Shadow Price Development Using Cincinnati Case as Example Kerry Krutilla I. Theory The shadow pricing issue on the cost side: what s the relationship between a payment for inputs or outputs (F) and the costs to the supplier (OC) of making inputs or outputs available? An input or output supplier is giving up something (their opportunity cost) by providing the input or output to the buyer. Thus, they need a payment to cover these costs. The question is: does the payment just cover these costs is it the same as these costs or is it greater than these costs? If the supplier is subsidized, the payments it receives from the purchaser of the input or output may actually be less than the supplier s opportunity cost, because the government, in addition to the purchaser, is paying part of the compensation the supplier needs through the subsidy payment. Thus, the purchaser s payment doesn t need to fully cover the supplier s costs for the supplier to be compensated, if the government subsidy makes up the difference. I ll deal with this kind of situation at the end of this note. For now, assume the question is whether payments made for output or input are equal to the supplier s opportunity costs, or are greater than the supplier s opportunity costs. If the payments for inputs and outputs are equal to the supplier s opportunity costs (F=OC), we can just use the payments (expenditures for inputs or outputs) to represent these costs. Shadow pricing is not needed in this case. However, if the payment is larger than the costs by Y (F=Y+OC), we must deduct Y from the payments to give an adjusted shadow expenditure, P-Y, which equals the cost. (F-Y=OC). The tableaus below illustrate some cases, using the inspection provision in the Cincinnati case as particular examples to illustrate. Note that in all cases the payments exchanged between purchasers and suppliers are transfers. The question is: what s the correlation between the transfer payment and costs? Is the correlation one to one, F=OC? Or does F overstate OC by some amount? Case 1: No need for shadow pricing: F=OC KHT1- default assumption Inspection purchaser Inspection supplier Payments for emissions inspection -F F 0 Costs -OC -OC Net -F F-OC=0 -OC= -F Shadow pricing implication: Since F=OC, you can use F the expenditure (payment)- to represent OC, the opportunity cost. I will show below that this case holds for the state-operated inspections program; the $4.00 fee for emissions inspections just covers the unit costs of providing the inspections. Thus, $4.00 can be taken to represent the unit cost to the inspection supplier of provisioning inspections. (NB. This is just the cost of provisioning the inspections by the inspection supplier, not also all the other costs associated with the inspections program, e.g., the inconvenience/transit costs incurred by customers who get the inspections. Inconvenience/transit costs are an externality of providing inspections which will not enter the inspection provider s fee structure. Such externalities MUST BE SHADOW PRICED incorporated into the social Net 1
cost assessment as shown in the Cincinnati model memo. But in the examples of this section, we are just examining the narrower question: are the inspection fees greater than or equal to the cost to the inspections provider supplying the inspections? (I will return to the larger question of an overall unit cost for inspections at the end of this note). Case 2. Shadow Pricing for some Degree of Monopoly Markup: F-X=OC KHT2 Inspection Inspection Net purchaser supplier Payments for emissions inspections -F F 0 Costs -OC -OC Net -F F-OC=X>0 -OC= -(F-X) In this example, X is a monopoly markup, a wedge between the Fee (F) and the cost (OC), i.e., the relationship is F=X+OC, not F=OC. In this case, F-X=OC. In short, to go from the Fee, F, to the opportunity cost, OC, we must deduct the monopoly rent X. Shadow pricing implication: Use shadow expenditure F-X to represent OC. Using F will OVERSTATE the cost by X. In the government contracting context, this kind of situation could result if the bid is not competitive. A competitive bid should induce suppliers to offer contracts at a price which just covers their cost, not at an inflated price which more than covers their cost. Case 3. Shadow pricing for tax on inputs (supply expansion case): F -T=OC