- Email: [email protected]
Shadow prices, market prices and project losses
Economics Letters 2 (1979) 363-367 0 North-Holland Publishing Company
SHADOW PRICES, MARKET PRICES AND PROJECT LOSSES * Peter G. WARR Monash Received
University, 18 July
Clayton,
Vict. 3168, Australia
1979
Provided that non-optimal taxes and subsidies are the only distortions project that is viable at the optimal set of shadow prices automatically nue indirectly to finance any losses it incurs at market prices.
affecting market prices, a generates enough tax reve-
1. Introduction Considerable attention has now been given to the problems of calculating shadow prices when market prices are affected by non-optimal taxes and subsidies. If these distorting taxes and subsidies must be treated as given for the purposes of benefitcost analysis, then the optimal vector of shadow prices for use in project selection typically differs from the vector of market prices. This raises an awkward problem. Suppose we have a project which is viable at shadow prices but not viable at market prices. If the project is implemented, how are its actual losses at market prices to be financed? If additional taxes have to be raised to finance these losses, and these additional taxes are distortionary, then surely this requires us to reconsider the vector of shadow prices previously calculated in disregard of this fact. Arguments along these lines have been advanced recently by several authors including Srinivasan and Bhagwati (1978, p. 114), Corden (1974, pp. 390-392), and Dasgupta and Stiglitz (1974, pp. 28-29). The present paper shows that provided non-optimal taxes and subsidies are the only distortions affecting market prices, this problem does not arise. A project that is viable at the optimal set of shadow prices is automatically self-financing in that its losses at market prices are at least matched by its indirect effects on tax revenue. There is no need to impose new taxes or to alter the levels or rates of existing ones in order to finance the project’s losses. Alternatively, the existence of a binding constraint on the size of the government’s budgetary deficit is in this case irrelevant for the calculation of the optimal set of shadow prices. ’ * Discussions
with W.M. Corden have been particularly helpful. ’ This result can be viewed as a generalization of the result obtained argument applied solely to traded commodities. 363
in Warr (1977),
where the
P.G. Warr / Shadow prices, market prices and project losses
364 2. The model
For convenience, we will take the standard trade-theoretic model employed by Srinivasan and Bhagwati, and largely adopt their notation. There are two primary factors, capital, k, and labor, 1, producing two traded outputs, denoted x1 and ~2, with fixed international prices p; and p;. These commodities are also consumed domestically. We are interested in evaluating a ‘small’ project which produces the output x3, an exported good with fixed international price p;, and uses k and 1 as inputs. Our problem is to find the appropriate shadow prices for evaluating this project when the domestic price ratio of the two traded goods produced elsewhere in the economy is distorted by a fixed tax-cum-subsidy. We will suppose, for simplicity, that this is a fixed ad vabrem tariff on imports of good 2. The domestic price ratio is then p;(l + t)/pF, where r is the rate of the tariff. There is a fixed supply of capital and labour, K and t respectively, each fully employed such that kl f k2 + k3 - K and 1, t l2 t l3 = E, where ki is the amount of capital used in producing commodity i, etc. We choose units of measurement such that the domestic market prices of commodities 1, 2 and 3 are p;, p;(l + t) and p;, respectively. The market prices of k and 1 are r^and G and their marginal input coefficients obtaining in the production of goods 1 and 2 are (k,, i,) and (kz, i2), the circumflex denoting the tax-distorted situation. Total government revenue is given by R = T + P, where T is total tax revenue and P is the net income at market prices earned by the public project, which, of course, is unrestricted in sign. Writing c2 for the consumption of commodity 2, R is given by R = tp;(c2
~ x2)
+
p;x3
- +13 - r^k3 .
(1)
3. Optimal shadow prices Since both consumed goods are traded with fixed domestic and international prices it is obvious that welfare maximization requires the maximization of the value of final output at international prices. This implies that the shadow price of the project output may be set at its international price, p;, and that the shadow prices of the primary factors are then in each case the marginal cost, in terms of the value at international prices of the final output of goods 1 and 2, of drawing a unit of the factor concerned into the public project. That is, writing Y =pyxl +p;x2 + p;xs, ..* W = -aY/X, and i* = aY/ak,. 2 Then ^*
W
=
-Max,Ia13
fp;ax,iai,),
’ The w*, F* notation follows Srinivasan ation of output change at international the tax-distorted position.
(2) and Bhagwati (1978). The asterisk refers to the evaluprices while the circumflex denotes its evaluation at
P.G. Warr /Shadow
365
prices, market prices and project losses
and -*
r
=
-(p;ax,iaka
+p;ax2/ak3).
