## Chris Jones

Print publication date: 2005

Print ISBN-13: 9780199281978

Published to Oxford Scholarship Online: July 2005

DOI: 10.1093/0199281971.001.0001

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# (p.278) Appendix

Source:
Applied Welfare Economics
Publisher:
Oxford University Press

# 1 DECOMPOSING THE CONVENTIONAL SHADOW PRICES

This section confirms the Sieper welfare decomposition in (2.12) by decomposing the actual and compensated shadow prices over endogenous changes in prices and income.

## 1.1 Actual (Uncompensated) Shadow Prices

The shadow price in (2.8) can be decomposed, as:

(A.1)
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with:
(A.2)
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Proof. Since private production is a function of market prices, we can write (2.8), as:

(A.3)
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The relative price changes must satisfy the market clearing conditions for the N markets, and they are solved using the following system of equations:
$Display mathematics$
with εk = 1, and εi = 0 ∀ik, where aggregate consumer income changes by:
$Display mathematics$
1After substitution, the price changes, are:
(A.4)
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2Equation (A.1) is obtained by using (A.4) to rewrite (A.3). □

## (p.279) 1.2 Compensated Shadow Prices

The compensated shadow price in (2.11) can be decomposed, as:

(A.5)
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where δjk is the compensated price change for each good j.

Proof In the compensated equilibrium the shadow price in (2.11) expands to:

(A.6)
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The equilibrium price changes will satisfy the N market clearing conditions, where:
$Display mathematics$
with εk = 1 and εi = 0 ∀ik. For each good j, we therefore have:
(A.7)
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which is used to rewrite (A.6) as the shadow price in (A.5). The generalized Hatta decomposition in (2.12) is confirmed by multiplying this shadow price in (A.5) by the revised shadow value of government revenue in (2.14) to obtain the uncompensated shadow price in (A.1). □

# 2 DERIVING THE CONVENTIONAL WELFARE EQUATION WHEN THERE ARE EXTERNALITIES

After totally differentiating the private sector budget constraint, and using the first-order conditions for price-taking consumers and firms, the dollar change in social welfare becomes:

(A.8)
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where 1 − g k is the share of the externality that falls on the private sector. The net change in government revenue is determined by:
(A.9)
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We obtain the conventional welfare equation in (4.20) by solving the revenue transfers dL using (A.9), and substituting them into (A.8).

# (p.280) 3 EFFECTIVE RATES OF PROTECTION AND DOMESTIC RESOURCE COST RATIOS

The relationship between the ERP (effective rate of protection) and the DRCR (domestic resource cost ratio) is obtained using the conventional shadow price in (5.6) and the price–tax relationships in (5.1) and (5.2) to write the shadow profit in (5.41), as:

(A.10)
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After rearranging terms, we have:
(A.11)
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where:
(A.12)
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This becomes equation (5.44) by using the ERPj in (5.40) and the DRCRj in (5.43).

# 4 ACTUAL WELFARE CHANGES IN THE ABSENCE OF (HYPOTHETICAL) LUMP-SUM TRANSFERS

The revised shadow price of any good k in (6.1) is obtained using the conventional welfare equation in (2.7), as:

(A.13)
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These are conventional Harberger terms isolated using (notional) lump-sum transfers to balance the government budget. The first term is the conventional shadow price of good k (S k) while the second term in (6.1) is obtained by dividing and multiplying the last term in (A.13) by the revenue transfers from the tax changes (∂L/∂t i) and using the marginal excess burden of taxation defined in (6.2).

