ShadowPricing, Direct Price Effects

Transcription

1 Week 3, Lecture 5 AGSM 2006 Page 1 ShadowPricing, Direct Price Effects 1 ShadowPricing [C&B Ch 5; DoF Ch 3; FP Ch 16, 6; S&W Ch 8] The NPV formula can be written as NPV = t Σ b(t) c(t) (1 + r ) t where b(t) = i Σ b i p i = b 0 p 0 + b 1 p 1 + b 2 p 2 + and c(t) = j Σ c j p j = Now, we assume here that we have the the quantities b i benefits and c j costs What of the prices p i and p j? >

2 Week 3, Lecture 5 AGSM 2006 Page 2 We want the true costs and benefits True prices reflect opportunities forgone (by suppliers, by consumers) shadow prices A shadow price better approximates the true oppor tunity cost or marginal valuation of a product or resource or service Five cases in which market prices are distorted, so that we must dig a little to obtain the shadow price, the true opportunity cost or valuation: 1 of atax, 2 of aprice chang e, 3 of atax with a price chang e, 4 of unemployment with minimum wages, 5 of atariff (a tax on imports)

3 Week 3, Lecture 5 AGSM 2006 Page 3 Efficiency Cost-Benefit Analysis To determine the true or shadow prices, use: Willingness To Pay (demand curve) for consumption Oppor tunity Costs (supply cur ve) for inputs But beware whether quantities increase or decrease in inputs or outputs C&B s Pricing Rule [C&B p93] Item to be valued Output Input Valued at equilibrium point on a: Demand Curve SupplyCur ve Satisfies additional Satisfies existing demand demand from alternative source Sourced from an Sourced from additional alternative market source supply (See also the Table on p30 of the DoF Handbook)

4 Week 3, Lecture 5 AGSM 2006 Page 4 The Area under the Demand Curve is the Value of that Quantity Ademand curve shows the maximum the buyer is prepared to pay for each unit bought As the price rises, buyerschoose to buy less They forgo the value to them of buying and using the forgone quantity: area = value forgone, as price P 1 P 2 P P 2 P 1 S D2 D 1 Q D 2 Q 1 Remember: the value of the last unit bought (the marginal value to the buyer) = the price of that unit Q

5 Week 3, Lecture 5 AGSM 2006 Page 5 The Area under the Supply Cur ve is the Cost of that Quantity To a firm in a competitive market, its supply cur ve shows the cost of supplying each unit A higher price induces more units to be offered for sale: the area under the supply cur ve is the total cost of supplying those units, as price P 1 P 2 P P 2 P 1 S D2 D 1 Remember: the marginal cost of supply (the cost of supplying the last unit sold) = the price of that unit Q 1 Q S 2 Q

6 Week 3, Lecture 5 AGSM 2006 Page 6 11 Example 1: ATax [C&B pp ; S&W Ch 83] Q: A remote electricity-generation project pays $1/litre for its fuel oil, the costliest input to the project The FA (financial appraisal) gives an NPV close to zero, but there is a tax on the fuel oil of 45 /litre What is the shadow price of fuel oil, to be used in the CBA NPV? A: Since the tax is a transfer (paying 45 /litre for nothing), ignore it in a CBA The shadow price is 55 /litre,and the CBA NPV will be positive, because of the lower oppor tunity cost of fuel oil at the shadow price A(specific) tax on a good supplied in a competitive market: places a wedge between the marginal cost (supply) and price (demand) the single (equilibrium) price p S 0 both valuation and cost Suppose the good is an input into a project: can no longer represent

7 Week 3, Lecture 5 AGSM 2006 Page 7 ATax (cont) Infinitely elastic supply $/unit p D 1 S 1 t p S 1 S 0 D 0 q 1 q 0 quantity q Because of the tax, the (tax-inclusive) demand price p D 1 is greater than the (tax-exclusive) supply price p S 0 The diagram shows the Tax Revenue and the Dead-Weight Loss as the tax is imposed, pushing up the effective supply, and reducing the quantity demanded, from q 0 to q 1

8 Week 3, Lecture 5 AGSM 2006 Page 8 ATax (cont) $/unit D p D 1 S 1 D 0 D 1 q 1 q 2 quantity q Let s say the project results in an expansion of demand, from D 0 to D 1 But there is no chang e in p D 1 with the increase in demand Because of the tax, the project pays the higher, tax-inclusive price p D 1 Is this the shadowprice? Does p D 1 reflect the opportunity cost associated with the extra quantity? No, in general, but it depends on the purpose of the tax (ie, revenue or green tax?)