(3)
We now focus on the shadow price of labour; the analysis for capital is exactly symmetrical. Proceeding along the lines of Srinivasan and Bhagwati (1978) it is now easily seen from the factor supply constraints that i, (axtiaz,)
t k,(ax,/ala)
= 0,
+ i,(ax,/az,)
=~ i ,
(4)
and i,(ax,/al,)
(5)
and it follows that ax, /al, = I&/D and ax2/ala Therefore, W - * = (p;kl
~
p;&)/D,
^^
kl/D, where D = k,l,
^^
- k$,.
(6)
and similarly, i* = (p;i,
- p;i,)/D.
(7)
This is the Srinivasan-Bhagwati result. It is now convenient to derive the relationship between the shadow price w* and the market price $. This is done by noting that profit maximization in the private sector implies p;
= tii,
+ A,
)
(8)
tik,.
(9)
and p;(l
tr)=L;,i,
Consequently, 6 = (p;(l
+ t)i,
- p;/&)/D
= 6* f tp;i,/D.
(10)
Similarly, ;=;*
_ tp&lD.
4. Shadow
prices and government
(11)
revenue
Consider the effect on aggregate government revenue of, on the one hand, drawing a unit of labour into the project and, on the other, of producing a unit of commod-
366
P.G. Warr/Shadow prices, market prices and project losses
ity 3 in the project. From (l), holdingx3
and k3 constant,
aRIai =tp;(ac,la13- ax,ja13)- G.
(12)
Since the relative prices of the consumer goods 1 and 2 are fixed, cz can be affected by a change in I3 only via a change in real income, which in this model is simply Y. Consequently, ac2/a13 =-mz”i, *, where m2 = ac,/aY and, of course, G* = -aY/al,. Using this, (10) and (12) we obtain
aRiai =-G *(i + tp;m2).
(13)
Similarly, 3
aR/ax,=p;(l + tp;m2),
(14)
noting that ax21ax3 = 0. We now have
-(aR/ai,y(aRfax,)=171~;.
(15)
The shadow price of labour relative to that of the project output is given by the general equilibrium effect on aggregate government revenue of reducing the project’s labour input by one unit relative to that of increasing its output by one unit. Similarly, it is now obvious that the same applies for capital,
-(aRlak,y(aRjax,)=r^*/pj.
(16)
This result can be shown to generalize to non-traded commodities as well as traded commodities and primary factors, and to hold regardless of the form of tax or subsidy employed, provided that such taxes and subsidies are the only distortions affecting market prices. 4 5. Shadow prices and project losses Now consider a small public project that is viable at shadow prices. We characterize this project by the vector (dx3, dka, dl3) such that p;dx3-i*dk3-G*d1320.
(17)
Then, from (1.5) and (16), dR = (aRlax3)dY3 +(aR/ak3)dk3 +
(aRlai3)di3> 0.
3 The quantity (1 + tp;m2) can be interpreted as the shadow price of ‘foreign although in non-monetary models this usage is possibly misleading. 4 See Harberger (1971) and Warr (1979).
(18) exchange’,
P.G. Warr /Shadow
Alternatively,
361
prices, market prices and project losses
since from (1) aRIax3 = p; + aT/ax3,
-aR/a13
= 6 - aT/a13, (17) implies that
@T/ax,)
dx, t (aT/i313) d13 + (aT/ak3)
dk, > Gdl,
-aRfak3
= i - aT/aks
f idk3 - p;dx3
.
and
(19)
The right-hand side of (19) is the project’s losses at market prices and the left-hand side is its indirect effect on tax revenue. That is, (17) implies that dT> -dP. If the project breaks even at shadow prices it is at least self-financing through its indirect effects on tax revenues. If the project exactly breaks even at shadow prices, both (17) and (19) become strict equalities and the project is automatically exactly selffinancing.
References Corden. W.M., 1974, Trade policy and economic welfare (Clarendon Press, Oxford). Dasgupta, Partha and Joseph E. Stiglitz, 1974, Benefit-cost analysis and trade policies, Journal of Political Economy 82, l-33. tiarberger, Arnold C., 1971, Three basic postulates for applied welfare economics: An interpretive essay, Journal of Economic Literature 9,785-797. Srinivasan, T.N. and Jagdish N. Bhagwati, 1978, Shadow prices for project selection in the presence of distortions: Effective rates of protection and domestic resource costs, Journal of Political Economy 86, 97-116. Warr, Peter G., 1977, On the shadow pricing of traded commodities, Journal of Political Economy 85, 8655872. Warr, Peter G., 1979, Properties of optimal shadow prices for a tax-distorted open economy, Australian Economic Papers, forthcoming.