It is also possible to derive this revised shadow price as direct changes in private surplus by ruling out lump-sum transfers to balance the government budget and (p.281) relying instead on endogenous changes in distorting taxes, where dollar changes in utility are obtained using (2.5) with dL = 0, as:

$Display mathematics$
By using this welfare equation we can derive the revised shadow price, as:
$Display mathematics$
The endogenous tax changes balance the government budget and are solved using (6.3). It is the same change in welfare as (A.13), which is confirmed by adding and subtracting the notional lump-sum transfers for good k (∂L/∂z k), where:
$Display mathematics$
This is the same welfare change as the shadow price in (A.13) because the Harberger terms are derived as:
$Display mathematics$

# 5 COMPENSATED WELFARE CHANGES IN THE ABSENCE OF (HYPOTHETICAL) LUMP-SUM TRANSFERS

The compensated revised shadow price of any good k in (6.5) is obtained from the conventional welfare equation in (2.9), as:

(A.14)
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These are conventional Harberger terms isolated using (notional) lump-sum to hold consumer utilities constant. The first term is the conventional shadow price of good k (Ŝk), while the second term in (6.5) is obtained by dividing and multiplying the last term in (A.14) by the compensating transfers for each tax $( ∂ L ^ / ∂ t i )$, and using the marginal excess burden of taxation defined in (6.6).

This welfare change can also be obtained by measuring the direct changes in net government revenue when notional lump-sum transfers are ruled out $d L ^ = 0$, where:

$Display mathematics$
(p.282) The revised compensated shadow price of good k becomes:
$Display mathematics$
where the welfare changes are no longer conventional Harberger terms. Instead, they are direct changes in net government revenue. The compensating tax changes are solved using (6.7), and as confirmation this provides the same welfare change in (A.14), we subtract and add the notional lump-sum transfers for good k (∂L/∂z k), where:
$Display mathematics$
The conventional welfare changes are derived as:
$Display mathematics$

# 6 DECOMPOSING THE REVISED SHADOW PRICES

The generalized Hatta decomposition in (6.9) can be confirmed by decomposing the shadow prices as functions of prices and money income when taxes have the same marginal excess burden, with mebi = mebdi.

The revised shadow price of any good k in (6.4) becomes:

(A.15)
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with:
(A.16)
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3

Proof By solving partial ∂L/∂z k using (2.6), we can write the revised shadow price in (6.4), as:

(A.17)
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(p.283) where the price changes in (A.4), and the welfare decomposition in (2.12) and (2.14), makes this:
(A.18)
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with:
(A.19)
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The price changes are solved using the goods market clearing conditions, as:
(A.20)
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4where this allows us to write the marginal excess burden for tax d in (A.19), as:
(A.21)
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5with:
(A.22)
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The revised shadow price in (A.15) is obtained by using the price changes in (A.20) and the welfare decomposition of the meb in (A.21) to rewrite (A.17). □

When taxes have the same marginal excess burden, with $m e ^ b i = m e ^ b d ∀ i$, the compensated shadow price in (6.8) can be decomposed as:

(A.23)
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proof By using (2.10) to solve the compensating transfers for extra output of good k we can write the revised shadow price in (6.8), as:

(A.24)
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The shadow price in (A.23) is obtained by using the compensated price changes in (A.7), and the welfare decomposition in (6.9) is confirmed by using the compensated shadow price for good k in (A.23) to rewrite its uncompensated shadow price in (A.15).□

# (p.284) 7 DECOMPOSING THE MCF

Using the measure of the marginal excess burden in (A.19) and the price changes in (A.20), we can write the conventional MCF, as:

(A.25)
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This collapses to the MCF in (7.9) when producer prices are constant and there is no public production.

# 8 PROJECT SHADOW PROFITS WITH PUBLIC PRODUCTION

In the presence of public production the revised shadow profit for the project in (7.13) is obtained using the revised shadow prices in (7.5), as:

$Display mathematics$
where z(dp/dz k) = z(∂p/∂z k) + z(∂p/∂R)MRT is the total change in public profit due to endogenous price changes from the project. The second term is the tax inefficiency when the change in profit is transferred to the private economy with distorting tax d. Since this exactly offsets direct changes in private surplus, only the tax inefficiency from making these transfers will impact on social welfare. By using the generalized Hatta decomposition in (7.15) the revised shadow profit becomes:
$Display mathematics$
where $( π ^ k S ) D = ( S ^ k ) D − MRT$ is the compensated revised shadow profit for the project. All the income effects are isolated in (S R)D.

