastic part of its demand curve. When demand is inelastic then $-1 < \epsilon < 0$ so $[1 + \frac{1}{\epsilon}] < 0$. And given that the price, P, is positive, it also follows that $P [1 + \frac{1}{\epsilon}] < 0$. So the marginal revenue will be negative, and no firm will produce an extra unit if it means it loses money.

## The monopoly firm

A monopoly firm is a price-maker, it can influence the market price through the quantity it produces. By producing less it will sell less but can sell at a higher price, by producing more it can sell more but only because the price falls. Ultimately the market price is determined by the interaction between the amount supplied by the monopoly firm, and the market demand (demanded by all the consumers in the market). The firm will usually face a downward sloping demand curve.

The monopoly firm will observe the same rules as any profit-maximising firm:

Marginal output rule – the firm will produce at an output where the price is equal to the marginal cost of production (MR = MC).

Shutdown rule – the firm will shut down if the average revenue is lower than the average cost at all output levels, so as the price equals average revenue, it will shut down if the price is lower than the average total cost at all levels.

We can look at this in terms of a graph:

It makes things easier if we consider linear demand curves, as then the marginal revenue curve is simply a straight line with twice the gradient intercepting the horizontal axis halfway along the way to the point the demand curve intercepts the horizontal axis. The quantity produced by the monopolist is that where MR = MC, the price is found by reading up to the corresponding point on the demand curve.

The profit for the monopolist will be the shaded area here:

The monopolist’s total revenue will be given by the price x quantity at the profit maximising amount. The total cost will be the average total cost x quantity at the profit maximising point. The profit will be the total revenue minus the total cost.

The firm’s profit function can be written as $\pi = TR - TC$. In order to maximise profit, we differentiate profit with respect to output, and set it equal to zero. As both total revenue and total cost are functions of Q, we can write this problem as $\pi = TR(Q) - TC(Q) = 0\Rightarrow \frac{d(\pi)}{dQ}=\frac{d(TR)}{dQ}-\frac{d(TC)}{dQ} = 0 \Rightarrow \frac{d(TR)}{dQ}=\frac{d(TC)}{dQ}$

The expression $\frac{d(TR)}{dQ}$ is the change in total revenue when output is increased by one unit – this is marginal revenue.

The expression $\frac{d(TC)}{dQ}$ is the change in total cost when output is increased by one unit – this is marginal cost.

To maximise profit a firm sells where its marginal revenue equals its marginal cost. Unlike the competitive market, marginal revenue does not just equal price.

We can make a model of a monopoly firm to see how this works.