KHT3 Inspection Inspection Govt Net purchaser supplier Payments for emissions inspections -F F 0 Taxes -T T 0 Costs -OC -OC Net -F F-T-OC=0 - OC = -(F-T) In this case, the Fees just cover taxes (T) and costs (OC), i.e. F=T+OC. Now, T gives a wedge between the fee payments (F) and Opportunity Cost (OC), just as X- the monopoly markup did in KHT2 above. Shadow pricing implication: deduct the tax payment (T) from the Fee (F) to come up with the shadow expenditure, F-T=OC. This shadow expenditure is substituted for F to represent opportunity costs Note: KHT3 shows that the tax payment is collected within the domain. If the government was not in the accounting domain, then T leaves the accounting domain. In that case, F=- (T+OC) gives the cost to the domain, with T being the tax which leaves the accounting domain, and OC the within-domain resource cost. That is, no shadow pricing is needed when the tax payment leaves the accounting domain. 2
Case 4: A combination of Cases 2 and 3 KHT4 (combination of Case 2 and Case 3) Inspection purchaser Inspection supplier Govt Net Payments for emissions -F F 0 inspections Taxes -T T 0 Costs -OC -OC Net -F F-T- OC=X -OC = -(F-T-X) In this case, there is both a tax and a monopoly markup, i.e., F=T+X+OC (the fee covers the tax (T), a monopoly markup (X) and costs (OC). Thus, to go from the fee to the costs, we would need to deduct both the tax and the monopoly markup from the fee to get the shadow expenditure F-T-X=OC. We would need to use this shadow expenditure to represent the projects costs, rather than the expenditure itself. II. Numerical Application to the Cincinnati Case. Case A. The State-Provided inspections from the Cincinnati Case assignment represents Case 1 above. (See Table 2 of the case assignment for data used in these examples). That is, no shadow pricing necessary. *Fees: $4.00 (inspection fee) * 800,000 (number of inspections)=$3,200,000 ** inspection costs: $700,000+$400,000+$1,800,000+$300,000==$3,200,000 (Data from Table 2 of the Cincinnati Case assignment) In sum, because the inspection fee of $4.00 per car inspected just covers the unit inspection costs, we could just as well use this fee to represent the unit cost of provisioning inspections. Or $4.00 per inspection multipled by 800,000 inspections gives inspection costs/expenditures as: $3,200,000. The KHT is below. KHT for contractor- provided inspections Inspection purchaser State Inspection Supplier Net Inspection Fees -$3,200,000=-F $3,200,000=F 0 Inspection Costs -$3,200,000=-OC -$3,200,000=-OC Net -$3,200,000=-F F-OC=0 -$3,200,000= -OC 3
Case B. The Contractor-Operated inspections provision represent Case 4 above, i.e., there is both an element of monopoly markup, and taxes. So we must shadow price the fee to subtract out taxes and the monopoly markup. *Fees: $4.50 (inspection fee) * 800,000 (number of inspections)=$3,600,000 ** inspection costs: $650,000+$350,000+$1,500,000+$300,000==$2,800,000 (Data from Table 2 of the Cincinnati Case assignment). In this case, the inspection fees of $3,600,000 overstate the costs by $800,000. This $800,000 overstatement pays property taxes of $600,000 (which can be assumed to collected within the accounting domain) with a $200,000 residual, which is a monopoly markup of payments over costs. (See Table 2 in Cincinnati case assignment, Table 3 in the model memo). In short, we have: F=X+T+OC <=> $3,600,000= $200,000 + $600,000+ $2,800,000. To go from the inspection fee payment of $3,600,000 to the cost of $2,800,000, we have to shadow price the inspection fee payment by deducting the $800,000. The KHT is as follows: KHT for contractor- Inspection Inspection supplier Gov Net provided inspections purchaser (Contractor Operated) Inspection Fees -$3,600,000=-F $3,600,000=F 0 Inspection Costs -$2,800,000=-OC -$2,800,000= -OC Taxes -$600,000=-T $600,000 0 Net -$3,600,000=-F F-T-OC1=X=$200,000 $600,000 -$2,800,000=-OC1 The conventional way to make this adjustment is to shadow price the inspection fee of $4.50. In this case, we would say that the inspection fee overstates unit costs because of the monopoly markup and tax payments. In particular, the shadow price for the unit costs would be $3.50, as opposed to the inspections fee itself of $4.50. Therefore, $3.5*800,000=$2,800,000. Conclude: Instead of using the emissions inspection fee of $4.5 to represent the unit costs of inspections, use the shadow price of $3.5 to represent the unit cost of inspections. Finally, we mentioned at the outset that payments/fees/expenditures on inputs or outputs by buyers might be less than costs, if the supplier was getting some degree of subsidy from the government which partially covered their costs. Actually, this relationship between fees and costs exists for private service stations in the Cincinnati case. *Fees: $6.00 (inspection fee) * 800,000 (number of inspections)=$4,800,000 ** inspection costs: 2,080,000 + 4,160,00 = $6,240,000 (Data from Table 3 in Cincinnati Model Memo) 4