SHADOW PRICES, MARKET PRICES AND PROJECT LOSSES * Peter G. WARR Monash Received
University, 18 July
Clayton,
Vict. 3168, Australia
1979
Provided that non-optimal taxes and subsidies are the only distortions project that is viable at the optimal set of shadow prices automatically nue indirectly to finance any losses it incurs at market prices.
affecting market prices, a generates enough tax reve-
1. Introduction Considerable attention has now been given to the problems of calculating shadow prices when market prices are affected by non-optimal taxes and subsidies. If these distorting taxes and subsidies must be treated as given for the purposes of benefitcost analysis, then the optimal vector of shadow prices for use in project selection typically differs from the vector of market prices. This raises an awkward problem. Suppose we have a project which is viable at shadow prices but not viable at market prices. If the project is implemented, how are its actual losses at market prices to be financed? If additional taxes have to be raised to finance these losses, and these additional taxes are distortionary, then surely this requires us to reconsider the vector of shadow prices previously calculated in disregard of this fact. Arguments along these lines have been advanced recently by several authors including Srinivasan and Bhagwati (1978, p. 114), Corden (1974, pp. 390-392), and Dasgupta and Stiglitz (1974, pp. 28-29). The present paper shows that provided non-optimal taxes and subsidies are the only distortions affecting market prices, this problem does not arise. A project that is viable at the optimal set of shadow prices is automatically self-financing in that its losses at market prices are at least matched by its indirect effects on tax revenue. There is no need to impose new taxes or to alter the levels or rates of existing ones in order to finance the project’s losses. Alternatively, the existence of a binding constraint on the size of the government’s budgetary deficit is in this case irrelevant for the calculation of the optimal set of shadow prices. ’ * Discussions
with W.M. Corden have been particularly helpful. ’ This result can be viewed as a generalization of the result obtained argument applied solely to traded commodities. 363
in Warr (1977),
where the
P.G. Warr / Shadow prices, market prices and project losses
364 2. The model
For convenience, we will take the standard trade-theoretic model employed by Srinivasan and Bhagwati, and largely adopt their notation. There are two primary factors, capital, k, and labor, 1, producing two traded outputs, denoted x1 and ~2, with fixed international prices p; and p;. These commodities are also consumed domestically. We are interested in evaluating a ‘small’ project which produces the output x3, an exported good with fixed international price p;, and uses k and 1 as inputs. Our problem is to find the appropriate shadow prices for evaluating this project when the domestic price ratio of the two traded goods produced elsewhere in the economy is distorted by a fixed tax-cum-subsidy. We will suppose, for simplicity, that this is a fixed ad vabrem tariff on imports of good 2. The domestic price ratio is then p;(l + t)/pF, where r is the rate of the tariff. There is a fixed supply of capital and labour, K and t respectively, each fully employed such that kl f k2 + k3 - K and 1, t l2 t l3 = E, where ki is the amount of capital used in producing commodity i, etc. We choose units of measurement such that the domestic market prices of commodities 1, 2 and 3 are p;, p;(l + t) and p;, respectively. The market prices of k and 1 are r^and G and their marginal input coefficients obtaining in the production of goods 1 and 2 are (k,, i,) and (kz, i2), the circumflex denoting the tax-distorted situation. Total government revenue is given by R = T + P, where T is total tax revenue and P is the net income at market prices earned by the public project, which, of course, is unrestricted in sign. Writing c2 for the consumption of commodity 2, R is given by R = tp;(c2
~ x2)
+
p;x3
- +13 - r^k3 .
(1)
3. Optimal shadow prices Since both consumed goods are traded with fixed domestic and international prices it is obvious that welfare maximization requires the maximization of the value of final output at international prices. This implies that the shadow price of the project output may be set at its international price, p;, and that the shadow prices of the primary factors are then in each case the marginal cost, in terms of the value at international prices of the final output of goods 1 and 2, of drawing a unit of the factor concerned into the public project. That is, writing Y =pyxl +p;x2 + p;xs, ..* W = -aY/X, and i* = aY/ak,. 2 Then ^*
W
=
-Max,Ia13
fp;ax,iai,),
’ The w*, F* notation follows Srinivasan ation of output change at international the tax-distorted position.
(2) and Bhagwati (1978). The asterisk refers to the evaluprices while the circumflex denotes its evaluation at
P.G. Warr /Shadow
365
prices, market prices and project losses
and -*
r
=
-(p;ax,iaka
+p;ax2/ak3).