9 Week 3, Lecture 5 AGSM 2006 Page 9 ATax (cont) eg oil at world price plus a local excise of t : $/unit p D 1 p S 1 t D S 1 S 0 D 0 D 1 q 1 q 2 quantity q Consumers value the increase in demand ( D)atthe tax-inclusive price p D 1 Suppliers price is unchang ed at p S 1 Shadowprice = p D 1 t = p S 1 =unchanging tax-exclusive price, ps 1 The tax revenue ( )isatransfer,and so chang es in the tax revenue (chang es in a transfer) are not chang es in cost ( ) This is only the case if there is no effect on existing purchasersofoutput, since there is no increase in price with perfectlyelastic supply the ex-tax price p S 1 is the shadowprice

10 Week 3, Lecture 5 AGSM 2006 Page Example 2: Shadow prices and opportunity costs when prices chang e [C&B pp 92 96; FP Ch 222, 232; DoF 34] Q: There is a local market for irrigation water The going price is $50/megalitre Anew cotton farm is planned, but its size and thirst for water are such that the going price of water will rise to $60/megalitre, given its demand of 10,000 megalitres/year Atthe lower price the NPV of the project is positive, but at the higher price negative Atthe higher price the incumbent users cut their consumption by 1000 megalitres/year A: The shadow price is between $50 and $60/megalitre,say $55 (assuming linear supply and demand curves) The existing users bear a cost of $ = $55,000/year for the water they can no longer afford (the displaced water) The shadow cost to the new farm is $550,000/year,which includes $55,000 to outbid the exiting usersfor 1000 megalitres/year, and the opportunity cost of $495,000 to induce the increased supply of9000 megalitres of water (the incremental water)

11 Week 3, Lecture 5 AGSM 2006 Page 11 Prices Change (cont) (Not to scale) P Q D 1 S Q 1 $50/ML D2 $60/ML Q Q value of displaced water =$55,000 cost of supplying incremental water =$495,000 total cost = $550,000 P S =$55 For a FA, cost = $60 10,000 = $600,000 For a CBA, cost = $55 10,000 = $550,000

12 Week 3, Lecture 5 AGSM 2006 Page 12 Prices Change (cont) S P 1 (2) S P 2 P 1 (1) (2) Q 1 Q s Q D Q 1 Q s G G Case 21 (No price chang e) Case 22 (Price chang e) Case 21: (No price chang e) resource opportunity cost = total social costs for increased factor supply (2) = P 1 G

13 Week 3, Lecture 5 AGSM 2006 Page 13 Case 22: (Price chang e) resource opportunity cost = total social costs P 1 G for increased factor supply (2) P 1 G <Area [(1) + (2)] < P 2 G P 1 < P s < P 2 P s is the effective or shadow price : P s G is the resource cost = area (1) + area (2) + value of reduced use of inputs in the rest of society as a response to higher prices, or opportunity costs (1)

14 Week 3, Lecture 5 AGSM 2006 Page 14 Prices Change (cont) Note: be conser vative if NPV >0with P 2 cost if NPV <0with P 1 cost if NPV (P 2 )<0 NPV (P 1 )>0 NPV (P s )? then GO then STOP then must find P s The point is so avoid the cost and effor t of deriving a better estimate of the shadowprice P s if it won t make any difference to the decision

15 Week 3, Lecture 5 AGSM 2006 Page 15 Some Equations and Harberger s Method Area under Demand Curve (1) area η P Q D P P D Area under SupplyCur ve (η P :price elasticity of demand) & Q D = Q D + Q 1 2 Q D Q D Q 1 Q (2) S area κ P Q s P P (κ P :price elasticity of supply) & Q S = Q 1 + Q S 2 Q 1 Q S Q Q S

16 Week 3, Lecture 5 AGSM 2006 Page 16 Case 22: Prices chang e (NFX: Not For Exam) From above, social cost (1) + (2) = P s G = P (ηq D +κq s ) P s = P (ηq D +κq S ) G shadow price = = P η Q 1 + Q D +κ Q 1 + Q S 2 2 G Pη Q 1 + Q D 2 + Q S 2 G (ifη=κ) ameans of obtaining the shadow price P s from Q 1, P,η,κ, Q D, Q S,and G

17 Week 3, Lecture 5 AGSM 2006 Page 17 The Project s Output Drives Prices Down The project produces an output which issold: FAbenefit = revenue = selling price quantity With no chang e in price, same with CBA If the output price falls, some marginal producers cut back orcease: a benefit FA revenue is too low P S 1 S2 P 1 P 2 D Q S 2Q 1 Q 2 value of the displaced production value of the incremental production P 1 <shadowprice < P 2 Q