Because there is no subsidy to make up the difference in this case, a question is raised: why would service stations voluntarily provide inspections when the fees they are allowed to charge don t cover their costs? In this case, EPA has underestimated the fee required to cover costs, and as a corollary, overestimated the level of expected participation (2000 service stations). The remedy is to allow service stations to charge an inspection fee of $7.8 per inspection. At that fee level, costs would just be covered. So in this case we would say that the $6.00 fee understates unit costs not because a subsidy payment also covers costs (the possibility raised at the beginning of this note), but rather, because the $6.00 administered price the government has established is too low to cover costs. Regardless, $7.8. is the correct shadow price for the unit costs of inspections provided by private service stations. As a final note, look at the Total column in Table 1 in the model Cincinnati memo, which gives the total costs of inspections for each of the three options. These total costs include not just the correctly-specified costs of provisioning inspections, our topic so far, but all other costs associated with the program options, specifically (1) the costs to the states of administering the programs (2) the costs of repairing vehicles which fail inspections and (3) the travel and time costs of the vehicle owners getting the inspections. If we divide the numbers in Table 1 by 800,000, we ll get the overall shadow price per car inspected for each of the program options (for this particular time and travel cost permutation). I have listed these shadow prices in the table below, and their decomposition by components. Table: Shadow prices per car inspected Private State Service Stations Contractor Overall Shadow Prices 26.31 26.81 22.12 Shadow prices for provisioning 3.5 4.00 7.8 inspections (discussed above) Shadow prices for time/transit costs 14.89 14.89 6.20 Unit costs for administrative + repair 7.93 7.93 8.11 Inspection Fees 4.5 4.00 6.00 The overall shadow price is the sum of the shadow prices for provisioning inspections, the shadow price for time/transit costs, and the shadow prices(unit costs) for administrative and repair. Looking at these figures shows why using the fees charged for inspections (bottom row) would give such a misleading picture of the unit social costs of the program alternatives (top row). First, as this handout has detailed, the inspection fees don t even accurately give the unit costs of provisioning the inspections themselves for two out of the three program options. Again, the inspection fee for the contractor option, 4.5, overstates the unit costs of inspections due to taxes and monopoly rents. The correct shadow price (unit cost) is 3.5. And for the private service option, the inspection fee of 6.00 understates the cost of provisioning inspections, which is actually $7.8. Again, that s because the EPA set the administered price too low. But beyond these issues, the vehicle inspection fees miss the unit social cost of the program due to all of the other cost components the fees are not covering. Thus, when assessing which of these options provides emissions inspections at lowest cost, we want to base our conclusion on the shadow prices in the top row, which accurately give the unit social costs of the program alternatives, not the inspections fees in the bottom, which do not. Using that criterion, service stations provide inspections at lowest social costs, at $22.12 per car inspected (for the travel time/cost permutation indicated). 5
