0.1 Tax Incidence

0.1. TAX INCIDENCE

0.1

1

Tax Incidence

An important implication of the demand/supply model is that a tax imposed on

the sale or purchase of the product will be born in part by sellers and in part by

buyers. Furthermore, the effects of the tax on buyers and sellers is independent

of whether the tax is nominally placed on selling or on buying. We begin with

the equilibrium established above, and we impose a tax on the selling of the

good. We consider an excise tax first, and then a value-added tax.

(%i40)

kill(all)$ fpprintprec : 5$

DQ(p, a, ED) := a*p^ED$

DP(q, a, ED) := (q/a)^(1/ED)$

SQ(p, b, ES) := b*p^ES$

SP(q, b, ES) := (q/b)^(1/ES)$

(%i6)

[a0, ED0, b0, ES0] : [1200, -4, 40, 0.75]$

qE : find_root(DP(q, a0, ED0) - SP(q, b0, ES0) = 0, q, 0.01, 500 );

pE : find_root(DQ(p, a0, ED0) - SQ(p, b0, ES0) = 0, p, 0.01, 500 );

(%o7)

68.437

(%o8)

2.0463

0.1.1

Excise Tax

The results of an excise tax, X0 = 0.50 are as follows: the quantity traded falls

from 68.437 units to 57.82 units, the price paid by buyers rises from 2.0463 to

2.1344, and the price received by sellers falls from 2.0463 to 1.6344. That is, the

price paid rises by 0.0881, and the price received falls by 0.412. Thus, for this

particular demand and supply pair, the tax falls disproportionately on sellers.

The equality in the second line below shows the willingness-to- pay function

on the left-hand side and the supply price on the right-hand side. The latter

contains two parts: the original function, which reflects non-tax costs, and the

per-unit excise tax.

(%i9)

X0 : 0.5$

qX0 : find_root(DP(q, a0, ED0) = SP(q, b0, ES0) + X0, q, 0.01, 500 );

pDX : DP(qX0, a0, ED0 );

pSX : SP(qX0, b0, ES0);

[pDX - pE, pSX - pE];

(%o10)

57.82

(%o11)

2.1344

(%o12)

1.6344

(%o13)

[0.0881, ?0.412]

2

If the tax had been placed on the buying of the good rather than the selling of

the good, the price that buyers would be willing and able to pay to sellers at

each quantity would be the original willingness to pay less the tax, X0. Thus

the left-hand side of the expression above would become DP (q, a0, ED0) ? X0

and the right-hand side would revert to SP (q, b0, ES0). All solution values

would be the same.

The graph shows the upward shift in the supply curve and the resulting equilibrium values. The supply curve shifts upward, parallel to the initial curve, by

the amount of the excise tax, X0.

If the tax had been placed on buyers, the demand curve would shift downward

by X0, reflecting the lower net price that buyers are willing and able to pay to

sellers.

(%i14)

pGrossNew : if q < qX0 then pDX else 0$

pNetNew: if q < qX0 then pSX else 0$

wxdraw2d( user_preamble = "set key bottom right",

xlabel = "Quantity", ylabel = "$ per unit",

yrange = [0, 1.25*pDX], explicit(pGrossNew, q, 0, 1.25*qE),

explicit(pNetNew, q, 0, 1.25*qE), key = "Demand",

explicit(DP(q, a0, ED0), q, 0, 1.25*qE),

key = "Initial Supply", line_width=2,

explicit(SP(q, b0, ES0), q, 0, 1.25*qE),

color = "dark-gray", key="Tax-laden Supply",

explicit(X0 + SP(q, b0, ES0), q, 0, 1.25*qE))$

kill(pGrossNew, pNetNew)$

(%t16)

0.1.2

Ad Valorem Tax

To illustrate the ad-valorem tax, we impose a 40% tax rate, V 0 = 0.4. We

impose this tax on the buyers. As a result, the tax-laden price is (1 + V 0 times

0.1. TAX INCIDENCE

3

the pre-tax price. As a result the amount that buyers are willing and able to

.

pay to sellers equals DP (q,a0,ED0)

1+V 0

The values that result from imposing this tax appear below. The quantity

traded falls to 55.335 units per time period. The tax-inclusive price rises to

2.1580, and the net price (received by sellers) falls to 1.5414. Again, given this

particular demand and supply curve pair, the tax falls disproportionately on

sellers.

(%i18)

V0 : 0.40$

qV0 : find_root(DP(q, a0, ED0)/(1 + V0) = SP(q, b0, ES0), q, 0.01, 500 );

pDV : DP(qV0, a0, ED0 );

pSV : SP(qV0, b0, ES0);

[pDV - pE, pSV - pE];

(%o19)

55.335

(%o20)

2.158

(%o21)

1.5414

(%o22)

[0.112, ?0.505]

The algebra above implies that the difference between the pre-tax (net) price

and the tax-laden (gross) price is proportional to the price. Thus the initial

demand curve and the tax-laden demand curve in the graph below are not

parallel.

If the tax had been imposed on the selling of the product rather than on the

buying of the product, then the tax-laden supply curve would be (1+V 0) higher

than the supply curve in the graph. Given the functions involve here, the fact

that the two are not parallel would be much more apparent than in the case of

the demand curves.

(%i23)

pNewGross : if q < qV0 then pDV else 0$

pNewNet : if q < qV0 then pDV/(1+V0) else 0$

wxdraw2d( yrange = [0, 2.5*pDV],

explicit(pNewGross, q, 0, 1.25*qE),

explicit(pNewNet, q, 0, 1.25*qE), key="Initial Demand",

explicit(DP(q, a0, ED0), q, 0, 1.25*qE),

line_width = 2, key = "Tax-laden Demand",

explicit(DP(q, a0, ED0)/(1 + V0), q, 0, 1.25*qE),

color = "dark-gray", key = "Supply",

explicit(SP(q, b0, ES0), q, 0, 1.25*qE) )$

kill(pNewGross, pNewNet)$

4

(%t25)

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