# (p.285) 9 THE MARGINAL SOCIAL COST OF DISTORTING TAXATION

The welfare loss from marginally raising tax m and using tax d to balance the government budget, is:

$Display mathematics$
The change in tax d solves:
$Display mathematics$
with: dt d/dt m = − (∂L/∂t m)/ (∂L/∂t d). After substitution we obtain the revised welfare change in (9.1).

# 10 THE RELATIONSHIP BETWEEN SHADOW AND PRODUCER PRICES

With a profit tax Φ the government budget constraint in (2.4) becomes:

(A.26)
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There is no change in the conventional welfare equation in (2.7) because revenue from the profit tax has been transferred back to the private sector as a lump-sum payment. The new transfer equation is obtained from (A.26), as:
(A.27)
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The relationships between the revised shadow prices of goods and their corresponding producer and consumer prices is obtained by solving changes in activity using demand and supply responses, respectively.

By solving changes in activity over demand responses the marginal excess burden for the representative tax d in (9.2) becomes:

(A.28)
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6The change in tax revenue can be decomposed as:
(A.29)
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(p.286) where the income effect is:
(A.30)
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and the price changes are:
(A.31)
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7After rewriting the change in tax revenue in (A.29) using the income effect in (A.30) and the price changes in (A.31) the marginal excess burden in (A.28) becomes:
(A.32)
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where the term inside the brackets is the compensated marginal excess burden for tax d which is also solved using demand-side responses. This can be rearranged as:
(A.33)
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By following the same approach we can decompose the revised shadow price of good k in (6.4), as:
(A.34)
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where:
(A.35)
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After rearranging terms this becomes:
(A.36)
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By using (A.36) it is possible to write the welfare change in (A.33) as:
(A.37)
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which is the welfare change in (9.9).

# (p.287) 11 WHY SHADOW AND PRODUCER PRICES ARE EQUAL WHEN AGGREGATE MARGINAL COSTS ARE CONSTANT

From (6.8) the revised compensated shadow price of any good k, is:

$Display mathematics$
Since producer prices do not change endogenously in economies with linear production possibility frontiers there are no compensating transfers from extra output of good k, where the compensating transfers are solved using (2.10) as $∂ L ^ / ∂ z k = 0$. This means that the revised shadow price is equal to the conventional shadow price, with:
$Display mathematics$
8In constant cost economies, extra public output will crowd out the same private output, where the change in tax revenue becomes:
$Display mathematics$
On this basis the compensated shadow price of good k is equal to its producer price, with:
$Display mathematics$

# 12 THE RELATIONSHIP BETWEEN SHADOW AND PRODUCER PRICES WITH DISTRIBUTIONAL EFFECTS

In the presence of distributional effects and the profit tax Φ the revised welfare equation becomes:

(A.38)
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where the change in the distorting tax(es) to balance the government solves:
(A.39)
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By decomposing changes in activity using demand responses we can write the marginal social cost of public funds for tax d, as:
(A.40)
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(p.288) where the marginal excess burden with distributional effects in (9.23), is:
(A.41)
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9The change in tax revenue in (A.41) solves:
(A.42)
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with income effect:
(A.43)
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and price changes:
(A.44)
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By using (A.43) and (A.44) we can write (A.41), as:
(A.45)
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with:
(A.46)
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(p.289) By following the same approach we can decompose the revised shadow price of good k with distributional effects in (7.21), as:
(A.47)
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where:
(A.48)
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Using (A.47) and (A.48), we have:
(A.49)
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where the welfare change in (A.45) becomes:
(A.50)
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By using the generalized Hatta decomposition in (7.24), with $( S ¯ i ) D = ( S ¯ R ) D ( S ^ i ) D ∀ i$, we obtain the welfare change in (9.23).

# 13 THE RELATIONSHIP BETWEEN SHADOW AND CONSUMER PRICES

When changes in activity are solved using supply responses the marginal excess burden for the representative tax d in (6.2), becomes:

(A.51)
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The change in tax revenue can be decomposed as:
(A.52)
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where the price changes are:
(A.53)
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(p.290) By using (A.52) and (A.53), we are able to write (A.51), as:
(A.54)
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where the term inside the brackets is the compensated marginal excess burden for tax d solved using supply responses. This can be rearranged as:
(A.55)
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We can decompose the compensated shadow price of good k in (A.34) by following the same approach, as:
(A.56)
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where:
(A.57)
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Using (A.57) allows us to write the welfare change in (A.55), as:
(A.58)
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The welfare change in (9.27) is obtained using the market clearing condition $y d + z d = x d − x ¯ d$.