Shadow Pricing Addendum Let Pd be the price paid by the project to purchase an input (the project s financial outlay to purchase one unit of input). Let Z be the unit social cost to the accounting domain of using the input. This is the relevant figure for the cost side of a benefit-cost analysis. Shadow pricing question. Does Pd =Z? If it does, shadow pricing is not necessary. We can just use Pd to represent the project s unit cost. And the total financial outlay to purchase the input, Pd(I1-Io) where I1-Io is the units of input purchased will represent project costs. However, if Pd Z, then we must find a shadow price, P*, which does equal Z. We then substitute P* for the market price Pd to represent unit cost to compute the costs for the accounting domain of using the input in the project. Often, market prices for inputs, Pd, will differ from the shadow price P* (the project s unit social cost) by some amount, call it θ. That is, Pd-P*= θ. That difference is most commonly introduced by taxes, subsidies, market power, and other kinds of market distortions, like price controls. The difference can also be created by externalities associated with the input s production which are not reflected in the price of the input. Rewrite Pd-P*= θ as Pd-θ=P*. Writing the expression as Pd-θ=P* shows what shadow pricing is essentially about: finding the distortion of the amount θ, and subtracting it from the market price Pd to give the shadow price P*. Again, we substitute P* for Pd, getting project costs as P*(I1-Io) rather than as financial outlays Pd(I1-Io). I illustrate below with a number of particular cases. 6
Case 1. Shadow Pricing with an input tax causing the market price distortion (See Figure 1 below) Case 1a. Tax levied by within-domain government. Price equilibrium: Pd=Ps+T, where Pd=demand price, Ps=supplier s net-of-tax price, T=Tax. Suppliers produces at the point where Ps=MC. So Price equilibrium can be rewritten: Pd=MC+T. (The demand price is equal to the producer s cost plus the markup to cover the tax payment). Shadow pricing question: what is the relationship of the price the project pays for the input (Pd) to the resource cost of the input (MC)? From price equilibrium we can see that: Pd=MC+T=> Pd-T=MC=Ps. Conclude: deduct the tax payment (T) from the price paid for the input (Pd) to get the supply price, Ps. Use Ps as the shadow price to compute the cost the project s use of the input imposes. That is, Ps(Io-I1) is the resource cost the project s input usage imposes -- the area C in Figure 1 rather than the financial outlay for the input, Pd(I1-Io), or the area A+B+C in Figure 1. The KHT is as follows: KHT1 Within accounting Net unit cost of Buyer Supplier domain governmental input usage entity s unit outlay on -Pd Pd 0 input Tax payment -T T 0 Marginal resource cost of input Net -Pd Pd-T=0 T =-(Pd-T)=-Ps 7
Case 1b. tax levied by government outside the accounting domain In this case, within-accounting domain costs fall into two parts: (a) the tax payment, T, which leaves the accounting domain; (b) the resource cost of producing the input, MC. Since price equilibrium is Pd=T+MC (again, see figure 1), the demand price, Pd, represents the costs of using the input for the accounting domain. So shadow pricing is not necessary in this case. The KHT is shown is as follows: KHT2 Buyer Supplier Net unit cost of input usage s unit outlay on input -Pd Pd 0 Tax payment -T -T Marginal resource cost of input Net -Pd Pd-T=0 -(T+MC)=-Pd Case 1c. A mix of the cases above. One tax levied by a within-the-accounting domain government (T1); the other tax levied by an entity outside the accounting domain (T2). Price equilibrium: Pd=T1+T2+MC Unit cost to accounting domain of input usage: T2+MC. From price equilibrium, shadow price, P*=Pd-T1=T2+MC This case is not graphed, but the KHT is below. KHT3 Buyer Supplier Within accounting domain governmental entity Net unit cost of input usage s unit -Pd Pd 0 outlay on input Tax payment 1 -T1 T1 0 Tax payment 2 -T2 -T2 Marginal resource cost of input Net -Pd Pd-T1-T2- MC=0 T1 -(T2+MC)=-(Pd- T1) 8
Case 1d. two taxes and some monopoly markup over cost. Price Equilibrium: Pd=X+T1+T2+MC, where X is the monopoly markup, and T1 and T2 and MC are as before. As before, the unit cost to accounting domain of input usage: T2+MC. From price equilibrium, the shadow price, P*= Pd-X-T1=T2+MC (deduct monopoly markup (X) and within-accounting domain tax (T1) to get the shadow price (P*) which equals T2+MC). This case is not graphed, and the KHT is identical to above, except the supplier s net instead of being: Pd-T1-T2=0 is Pd-T1-T2=X. In short, the net impact on the project supplier is greater than zero by the amount X, rather than equal to zero. 9
