Theory of Measurement of Dead Weight Loss

1 year ago

Atiqur Rahman

24 minutes

Introduction to Dead Weight Loss

At the very centre of public finance analysis lies the distortions introduced by taxation. Tax-induced diminutions in economic efficiency are termed deadweight losses or the additional burdens of taxation, the latter representing the added cost to taxpayers and society of raising revenue through taxes that distort economic decisions.

It explains how a society loses in terms of its total surplus when it is taxed. In the general context, the excess burden can be caused by market conditions other than perfect competition, like monopoly.

Whenever a commodity is taxed, it distorts the market in such a way that consumers are discouraged from consuming and producers are discouraged from producing, and the equilibrium quantity in the market falls, hence reducing the total surplus in the market.

The tax leads to an excess burden in the market due to decreased quantity traded. The excess burden is clearly defined only in the framework of an exact evaluation or conceptual experiment. If one simply seeks “the” excess burden of a particular tax policy, there are many equally plausible answers, so in order to obtain a unique meaning, it is necessary to be more specific.

E.g. the extra burden of a 10% tax on retail sales differs not only with the initial conditions of the tax system but also with the direction of change, i.e. whether the tax is being added or removed.

Taxes almost perpetually have excess burdens because obligations of tax are functions of individual behaviour. The alternative, pure lump-sum taxes, are attractive from an efficiency perspective but are of limited usefulness precisely because they do not vary with indicators of ability to pay, such as income or consumption, that are functions of taxpayer decisions.

Thus, even though analysis of tax often starts with the simple case of a representative household, it is household heterogeneity and the inability fully to observe individual differences that justify the restrictions commonly imposed on the set of tax instruments. Designing an ideal tax system means keeping tax distortions to a minimum, subject to limitations presented by the need to raise revenue and keep an equitable tax burden.

In the theory of microeconomics, we study the interactions between different economic agents like consumers, producers, social planners etc., to come to a general equilibrium in society. This determines the allocation of resources in such a manner that the scarce resources, having alternative uses, can best satisfy the wants of society. The allocation, in theory, subjects itself to answer three central questions, i.e., what to produce, how to produce and how much to produce.

Economists and the one who makes policy often want to recognize not only whether particular policies make people better off or worse off — they also need to quantify how much better off or worse off different consumers are. The tools are developed that permit us to measure the welfare of consumers in objective terms without us having to measure happiness directly.

This way of thinking about welfare effects from institutional or policy changes allows us then to address the following question: Is it at least in principle possible to compensate those who lose from the policy with part of the gains accruing to those who gain from the policy?

If the answer is yes, then, at least in principle, there is a way to make the world more efficient — to make some people better off without making anyone worse off. If the answer is no, on the other hand, then we know that the new situation will be less efficient.

Put differently, if the victors from a policy gain more than the failures lose, the policy could, in principle, be accompanied by a compensation scheme that would result in unanimous approval of the policy.

This part of the microeconomic theory connects to the optimal allocation of resources. An allocation is said to be optimal or best if it satisfies the condition of being Pareto. Real-world policies come, at best, with imperfect compensation schemes – and thus rarely enjoy unanimous approval.

We should not accept a policy right away without comparing the benefits to costs associated with it – because, in some instances, we may, in fact, place more weight on the decline in the welfare of those who lose than on the gains in the welfare of those who win.

The Pareto optimality condition states that a given allocation is Pareto optimal if there is no other allocation where anyone can be made better off without making someone else worse off. The Pareto optimal allocation in general equilibrium in a two good two consumer case is represented by the Edgeworth box diagram.

The efficiency of an allocation can thus be measured using the Pareto optimality criterion. The implementation of a certain policy thus depends on weighing who loses and who gains.

Inefficiency Of The Taxes

Measuring Consumer Welfare in Monetary Terms

We will begin our analysis of the measurement of consumer welfare by quantifying how much better off or worse off consumers are for being able to purchase goods voluntarily at given market prices. Put differently; we will ask how much better off a consumer is when he participates in the market rather than being excluded from it.

This will lead us to define the terms like marginal and total willingness to pay as well as consumer surplus. We will then proceed to demonstrate how policymakers might analyze the impact of particular proposals on consumers when those proposals change the relative prices in an economy.

In the process, we will also learn the importance of recognizing the difference between income and substitution effects — and how the substitution effect contributes to dead weight losses for society while the income effect does not.

Consumer Surplus

Marginal Willingness to Pay:

Let’s ask how much a consumer would be willing to pay for that gallon given that he ended up at an optimal bundle, say A.

For the first gallon, we can measure this willingness to pay by finding the slope of his indifference curve — the marginal rate of substitution — at 1 gallon. Suppose that this slope is −20. This tells us that the consumer was willing to trade $20 worth of other consumption for the first gallon of gasoline.

We can then proceed to the second gallon and find the marginal rate of substitution at 2 gallons. Suppose that it is −19. This tells me that he would have been willing to give up $19 of other consumption to get the second gallon of gasoline and so on.

For each gallon of gasoline, with the marginal rate of substitution at bundle A is equal to the price of gasoline. At the end of this exercise, we will get the consumer’s marginal willingness to pay (MWTP) for each of the gallons of gasoline he consumed and all the additional gallons that he chose not to consume.

In the lower panel of the above diagram, we can then simply plot gallons of gasoline on the horizontal axis and dollars on the vertical. The marginal willingness to pay curve for a consumer who ends up on the indifference curve containing bundle A is then simply plotted by plotting the dollar values of the MRS at each gallon of gasoline.

Marginal Willingness to Pay Curves and Substitution Effects:

There is, however, a slightly different way of deriving marginal willingness to pay curves that builds more directly on the material we have covered above. The top panel of the diagram below begins with the same initial budget and optimal bundle A as we started with in diagram 1 above.

Instead of directly identifying the marginal rates of substitution on the indifference curve that contains bundle A, however, we now imagine a price increase from p to p’ and then illustrate the compensated budget and get the substitution effect. We then illustrate bundle B and bundle C either to the right or left of B, depending on whether the good on the horizontal axis is a normal or inferior good.

Here we are assuming that gasoline is a normal good and thus place bundle C to the left of B. Consider the diagram below. Specifically, in the lower panel of diagram 2, we illustrate the quantity consumed at bundle A at the original price p and the quantity consumed at bundle B at the new price p′ (when the consumer receives compensation to make me as well off as he was originally).

But notice that all we are doing is plotting the slope of the indifference curve that contains bundle A at two different quantities. This can be done for many different price changes — each time finding the corresponding compensated budget and the new optimal bundle on that compensated budget. When we do this, the marginal rates of substitution are plotted at the different quantities, which leaves us with the same marginal willingness to pay curve as shown in the lower panel of diagram 1.

For this reason, the marginal willingness to pay curve is often mentioned as the compensated demand curve, while the regular demand curve is called as the uncompensated demand curve sometimes. The only time when the two curves that are the own-price demand curve and the marginal willingness to pay curve are the same is when there are no income effects with respect to the good whose demand curve we are drawing — and that is true only for tastes that are quasilinear in that good.

Marginal Willingness to Pay Curves and Substitution Effects

Total Willingness to Pay and Consumer Surplus:

We began by asking how much the consumer might be willing to pay for the opportunity to be able to purchase gasoline at the market price rather than not being able to get access to the gasoline market. The response can now be read off the marginal willingness to pay curve we have just derived once we have identified two further concepts in the marginal willingness to pay graph.

First, we need to identify my total willingness to pay for all of the gasoline the consumer is purchasing in the market, and second, we need to subtract from this the amount that the consumer actually had to pay in the market. The difference between these two amounts is how much better off he is for being able to participate in this market — how much more he would have been willing to pay than he actually had to pay.

Diagram 3 replicates the marginal willingness to pay curve we just derived — illustrating the consumer’s marginal willingness to pay for each of the gallons of gasoline that he is consuming (and for each of the gallons that he is not consuming), given that he ends up consuming at bundle A when he faces the market price p. His total willingness to pay is equal to his marginal willingness to pay for the first gallon plus his marginal willingness to pay for the second gallon, etc.

Total Willingness to Pay and Consumer Surplus

Distortions of the Taxes

Governments use taxes to raise revenues that, in turn, fund expenditures on a variety of government programs. These programs may have enormous benefits, but — to the extent that they are funded through taxes — they come at an economic cost which economists refer to as the dead weight loss from taxation.

Often students think that pointing this out makes all economists raving anarchists – that being an economist means being against all taxes and all government expenditures that are funded through taxes.

But recognizing the economic cost of taxation does not mean that one has to oppose to all taxes. Like recognizing a cost to going to the movies implies that one is against going to the movies.

After all, the benefits from certain government programs may well outweigh these costs, just as the enjoyment of the movie might outweigh the cost of watching it. It does, however, lead us to think more carefully about the relative cost of different kinds of taxes, and we can now use the tools we have developed to illustrate how such costs can be measured.

To see what makes a particular tax costly and to see how we can measure this cost objectively, we will try to answer the following question: How much would a taxed individual be willing to bribe the government to get the tax reduced?

We will then compare this amount to the amount that the individual is actually paying in tax. If the maximum size of the bribe the individual is willing to pay is larger than her actual tax payment, then we know that there exists, at least in principle, a way to raise more revenue from the individual without making her worse off.

The variance between the hypothetical bribe and the payment of tax is a measure of how much more the government could have raised without making anyone worse off — and it is our measure of deadweight loss.

One way to think of deadweight loss from taxation is to imagine the government collecting taxes in a bucket that has a hole in it – and as the government passes the bucket, the bucket leaks. What remains in the bucket is what the government gets to use to provide public programs and services – and what leaks from the bucket is the dead weight loss that no one gets but that we could get to if we just found a better bucket. The challenge is to find a bucket – a tax – that has a small hole so that the leakage is minimized. But why is there a hole in the first place?

Some Intuition on the Dead Weight Loss and Inefficiency of Taxation- The question is not rhetorical – and the answer is not immediately obvious. In fact, students are often puzzled at this point. Why would anyone ever be willing to pay more in a bribe to get rid of a tax than she is paying in taxes when the tax is in place? Why do we think that we can find another tax that will raise more revenue while not making people worse off?

Consider the following extreme example. I like to drink soda, and I especially like to drink the imported soda variety. Suppose the domestic soda company X convinces the government to impose a large tax on imported soda and suppose that this leads to a sufficient increase in the domestic price of imported soda to cause me to switch to domestic.

Notice that because I have substituted away from imported and towards domestic soda, I end up paying no tax at all. At the same time, I have clearly been made worse off by the imposition of a tax on imported soda and would therefore be willing to pay something to get the government to abolish this tax – despite the fact that I do not pay any of the tax when it is imposed.

With the government not raising any revenue and me being made worse off, we have identified a “bucket” that has not bottom – no tax revenue from me is actually reaching the government even though the imposition of the tax is making me worse off. Stated otherwise, society has been made worse off without anyone getting a benefit — and that is called deadweight loss. It is also what makes taxes inefficient.

In theory, we defined a situation to be inefficient if there is a way to change the situation and thus make someone better off without making anyone worse off. The tax on imported soda is inefficient because the government could have raised more money from me without making me any worse off by thinking of a different way of raising money, finding a different “bucket” that doesn’t leak so much.

For instance, they could have just come by my house and taken some money, leaving the price of imported soda unchanged and thus not giving me the incentive to switch to a domestic brand just to avoid a tax.

The example, though extreme, gives us an initial insight into what it is about taxes that makes taxes costly. By altering the relative prices in an economy, taxes cause consumers, workers and savers to substitute away from taxed goods and services and toward untaxed goods and services.

To the extent that this substitution activity happens solely because of a change in opportunity costs — i.e. to the extent to which taxes give rise to substitution effects, taxes are distortionary and inefficient ways of raising revenues.

Many real-world examples may be less extreme — they may lead us to consume less of the taxed good and more of other goods without causing us to eliminate our consumption of particular taxed goods.

But the basic intuition remains: To the extent to which taxes change opportunity costs and thus cause us to alter our consumption plans solely because of those changed opportunity costs, we are worse off without contributing to the government’s effort to raise revenues — and society has incurred a dead weight loss.

We can now use the tools we have developed to show more formally that this entire dead weight loss happens because of substitution effects – which are, therefore, the underlying cause of the leak in the “bucket”.

Theory of Measurement of Dead Weight Loss

Measuring Deadweight Loss

The theory of consumer surplus is very tidy in the case of quasilinear utility. Even if utility is not quasilinear, the consumer’s surplus may still be a reasonable measure of the consumer’s welfare in many applications. Usually, the errors in measuring demand curves outweigh the approximation errors from using consumer surplus.

In the Figure below, we have illustrated the change in consumer surplus associated with a change in price. The change in consumer surplus is the difference between two roughly triangular regions and will therefore have a roughly trapezoidal shape. The trapezoid is further composed of two sub-regions, the rectangle indicated by R and the roughly triangular region indicated by T.

Change in Consumer’s Surplus:

The change in consumer surplus will be the difference between two roughly triangular areas and thus will have a roughly trapezoidal shape.

The rectangle measures the loss in surplus due to the fact that the consumer is now paying more for all the units he continues to consume. After the price increases, the consumer continues to consume x“ units of the good, and each unit of the good is now more expensive by p“−p`. This means he has to spend (p“-p`)x“ more money than he did before just to consume x“ units of the good.

But this is not the entire welfare loss. Due to the increase in the price of the x-good, the consumer has decided to consume less of it than he was before. The triangle T measures the value of the lost consumption of the x-good. The total loss to the consumer is the sum of these two effects: R measures the loss from having to pay more for the units he continues to consume, and T measures the loss from the reduced consumption.

Producer’s Surplus

The demand curve measures the amount that will be demanded at each price; the supply curve measures the amount that will be supplied at each price. Just as the area under the demand curve measures the surplus enjoyed by the demanders of a good, the area above the supply curve measures the surplus enjoyed by the suppliers of a good.

We’ve referred to the area under the demand curve as the consumer surplus. By analogy, the area above the supply curve is known as the producer’s surplus. The terms consumer’s surplus and producer’s surplus are somewhat misleading since who is doing the consuming and who is doing the producing really doesn’t matter. It would be better to use the terms “demander’s surplus” and “supplier’s surplus,” but we’ll bow to tradition and use the standard terminology.

Suppose that we have a supply curve for good. This simply measures the amount of a good that will be supplied at each possible price. The good could be supplied by an individual who owns the good in question, or it could be supplied by a firm that produces the good. We’ll take the latter interpretation so as to stick with the traditional terminology and depict the producer’s supply curve in the diagram below.

If the producer is able to sell x∗ units of product in a market at a p∗ (price), what would be the surplus she enjoys? It is most convenient to conduct the analysis in terms of the producer’s inverse supply curve, ps(x). This function measures what the price would have to be to get the producer to supply x units of the good.

Calculating Gains and Losses

If we have estimates of the market demand and supply curves for goods, it is not difficult, in principle, to calculate the loss in consumers’ surplus due to changes in government policies.

For example, suppose the government decides to change its tax treatment of some goods. This will result in a change in the prices that consumers face and, therefore, a change in the amount of the good that they will choose to consume. We can calculate the surplus of consumers related to different tax proposals and see which tax reforms generate the smallest loss.

This is often useful information for judging various methods of taxation, but it suffers from two defects.

First, as we have indicated earlier, the surplus of consumer’s calculation is only valid for special forms of preferences— namely, preferences representable by a quasilinear utility function. We claimed earlier that this kind of utility function may be a rational approximation for goods for which fluctuations in income lead to small changes in demand, but for goods whose consumption is closely related to income, the use of consumer surplus may be inappropriate.

Second, the calculation of this cost effectively bumps together all the consumers and producers and generates an estimate of the “cost” of a social policy only for some mythical “representative consumer.” In many cases, it is desirable to know not only the average cost across the population but who bears the costs. The political success or failure of policies often depends more on the distribution of gains and losses than on the average gain or loss.

Background of Theory of Measurement of Deadweight Loss

Although the theory of measurement of deadweight loss has a long and colourful history that dates back to the 19th-century contributions of Jules Dupuit (1844) and Fleeming Jenkin (1871/72), economists seldom measured actual deadweight losses prior to the pioneering work of Arnold Harberger in the 1950s and 1960s.

In two significant papers published in 1964, Harberger (1964) derived the approximation used to measure deadweight loss and (1964b) applied the method to estimate deadweight losses due to income taxes in the U.S. Harberger shortly thereafter (1966) produced estimates of the welfare cost of U.S. capital taxes.

A major practical difficulty in measuring the excess burden of a single tax or of a system of taxes is that excess burden is a function of demand interactions that are potentially very difficult to measure.

For example, a tax on labour income is expected to affect hours worked but may also affect the accumulation of human capital, the intensity with which people work, the timing of retirement, and the extent to which compensation takes tax-favoured (e.g., pensions, health insurance, and workplace amenities) in place of tax-disfavored (e.g., wage) form.

In order to estimate the excess burden of a labour income tax, it is, in principle, necessary to estimate the effect of the tax on these and other decision margins. Similar difficulties are related to assessing the excess burdens of most other taxes. In practice, it can be hard to get reliable estimates of the impact of taxation on just one of these variables.

The analysis of excess burden and optimal taxation is one of the oldest subjects in applied economics, yet research continues to offer important new insights that build on the original work of Dupuit, Jenkin, Marshall, Pigou, Ramsey, Hotelling, and others.

Fundamentally, it remains accurate that departures from marginal cost pricing are related to excess burden, that the magnitude of excess burden is roughly proportional to the square of any such departure, and that efficient tax systems are ones that minimize excess burden subject to achieving other objectives.

The contribution of modern analysis is to identify new and important reasons for prices and marginal costs to differ, to assess their practical magnitudes, and to consider their inferences for taxation.

One of the major developments of the last fifty years is the widespread application of rigorous empirical methods to analyze the efficiency of the tax system. Empirical work not only assists the formation and analysis of economic policy but also plays a critical role in distinguishing important from less-important theoretical considerations, thereby contributing to further theoretical development.

Properly executed, empirical analysis is not only consistent with the welfare theory that underlies normative public finance but also takes the theory further by testing its implications and offering reliable measurement of parameters that are critical to the assessment of tax systems.

Theory of Measurement of Dead Weight Loss

Introduction to Dead Weight Loss

Inefficiency Of The Taxes

Measuring Consumer Welfare in Monetary Terms