## Some Carbon Tax Scenarios

How does a competitive industry respond to an emissions tax in the short run and the long run?  What if the industry is a monopoly?

In this post I bring together two standard pieces of microeconomic analysis: the effect of an emissions tax to address a pollution externality; and the behaviour of profit-maximising firms in different market structures.  It’s a straightforward exercise, but some of the results may be found a little surprising.

My method here is the exploration of numerical examples.  Therefore the only claim I make for the results is that they demonstrate possibilities: to infer any sort of generalities would be an obvious fallacy.  The numbers from which the examples begin have been chosen for ease of calculation: it is no accident that many of the output and other figures to which they lead are round numbers.

I consider two industries, one with many firms in perfect competition, and one a monopoly.  The following assumptions are common to both:

1. Emissions are uniformly mixed and very large in total (as is the case for CO2 and some other pollutants).  Hence the damage due to any one firm’s emissions is independent of its location, and its contribution to total emissions is too small to affect the marginal damage per unit of emission.
2. Marginal damage m from the pollutant (in terms of the local currency) is 1 per unit of emission.
3. In the absence of an emissions tax, with firms taking no particular measures to limit their emissions, the emissions ratio e (the ratio of emissions to output) is assumed to be 4.
4. The emissions tax is introduced with minimal notice.  Therefore all adjustment to the tax takes place after its introduction.
5. Firms’ costs consist of four components: a) a fixed component; b) a component proportional to the square of output (in conjunction with (a) this yields the characteristic U-shaped average cost curve); c) a component reflecting, for any level of output, higher costs for a lower emissions ratio; d) a component for the cost of the tax, where applicable.  I take costs to include ‘normal’ profit: all references below to profit should be understood to mean economic or supernormal profit.

Outcomes are assessed from several points of view, of which perhaps the most important is net welfare, calculated as consumer surplus plus producer surplus minus damage due to emissions plus tax receipts.

An Industry in Perfect Competition

The industry is assumed to be a constant-cost industry, that is, the entry or exit of firms does not affect the cost functions of its firms.  Writing q for output volume and t for the tax rate, the cost function per period of each firm is: $c(q,e) = 10 + 0.1q^2 + 4q/e + qet$

Writing P for the market price and Q for total industry output volume, the market demand function per period (in inverse form) is: $P = 11 - 0.01Q$

Initial Position with No Emissions Tax

We assume that the industry is in equilibrium, with competition having driven the market price to the minimum point of the firms’ average cost (AC) curves so that their profit will be zero.  We have to find the output q of each firm at which average cost is minimised. Price P is then equal to average cost at that point.  Using the market demand function we can then find industry output Q, from which we can infer the number of firms (Q/q) and the total value of the industry’s sales (PQ).  We can also calculate the industry’s total variable costs excluding tax, which is of interest as an indicator of the total employment supported by the industry and its suppliers.

The industry’s emissions are simply Qe.  To calculate producer surplus we need the average variable cost (AVC) of one firm at the output determined above.  We then have everything needed to calculate the components of net welfare.

With Emissions Tax: Short Run

I take the short run to be a period in which no firm has changed its emissions ratio or exited the industry.  Thus any reduction in emissions resulting from the tax must be due to a reduction in output.  In a previous post, I noted that a reduction in emissions in response to a tax could be due to the introduction of abatement technology, to a reduction in output, or to a combination of the two.  Here I consider the implications of the timing of such a combined response: a reduction in output can usually be almost immediate, but the introduction of abatement technology will normally take time.

In the short run, having not fully adjusted to the tax, firms will not set their output to the minimum point of their average cost curves.  Instead, we must start from the more fundamental principle that they will set their output to the point at which their marginal revenue (the market price P) equals their marginal cost.  So from the cost function we obtain marginal cost in terms of firm output q and set this equal to P: this yields the inverse supply function for a firm.  Since the number of firms is known from the initial position, we can infer the market supply function relating P and Q.  From this in conjunction with the market demand function we can infer the values of P and Q, and hence q.  The remaining calculations are just as for the initial position.

With Emissions Tax: Long Run

I use the term ‘long run’ in a special sense: a period in which all firms have adjusted to the tax as fully as possible by changing their emissions ratio or exiting the industry.  This is not quite the Marshallian long run since the fixed component of the firms’ cost functions is assumed unchanged from the initial position (I leave for another day the important case in which abatement of emissions involves investment in fixed capital).

The method of calculation is as for the initial position except that the average cost curve now contains two unknowns: firm output q and the emissions ratio e.  So we must find the combination of values of those two variables which minimises average cost. Once we have found that minimum point, yielding q, e and P, the calculations proceed in the familiar way.

Results

Table 1 below sets out the results of the above calculations.  It can be seen that the industry’s emissions are reduced in the short run and further reduced in the long run.  Thus the primary purpose of the tax is achieved.  Also on the positive side, net welfare is increased in the short run and further increased in the long run.