Case 2. Shadow Pricing with a subsidy causing the market price distortion (see figure 2 below) Case 2a. government making the subsidy assumed within the domain Price equilibrium: Ps=Pd+S=MC. (see figure 2). Interpretation: demand price, Pd, is below producer cost (MC) by the amount of the subsidy (S), i.e. Ps-S=Pd=MC-S. The producer can sell its product for a price (Pd) less than unit costs (MC) since the government pays the unit subsidy (s) to make up the difference (S=MC-Pd). Unit cost to accounting domain of input usage: MC From price equilibrium, you can again see that Ps=Pd+S=MC. Hence, we need to shadow price, adding the subsidy S to the price paid for the input, Pd, to get a price that accurately reflects marginal cost, P*=Ps=MC. Thus, from figure 2, Ps(I1- Io)=A+B+D+C, which is the total resource cost of the project s input usage. Using the price the project paid for the input, Pd, would give Pd(I1-Io)=D+C. In short, the financial outlay for the input, Pd(I1-Io) understates the total resource cost of using the input by the amount A+B. The KHT is below: KHT4 Buyer Supplier Within accounting domain governmental entity Net unit cost of input usage s unit -Pd Pd 0 outlay on input Unit subsidy +S -S 0 Marginal resource cost of input Net -Pd Pd+S -S =-(Pd+S) 10
Case 2b. government making the subsidy assumed outside the domain Again, price equilibrium is Ps=Pd+S=MC. Again see figure 2. Unit cost to accounting domain of input usage is only MC-S, because the subsidy, S, is a gift from outside the accounting domain (which helps defray part of the resource cost). This unit cost, MC-S, is equal to Pd, a fact we get by rewriting the price equilibrium condition Pd+S=MC as Pd=MC-S (again see figure 2). In this case then, the price the project pays for the input accurately reflects the within-accounting domain cost. So, Pd(I1-Io) the financial outlay for the input is equal to the project s net cost to the accounting domain (D+C), which is A+B+D+C minus the subsidy payment A+B. The KHT is shown below: KHT5 Buyer Supplier s unit outlay on input -Pd Pd 0 Unit subsidy +S +S Marginal resource cost of input Net -Pd Pd+S S=-Pd Net unit cost of input usage 11
Case 3. Shadow Pricing with a negative externality causing the market price distortion Case 3a. input usage subsidized by uncompensated or otherwise unpriced environmental damages (see figure 3 below). Price equilibrium: Ps=Pd=P=MC (see figure 3) Unit cost to accounting domain of input usage: MC+ED Since Pd=MC, and the social unit cost is MC+ED, we must compute ED to add to Pd to get a shadow price, which reflects the environmental damages, i.e., P*=ED+Pd=ED+MC. Then, P*(I1-I1) in figure 3 => A+B+D+C the social cost of the project s input usage. Using the market price paid for the input, Pd, would give the financial outlay for the input usage of D+C, which understates social cost by the amount ED. The KHT is shown below: KHT6 Buyer Supplier Stakeholders impacted by environmental damage Net unit cost of input usage s unit outlay -Pd Pd 0 on input Marginal resource cost of input Environmental -ED -ED Damage Net -Pd Pd -ED -(MC+ED)=P* 12
Case 3b. environmental damage priced through a tax on the input producer equal to marginal environmental damages, i.e., T=ED. See figure 4 below. Price equilibrium: Pd=Ps+T=MC+ED, with T=ED (again see figure 4) Unit cost to accounting domain of input usage: MC+ED Since Pd=Ps+T=MC+ED, using Pd as the unit cost gives the correct measure. Thus, Pd(I1- Io)=A+B+C in Figure 4. In short, public policy the pollution tax has been used to correct the market distortion and get the prices right. Thus, the corrected market price, Pd, can be used to estimate the unit social cost of the project, without any shadow pricing. The KHT is shown below: KHT7 Buyer Supplier Stakeholders impacted by environmental damage State gov Net unit cost of input usage s unit outlay on -Pd Pd 0 input Marginal resource cost of input Environmental Damage -ED -ED Environmental tax -T T 0 Net -Pd Pd-T -ED -(MC+ED) =-Pd 13