# Monopoly power and corporate taxes

There has been a fair amount of debate about corporate taxes in the econ blogosphere. The debate was framed early on by a cute little exercise on Greg Mankiw’s blog which was supposed to  show that, in a small-open economy with perfect competition, a 1 dollar cut in capital taxes raises wage income by more than 1 dollar.

Paul Krugman and others have rightly pointed out that Mankiw’s toy example, its cuteness notwithstanding, provides little to no insight into the real policy debate now going on in the US, because (i) the US is not a small open economy and (ii) there is evidence that much of corporate profits are monopoly rents rather than returns to capital, which casts doubt on the relevance of perfect competition models.

Indeed, there’s a new paper documenting that mark-ups (difference between price and marginal costs) have increased in practically every industry in recent decades. The paper has not yet gone through peer review, so it’s probably wise not to jump to conclusions from it. Nevertheless, it’s useful to think about potential implications.

One of the basic results in public finance is that taxes on rents produce no deadweight loss. So if corporate profits are just monopoly rents, we can tax them away at zero social cost. Right?

Wrong.

Consider the textbook model of monopolistic competition in that Nobel-price winning 1980 paper by Krugman. Each individual producer has a monopoly over her variety of a differentiated good (think VW having a monopoly over VW Beatle cars or Apple over iPhones). She faces the demand curve
$\displaystyle q = Q\left[\frac{p}{P}\right]^{-s},$
where q is her individual output, Q is the aggregate output of the industry as a whole (i.e. the „size of the market“), p is the price of the individual firm’s output and P the aggregate industry price index. Finally, s is the elasticity of substitution across varieties.

The total cost curve of the individual producer is
$\displaystyle f + q/a,$
where f is fixed costs and 1/a are marginal costs (a being marginal productivity) which we assume to be constant. Let t be the corporate tax rate and pi gross profits, then net profits of the individual monopolist are given by
$\displaystyle \pi = (1-t)[pq - (f + q/a)].$

Our monopolist, being the ruthless corporate egomaniac that she is, strives to choose q such as to maximize net profits. It should be clear from looking at the expression above, that her maximization problem is unaffected by t. That is, the corporate tax doesn’t matter for her decision at all. She would produce just the same q and charge just the same p whether t would be 0 or 0.5 or 0.9, everything else equal.

The last phrase is of the essence here. Everything else will not remain equal. But I’m getting ahead of the story here.

So what will be the profit-maximizing decision? Setting marginal revenue equal to marginal costs and solving for p yields
$\displaystyle p(a) = \frac{s}{s-1}\frac{1}{a}.$

The revenue is
$\displaystyle r(a) = PQ(Pa)^{s-1}$

and net profit is
$\displaystyle \pi(a) = (1-t)[r(a)/s-f].$
Notice that the price is increasing and profits are decreasing in marginal costs.

What will the industry equilibrium look like? As is well known, in an industry with free entry and identical costs, every firm will produce the same quantity, charge the same price and make zero profits in equilibrium. But to analyze the effect of taxing firm profits, we better have a model where firms make non-zero profits.

Let’s do it the way Melitz did it in his famous 2003 paper. Let’s assume first that firms do not know their their productivity before entering the market, but only the distribution of productivities. Let G(a) be that distribution. Second, firms need to pay fixed entry costs z to set up shop in the industry. Third, once a firm has established itself, it runs a constant risk of becoming bankrupt and exiting the market given by the probability d.

As Melitz has shown, the industry equilibrium can be represented by two equations in two unknowns: the average profit of active firms, $\displaystyle \bar{\pi}$, and the critical level of productivity, a*, at which a firm just breaks even. That critical level a* is implicitly defined by
$\displaystyle r(a^\ast) = sf.$

All firms with higher productivity stay in the market, while all others exit immediately. We can define $\displaystyle \bar{a}$ as the (weighted) average productivity among active firms. This average turns out to be uniquely determined by a* for any given distribution G(a). What’s more, it can be shown that the average profit of active firms is equal to the profit of a firm operating with the average productivity abar (sounds like a trivial point, but requires some algebra). Under some harmless conditions on G(a), the average net profit of active firms is a decreasing function of the cut-off productivity:
$\displaystyle \bar{\pi}(a^\ast) = (1-t)[r(\bar{a}(a^\ast))/s-f].$

Notice what an increase in the tax rate does to the average net profit function: it makes the whole thing shift downward in pi-a space. This is going to be the key for all later results, so let’s repeat it in plain words. Raising the corporate tax rate reduces net profits for any given cut-off productivity level.

The second condition to pin down the industry equilibrium comes from the entry/exit dynamics. A potential monopolist contemplating entering the industry compares the fixed entry cost z to the present value of expected net profits which she would earn now and in the future. The latter is given by the product of the probability of drawing an above-critical productivity level [1-G(a*)] and the present value of net profits pibar/d. In equilibrium, no potential monopolist wants to enter, so net profits of surviving firms must be
$\displaystyle \bar{\pi}(a^\ast) = \frac{dz}{1-G(a^\ast)}.$

This defines an increasing relationship between average net profits and the productivity cut-off. Notice that this relationship is unaffected by the tax rate.

The last two equations pin down $\displaystyle \bar{\pi}$  and a*. The first is called the „zero cut-off profit condition“, the second the „free-entry condition“. All endogenous variables can be calculating from those. In particular we can calculate the price index and the number of firms in equilibrium:
$\displaystyle P = \frac{s}{s-1}\frac{1}{\bar{a}(a^\ast)}N^{1/(1-s)}$
$\displaystyle N = \frac{L}{s(\bar{\pi}+f)},$

where L is total labor supply (the size of the economy). Notice that P decreases in N and N decreases in pibar.

So what happens if t increases?

The zero cut-off profit curve shifts downwards while the free-entry curve remains the same. Therefore, in the new equilibrium, we get both a lower net profit $\displaystyle \bar{\pi}$ and a lower cut-off productivity level a*. How is that possible? Well, suppose a* would remain the same initially. Those firms with productivity above a* would earn lower profits than before the tax increase. But then, no would-be monopolist would want to enter the market anymore while active firms still exit at rate d, which implies that the number of competitors decreases. Due to the decrease in competition, the aggregate price level and hence the revenue of active firms increases. This means that some firms whose productivity is slightly below a* are now able to make positive net profits. The result is a drop in the cut-off productivity that is sufficiently low as to induce potential firms to enter the industry.

Notice that the fall in the cut-off productivity counteracts the negative effect of the tax increase on average net profits such that the latter falls less than one-for-one with the tax hike.

Summing up, as a result of the corporate tax we have

• fewer firms
• lower average productivity
• a higher price index, which implies lower real wages
• which implies lower social welfare.

I have not found a cute formula to calculate the precise welfare effects, but I would venture a guess that the effect depends mostly on the elasticity of substitution s and the shape of the productivity distribution G(a).

Anyway, I think I have shown that a corporate tax is socially costly even if all corporate profits are due to monopoly rents. The negative effects on market entry are the key here. Fairly obvious extensions to an open economy are left for the reader.

Update: Here’s the story in a graph: