Public Economics UC3M 2015 DEADWEIGHT COST AGZ 2.1 and Gruber

Deadweight cost: effect of policies on size of the pie Focus in efficiency analysis is on quantities, not prices

Government raises taxes for one of two reasons: 1.To raise revenue to finance public goods 2.To redistribute income But to generate $1 of revenue, welfare of those taxed falls by more than $1 because the tax distorts behavior How to implemente policies that minimize these efficiency costs?

Size of the welfare loss? Deadweight loss: the equivalent in $ to the utility reduction due to the tax, above the reduction cretaed by a lum-sump tax. We assume perfect competition. Remember the link between price, marginal utility and marginal cost. Deadweight loss related to the substitution effect. First, we analyze it using indifference curves. Suppose constant marginal cost, and a numerarie, Y, with price 1.

Y X G H slope: -p x A xAxA We introduce tax t Slope: -p x (1+t) B xBxB U0U0 U1U1

Equivalent variation: How many units of Y (i.e. money) is the consumer willing to pay to avoid the tax t?

Y X G H A xAxA Tax t B xBxB U0U0 U1U1 C xCxC D W What would happen with a lump-sum tax? F U2U2 E EgEg

In general, the higher the slope of the indifference curve the higher the deadweight loss, bacuse of the substitution effect We study next the same welfare loss but using the demand and supply curves.

p X O0O0 pxpx X(p,R) xAxA We introduce tax t p x (1+t)O1O1 xBxB B P N M

Consider the following demand curve: X c (p, U 1 ). This is the compensated demand function (Hicksian demand): demand of x that minimizes the expenditure while delivering a fixed level of utility U 1

p X O0O0 pxpx X(p,R) xAxA p x (1+t)O1O1 xBxB B P N M X c (p,U 1 ) xCxC Change in welfare: NBCM C Tax revenue: NBPM Excess burden : BCP EgEg

In empirical work: we take the regular demand functions as an approximation to the compensated demand function. This approximation is a good one if : small income effect-> small difference between X A and X C.

Method often used in empirical work: Assume that the segment BC in the compensated demand function is linear. Deadweight loss: the triangle of area We can write it as: The price elasticity of the compensated demand at C is

So that: Then deadweight loss is higher when: The higher the “compensated” price-elasticity The higher the before-tax expenditure in X The higher the tax rate (notice that we have t square)

P Q P2P2 P1P1 Q1Q1 Q2Q2 D1D1 S1S1 S2S2 B A C DWL P Q P2P2 P1P1 Q1Q1 Q2Q2 D1D1 S1S1 S2S2 B A C (a) Inelastic Demand(b) Elastic demand 50¢ Tax 50¢ Tax Figure 2 Demand is fairly inelastic, and DWL is small. Demand is more elastic, and DWL is larger.

This point about DWL rising with the square of the tax rate can be illustrated graphically Marginal deadweight loss is the increase in deadweight loss per unit increase in the tax.

P Q P2P2 P1P1 Q1Q1 Q2Q2 D1D1 S1S1 S2S2 B A C S3S3 Q3Q3 P3P3 D E $0.10 Figure 3 The first $0.10 tax creates little DWL, ABC. The next $0.10 tax creates a larger marginal DWL, BCDE.

The insight that deadweight loss rises with the square of the tax rate has implications for tax policy with respect to: –Preexisting distortions –Progressivity –Tax smoothing

Taxation and economic efficiency Deadweight loss and the design of efficient tax systems Preexisting distortions are market failures that are in place before any government intervention. –Externalities or imperfect competition are examples. Figure 4Figure 4 contrasts the use of a tax in a market without any distortions and in one with positive externalities.

P Q Q1Q1 D1D1 S1S1 S2S2 B A C P Q Q1Q1 D1D1 S1S1 S2S2 E D F SMC G H Q0Q0 No positive externalityPositive externality Q2Q2 Q2Q2 Figure 4 In a market with a preexisting distortion, taxes can create larger (or smaller) DWL.

Imposing the tax in the first market, without externalities, results in a modest deadweight loss triangle equal to BAC. When an existing distortion already exists where the firm is producing below the socially efficient level, the deadweight loss is much higher. The marginal deadweight loss from the same tax is now GEFH. Of course, if there were negative externalities, such a tax would actually improve efficiency.

Taxation and economic efficiency Deadweight loss and the design of efficient tax systems This insight about deadweight loss also demonstrates that a progressive tax system can be less efficient. Consider two tax systems – one a proportional 20% payroll tax, and the other a progressive tax that imposes a 60% rate on the rich, and a 0% rate on the poor. Figure 5Figure 5 shows these cases.

Wage (W) Hours (H) W 2 =11.18 W 1 =10.00 H 1 =1,000H 2 =894 D1D1 S1S1 S2S2 B A C Wage (W) Hours (H) W 2 =22.36 W 1 = H 1 =1,000H 2 =894 D1D1 S1S1 S2S2 S3S3 W 3 =23.90 H 3 =837 E D F G I Low Wage WorkersHigh Wage Workers Figure 5 DWL increases with the square of the tax rate. Smaller taxes in many markets are better.

Taxation and economic efficiency Deadweight loss and the design of efficient tax systems Under the proportional system the efficiency loss for society is the sum of two deadweight loss triangles, BAC and EDF. Under the progressive system, the efficiency loss is the triangle GDI – that is, it adds the area GEFI but does not include BAC. Table 1Table 1 puts actual numbers to the picture.

Table 1 Low wage worker Panel A High wage worker Panel B Tax Rate Below $10,000 Tax Rate Above $10,000 Hours of labor supply Deadweight Loss from Taxation Hours of labor supply Deadweight Loss from Taxation Total Deadweight Loss No Tax (H 1 )0 00 Proportional Tax20% 894 (H 2 )$ (area BAC) 894 (H 2 )$ (area EDF) $ (BAC + EDF) Progressive Tax0%60%1000 (H 1 )0837 (H 3 )$ (area GDI) $ (EDF + GEFI) A lower proportional tax creates less DWL than the higher progressive tax.

In this case, a proportional tax is more efficient. The large increase in deadweight loss arises because the progressive tax is levied on a smaller tax base. In order to raise the same amount of revenues on a smaller base, the tax rate must be higher meaning a higher marginal DWL. This illustrates the larger point that the more one loads taxes onto one source, the faster DWL rises. The most efficient tax systems spread the burden most broadly. Thus, a guiding principle for efficient taxation is to create a broad and level playing field.

The fact that DWL rises with the square of the tax rate also implies that government should not raise and lower taxes, but rather set a long-run tax rate that will meet its budget needs on average. For example, to finance a war, it is more efficient to raise the rate by a small amount for many years, rather than a large amount for one year (and run deficits in the short-run). This notion can be thought of as “tax smoothing,” similar to the notion of individual consumption smoothing.

Example It may be easier to approximate the DWL from point B. In this case, the equivalent eq. to (11) is Where the elasticity is calculated at B and q x =p x (1+t) and t~=t/(1+t) Example on taxes on cigarretes. Demand (in 2003) $7.625 m Tax rate 250% Price-elasticity (compensated demand) Using the fromula we get E g =$1731 m This is 32% of the total tax revenue

Application: savings, labor supply Y= consumption today, X= consumption tomorrow. If we know the elasticity of savings we can apply the above formula. Y= total consumption. H=leisure. We can estimate the DWL from labor taxation Economists often find that the DWL associated to female labor supply is higher than the DWL from men’s labour supply. Do you know why?

What happens if there are several taxes Things change 1.The total DWL is not the sum of the DWL from each tax. 2.If there is already a tax, introducing a new one might improve things (reduce total DWL)

EXAMPLE. Two goods x and y. They are substitutives Constant marginal costs. Demand curves Suppose zero income effect so that they coincide with the compensated demand curves The good x is taxed at the rate t x. We are considering to introduce a tax on y, t y.

pxpx xy pypy X( p x (1+ t x ), p y ) x0x0 pxpx P x (1+t x ) x1x1 Y( p x (1+ t x )) y1y1 y2y2 pxpx P y (1+ t y ) X( p x (1+ t x ), p y (1+ t y )) x2x2

Do we have a similar formula to (11) to compute DWL here? We need a linear approximatio to the area ABC, i.e. we need to assume that the demand is linear. In this case area ABC is We do the same with good y. One can prove that a good approximation is:

In general, if there are n goods the formula is