(3)
We now focus on the shadow price of labour; the analysis for capital is exactly symmetrical. Proceeding along the lines of Srinivasan and Bhagwati (1978) it is now easily seen from the factor supply constraints that i, (axtiaz,)
t k,(ax,/ala)
= 0,
+ i,(ax,/az,)
=~ i ,
(4)
and i,(ax,/al,)
(5)
and it follows that ax, /al, = I&/D and ax2/ala Therefore, W - * = (p;kl
~
p;&)/D,
^^
kl/D, where D = k,l,
^^
- k$,.
(6)
and similarly, i* = (p;i,
- p;i,)/D.
(7)
This is the Srinivasan-Bhagwati result. It is now convenient to derive the relationship between the shadow price w* and the market price $. This is done by noting that profit maximization in the private sector implies p;
= tii,
+ A,
)
(8)
tik,.
(9)
and p;(l
tr)=L;,i,
Consequently, 6 = (p;(l
+ t)i,
- p;/&)/D
= 6* f tp;i,/D.
(10)
Similarly, ;=;*
_ tp&lD.
4. Shadow
prices and government
(11)
revenue
Consider the effect on aggregate government revenue of, on the one hand, drawing a unit of labour into the project and, on the other, of producing a unit of commod-
366
P.G. Warr/Shadow prices, market prices and project losses
ity 3 in the project. From (l), holdingx3
and k3 constant,
aRIai =tp;(ac,la13- ax,ja13)- G.
(12)
Since the relative prices of the consumer goods 1 and 2 are fixed, cz can be affected by a change in I3 only via a change in real income, which in this model is simply Y. Consequently, ac2/a13 =-mz”i, *, where m2 = ac,/aY and, of course, G* = -aY/al,. Using this, (10) and (12) we obtain
aRiai =-G *(i + tp;m2).
(13)
Similarly, 3
aR/ax,=p;(l + tp;m2),
(14)
noting that ax21ax3 = 0. We now have
-(aR/ai,y(aRfax,)=171~;.
(15)
The shadow price of labour relative to that of the project output is given by the general equilibrium effect on aggregate government revenue of reducing the project’s labour input by one unit relative to that of increasing its output by one unit. Similarly, it is now obvious that the same applies for capital,
-(aRlak,y(aRjax,)=r^*/pj.
(16)
This result can be shown to generalize to non-traded commodities as well as traded commodities and primary factors, and to hold regardless of the form of tax or subsidy employed, provided that such taxes and subsidies are the only distortions affecting market prices. 4 5. Shadow prices and project losses Now consider a small public project that is viable at shadow prices. We characterize this project by the vector (dx3, dka, dl3) such that p;dx3-i*dk3-G*d1320.
(17)
Then, from (1.5) and (16), dR = (aRlax3)dY3 +(aR/ak3)dk3 +
(aRlai3)di3> 0.
3 The quantity (1 + tp;m2) can be interpreted as the shadow price of ‘foreign although in non-monetary models this usage is possibly misleading. 4 See Harberger (1971) and Warr (1979).
(18) exchange’,
P.G. Warr /Shadow
Alternatively,
361
prices, market prices and project losses
since from (1) aRIax3 = p; + aT/ax3,
-aR/a13
= 6 - aT/a13, (17) implies that
@T/ax,)
dx, t (aT/i313) d13 + (aT/ak3)
dk, > Gdl,
-aRfak3
= i - aT/aks
f idk3 - p;dx3
.
and
(19)
The right-hand side of (19) is the project’s losses at market prices and the left-hand side is its indirect effect on tax revenue. That is, (17) implies that dT> -dP. If the project breaks even at shadow prices it is at least self-financing through its indirect effects on tax revenues. If the project exactly breaks even at shadow prices, both (17) and (19) become strict equalities and the project is automatically exactly selffinancing.
References Corden. W.M., 1974, Trade policy and economic welfare (Clarendon Press, Oxford). Dasgupta, Partha and Joseph E. Stiglitz, 1974, Benefit-cost analysis and trade policies, Journal of Political Economy 82, l-33. tiarberger, Arnold C., 1971, Three basic postulates for applied welfare economics: An interpretive essay, Journal of Economic Literature 9,785-797. Srinivasan, T.N. and Jagdish N. Bhagwati, 1978, Shadow prices for project selection in the presence of distortions: Effective rates of protection and domestic resource costs, Journal of Political Economy 86, 97-116. Warr, Peter G., 1977, On the shadow pricing of traded commodities, Journal of Political Economy 85, 8655872. Warr, Peter G., 1979, Properties of optimal shadow prices for a tax-distorted open economy, Australian Economic Papers, forthcoming.