Tuesday, February 06, 2007

Bad Economics: Businesses Simply Pocket a Tax Cut

Let's take an economically ignorant statement that I just read and analyze it:

"If a company makes more money per unit sold as a result of lower corporate taxes, that doesn't automatically mean that they will drop their prices."

There are several ways to tax a company. We could tax their profits, tax their output, or assign a lump-sum tax. With two of these taxes, it will always be the case that a lower tax means lower prices, even if there is no competition in the industry at all. If the industry is competitive, it just means that prices will be lower still. So let's see what happens to a monopolist under these different taxes. We'll use some simple calculus to get our results. You could also do this graphically, although it would be a lot more work for a blog like this.

No Tax:
Suppose that the (inverse) demand function is P=100-10Q. Suppose that the monopolist's total costs are C(Q)=50+10Q. We can write the profit function as Total Revenue minus Total Cost, or:

Profit=PQ-C(Q)=(100-10Q)Q-50-10Q

To maximize this, simply take a derivative with respect to Q and set it equal to zero (extra credit: check the second derivative to verify this is a maximum). You get:

100-20Q-10=0

Solving for Q, we get Q=4.5 as the amount of output that maximizes profits. What is the price? Simply substitute Q back into the demand function. You get P=55. So to summarize, with no tax, we get a price of 55 and a quantity of 4.5.

The Output Tax:
Suppose the government imposes a tax of $10 on each unit of output the firm produces (perhaps because the firm's output produces pollution--a Pigovian Tax).

Profit=PQ-C(Q)-Tax=(100-10Q)Q-50-10Q-10Q

Note the new term on the end. That's the tax. Maximizing this gives us:

100-20Q-10-10=0

This gives us a profit maximizing Q=4. The profit maximizing price is therefore P=60. So look at what we have so far: In response to the tax, the monopolist charges a higher price and produces a lower quantity. Take away the tax, and the monopolist charges a lower price and produces a higher quantity (i.e., we're back in our first example).

The Profit Tax:
Suppose the government puts a tax of 5.2632% on the firm's profits (I've picked this odd number for a reason which will later become apparent). The profit function becomes:

Profit=PQ-C(Q)-Tax=(100-10Q)Q-50-10Q-0.052632*[(100-10Q)Q-50-10Q]

The last term is the profit tax. The government takes ten cents out of every dollar of profit the monopolist makes. What happens? Let's check the first order condition:

100-20Q-10-0.052632*[100-20Q-10]=0

Solve this for Q and you get approximately Q=4, and P=60. Do those numbers sound familiar? They're the same numbers we got with the tax on output in the previous example. A tax on profits and a tax on output can be identical, if the taxes are chosen appropriately. And again, eliminating the tax takes us back to the first case with no tax--with a lower price and higher output.


The Lump-Sum Tax:
Now suppose the government simply slaps a fixed tax of $42.5 on the monopolist.

Profit=PQ-C(Q)-Tax=(100-10Q)Q-50-10Q-42.5

Because the tax is not dependent on output, when we maximize profit it will simply disappear (the derivative of a constant is zero). So we're back to our intial first order condition:

100-20Q-10=0

And we get Q=4.5 and P=55 . We're back to the same price and quantity from the no tax situation.

Profitability:
What happens to the profits of the monopolist under these three taxes? Just plug in the four quantities we calculated:

No tax:
Profit=PQ-C(Q)=(100-10Q)Q-50-10Q=(100-10*4.5)*4.5-50-10*4.5=152.5

Output Tax:
Profit=PQ-C(Q)-Tax=(100-10Q)Q-50-10Q-10Q=(100-10*4)*4-50-10*4-10*4=110

Profit Tax:
Profit=PQ-C(Q)-Tax=(100-10Q)Q-50-10Q-0.052632*[(100-10Q)Q-50-10Q]
=(100-10*4)*4-50-10*4-0.052632*((100-10*4)-50-10*4)=151.58

Lump-Sum Tax:
Profit=PQ-C(Q)-Tax=(100-10Q)Q-50-10Q-42.5=(100-10*4.5)*4.5-50-10*4.5-42.5=110

What lessons can we draw from this?
-Even a monopolist will automatically reduce price and increase output when a tax is eliminated. It doesn't matter what form the tax takes, unless it's a lump-sum tax (which doesn't affect output, although it might affect the decision of a firm to start up or shut down). . And they don't do it because they're nice people. Reducing price and increasing output increases profit.
-Different taxes can yield the same output and price, and different profits. Or we could rearrange all the taxes to give the same profits, but different outputs and prices.

Some other things to note:
-If this were a perfectly competitive industry, profits would be competed down to zero (remember, these are economic profits, not accounting profits). This would result in lower prices and higher output, but it would not change the fundamental result that the removal of a tax on a producer will lower prices. Anyone familiar with Supply and Demand should be able to demonstrate this on a graph.
-If this monopolist is the owner of a nonrenewable resource, the price should rise over time as the resource becomes more scarce. The removal of a tax will still have similar short-run effects, however, even if the (untaxed) price eventually rises back to its former (taxed) level due to scarcity.
-Market adjustment of prices sometimes takes time; we may not observe instant and complete price adjustment in some markets. Firms and consumers are working through a discovery process, finding out new information and incorporating it into their production and consumption plans.