Output per firm in the long run is the same as in the initial position.  Thus the long run reduction in emissions is achieved via a combination of a lower emissions ratio and a reduction in the number of firms.

Although the tax reduces the volume of output and increases its price per unit, these may be regarded as necessary side-effects of the emissions reduction.  However, the fact that both these changes slightly overshoot in the short run may be considered to impose an unnecessary (albeit temporary) detriment on consumers.  The need for the losses incurred by firms in the short run is questionable: by providing an incentive for firms to exit the industry they hasten the arrival of long-run equilibrium with few firms and profits restored to nil, but perhaps that process could be facilitated by other means.  These features of the short-run position after introducing a tax with minimal notice suggest that there could be advantage in giving a longer period of notice allowing firms to adjust before the tax comes into effect.  However, the way in which firms would respond during such a notice period would be difficult to predict.  It would depend on, among other things, the degree of certainty with which firms believe that the tax will be introduced, and the judgments firms make as to how many of their competitors will exit the industry.

It is important to note that the industry will not leave the short run one day and arrive at the long run the next.  Between the two is a transitional process in which some firms introduce abatement technology and others exit the industry.  Again, firms’ behaviour during this period is difficult to predict.  Perhaps some firms will make an early strategic decision to exit.  Alternatively, all firms may begin incurring the extra costs of abatement technology, and only as losses accumulate will some firms decide to leave the industry.

How does the tax effect employment in the long run?  To the extent that industry variable costs excluding tax are a good proxy for the employment supported by the industry, the direct effect is only a small reduction. Although many firms leave the industry, the effect on employment is largely offset by the extra costs per firm of reducing their emissions (staff made redundant by exiting firms may be re-employed by other firms).  Taking a broader view, however, the significant increase in industry sales value implies, given constant aggregate demand, a corresponding reduction in demand for other goods, adding to any reduction in employment.  Much therefore depends on how the government uses the tax receipts. If it uses them in ways which raise employment, either via government expenditure on goods and services, or via a cut in another tax, then the overall effect on employment could be neutral or even positive.

A Monopoly

The single firm’s cost function is: $C(Q,e) = 800 + 0.01Q^2 + 4Q/e + Qet$

Its inverse demand function is: $P = 13 - 0.01Q$

Initial Position with No Emissions Tax

Here e = 4 and t = 0.  Using the demand function we can express profit Pr as a function of Q only and then find the level of Q that maximises profit.  Price P, sales value and profit follow immediately.  We can also calculate variable costs, emissions (Qe), and then the components of net welfare.

With Emissions Tax at Rate Equal to Marginal Damage: Long Run

For this industry I omit the short-run analysis and proceed directly to the long run.  Here t = 1 while e, along with Q, is an unknown to be found.  So we find the levels of Q and e which maximise profit.  The only other difference from the calculations for the initial position is that we need both total variable costs (in order to calculate producer surplus) and variable costs excluding tax (as an indicator of employment).

With Emissions Tax at a Rate Less Than Marginal Damage: Long Run

We take the case t = 0.7.  The method of calculation is exactly as for t = 1.

Results

Table 2 shows the results of the above calculations.  As expected, the tax reduces emissions, partly by reducing output and partly by reducing the emissions ratio, and the higher tax rate reduces emissions by more.

The tax considerably reduces the firm’s profits, but they are still positive, and a reduction in the profits of a monopoly may be considered of little concern.  The small increase in price represents only a modest additional burden to consumers.  Since the reduction in sales value exceeds that in variable costs excluding tax, the net effect on employment may well be positive, even before consideration of how the government uses the tax receipts.

Net welfare is increased at either of the two tax rates, but is slightly higher when the rate is somewhat lower than the rate of marginal damage.  The reason for this is that, leaving aside the emissions damage, the initial position is sub-optimal relative to what could be achieved if output were set to equate price and marginal cost, rather than restricted so as to maximise the monopolist’s profit.  The theory of second best implies that a policy measure that would otherwise be optimal to address a market failure may not be optimal if another form of market failure is also present (1).  For a theoretical treatment of taxes to address externalities in the context of monopoly see Barnett (1980) (2).

A policy-maker selecting a tax rate in this situation might nevertheless want to look not only at net welfare but also at its separate components.  These are shown in Table 3 below.

It can be seen that the extra net welfare at the lower tax rate is due to an increase in producer surplus plus a smaller increase in consumer surplus, offset by an increase in damage due to emissions and a reduction in tax receipts.  The increase in producer surplus is exactly reflected in increased profits.  A policy-maker might reasonably conclude that, although it does not maximise net welfare, the tax rate equal to the rate of marginal damage is to be preferred.

The workings supporting the above results may be downloaded below (MS Word 2010 format).

References

1. Wikipedia Theory of the Second Best  https://en.wikipedia.org/wiki/Theory_of_the_second_best
2. Barnett, A H (1980) The Pigouvian Tax Rule under Monopoly  American Economic Review 70(5) pp 1037-41

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