Distributional Effects are included by using the conventional welfare equation in (3.4) with dL = 0 and including the profit tax Φ to write the marginal excess burden for the representative tax d, as:

(A.59)
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10 (p.291) The change in tax revenue in (A.59) becomes:
(A.60)
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where the price changes are:
(A.61)
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By using (A.60) and (A.61) the welfare change in (A.59) expands as:
(A.62)
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The revised shadow price of good k can be decomposed using the same approach, as:
(A.63)
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where:
(A.64)
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This allows us to write the revised shadow price with distributional effects in (A.63), as:
(A.65)
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By using (A.65), the welfare change in (A.62) becomes:
(A.66)
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This extends the welfare decomposition of the marginal excess burden of taxation in (9.28) by including distributional effects.

# (p.292) 14 DECOMPOSING THE SAMUELSON CONDITION

The change in tax revenue in the shadow price of the public good in (10.5) can be decomposed, as:

(A.67)
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where the price changes are solved using the market clearing conditions for the private goods markets, as:
(A.68)
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11By using (A.68) the change in tax revenue in (A.67) becomes:
(A.69)
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with: θ = ∑i t ij (∂y i(p)/∂p j) αj and αj = −∑i δji (∂x i(q,I,G)/∂I).

Using the Slutsky decomposition, we have:

(A.70)
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with: (∂I(q, I, G)/∂G) = − ∑ MRS, where this makes the change in tax revenue in (A.69):
(A.71)
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After substituting (A.71) into (10.5) we obtain the welfare decomposition in (10.9).

# 15 DECOMPOSING THE REVISED SAMUELSON CONDITION

The spending effect is defined in (10.8) as the total change in tax revenue for the project, with:

(A.72)
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From (A.71), we have:
(A.73)
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(p.293) and using the decomposition in (2.14), we have:
(A.74)
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After substituting (A.73) and (A.74) into (A.72) we obtain the welfare decomposition for the spending effect in (10.19).

## Notes:

(1.) In general equilibrium aggregate consumer income is:

$Display mathematics$

(2.) This is obtained using the Slutsky decomposition, where δji is defined in Chapter 2 as the element of matrix:: |δji| = |◯i(q)/∂q j−∂y i(p)/ ∂p j|−1, and αi = −∑j δij(∂xi(q, I)/ ∂ I).

(3.) Using (2.6) to solve the lump-sum transfers allows us to write the shadow value of government revenue in (6.10), as:

$Display mathematics$
where: ∂pi/∂R = −αi SR. The revised shadow value of government revenue in (A.16) is obtained using the welfare decomposition in (2.14).

(4.) These price changes are solved using the approach employed in the Appendix 1 where the change in income is:

$Display mathematics$

(5.) Using the price changes in (A.20) allows us to write (A.19) as:

$Display mathematics$
The decomposition in (A.21) is obtained by using the revised shadow value of government revenue in (A.16).

(6.) We use the conventional welfare equation in (2.7) and the transfer equation in (A.27) to solve the marginal excess burden for tax d in (9.2). Also, from the market clearing conditions we have:

$Display mathematics$

(7.) The price changes are solved using the market clearing conditions, and this is demonstrated in the Appendix 1. We also use the price–tax relationships in (2.3), where:

$Display mathematics$

(8.) With constant marginal production cost, we have: ∂ ŷk/∂zk = −1 and ∂ ŷi/∂zk = 0 ∀i≠k.

(9.) When consumers have the same distributional weights this collapses to the marginal excess burden in (A.28).

(10.) When consumers have the same distributional weights this collapses to the marginal excess burden in (A.51).

(11.) The price changes are solved using the approach demonstrated in the Appendix 1 and using the income effects obtained from the economy's budget constraint, as:

$Display mathematics$