18 Week 3, Lecture 5 AGSM 2006 Page Example 3: Prices chang e with a tax wedge [DoF 35] Q: In the cotton-farm example, assume that the prices of $50 (before) and $60/megalitre (after) include a tax of $4/megalitre, to raise revenue What now isthe shadow cost of water to the new farm? A: Wehavetoadjust for both the induced price increase and the tax wedg e between suppliers ofwater and users ofwater The value of the displaced 1000 megalitres of water for the existing farmerswho cannot afford topay $60/megalitre is still $55,000 year: we use the tax-inclusive price of $55/megalitre because theyevidently value this water at $50/megalitre at least, butnot at $60/megalitre The shadowcost of the incremental water is between $46 and $56/megalitre (the tax-exclusive prices), since that s what the suppliersofwater receive to induce them to increase supply; say $ megalitres/year = $459,000/year Total shadowcost = $55,000 + $459,000 = 514,000/year Shadowprice = $5140/megalitre

19 Week 3, Lecture 5 AGSM 2006 Page 19 Price Change&Tax (cont) (Not to scale) P Q D 1 S S Q 1 $50/ML D2 $60/ML Q Q value of displaced water =$55,000 (still) cost of supplying incremental water =$459,000 total cost = $514,000 P S =$5140 For a FA, cost = $60 10,000 = $600,000 For a CBA, cost = $ ,000 = $514,000

20 Week 3, Lecture 5 AGSM 2006 Page 20 Price Change&Tax (cont) Elastic supply $/unit S 1 p D 1 B S 0 p 0 p S 1 t A D 0 q 1 q 0 Figure 1 quantity q

21 Week 3, Lecture 5 AGSM 2006 Page 21 Price Change&Tax (cont) In Figure 1 above: p 0, q 0 is the initial price at A Aspecific tax of p D 1 p S 1 = t is imposed This is perceived by consumers asashift in supply from S 0 to S 1 buyers pay (tax-inclusive) p D 1 and producers receive (taxexclusive) p S 1 The tax revenue is (p D 1 p S 1 )q 1 = tq 1 Consumption falls from q 0 to q 1 (by b) The tax revenue is a transfer from consumers ofthe product to consumersingeneral (via tax receipts and government expenditure)

22 Week 3, Lecture 5 AGSM 2006 Page 22 Price Change&Tax (cont) $/unit p D 2 p D 1 D= 1 S 1 a 1 a q 1 q 2 Figure 2 D 0 D 1 quantity q

23 Week 3, Lecture 5 AGSM 2006 Page 23 Price Change&Tax (cont) In Figure 2: p 1 is the initial tax-inclusive price = p D 1 because supply S 1 includes the tax t The project to be evaluated shifts the demand for the input to the right from D 1 to D 2 (assume D =1) the tax-inclusive price is forced up from p 1 to p 2 = p D 2 and production goes up in total by 1 a The price increase induces other uses of the good to release an amount a which isabsorbed bythe project total usage of the input is a + 1 a = 1 in the project The expansion in output takes place at the tax-exclusive cost S 1 t or p 2 t The gain to the taxpayerissimplyatransfer ignore it Existing consumers value the reduction in a at the tax-inclusive price p D 2 that they pay unit social cost = a(gross-of-tax price) + (1 a)(net-of-tax price) = ap D 2 + (1 a)(p D 2 t ) = shadow price Hence shadow price = a weighted average ofthe tax-inclusive and tax-exclusive prices, p 2 and p 2 t,respectively

24 Week 3, Lecture 5 AGSM 2006 Page 24 Price Change&Tax (cont) How dowecalculate the weights a and 1 a?: η D = initial price elasticity of demand at p 1 a/q η D = 1 (using initial-point convention) (p 2 p 1 )/p 1 =%chang e in quantity %chang e in price Similarly: κ S = supplyelasticity = (1 a)/q 1 (p 2 p 1 )/p 1 Hence η D = a κ S 1 a η D and a = κ S η D and ShadowPrice = ap 2 + (1 a)(p 2 t): foranincreased demand for the input also for an increase in supply ofthe good if the project results in more of the good

25 Week 3, Lecture 5 AGSM 2006 Page 25 Price Change&Tax (cont) Figure 3 $/unit p D 2 D = 1 S 1 p D 1 S 0 p S 1 t a 1 a q 1 q 2 D 0 D 1 quantity q Valuation of the increased supply 1 a depends on S 0 (taxexclusive) Valuation of the demand shifted to the project a depends on the shaded area under D 1 Figure 3 adds the tax-exclusive supply cur ve S 0 to Figure 2 So D p S =the sum of the two areas

26 Week 3, Lecture 5 AGSM 2006 Page Example 4:Shadowpricing of labour [C&B pp 96 98; FP Ch 641, Ch 10; DoF 39] In a competitive labour market the shadow price of labour is simplythe market wage: $/unit D S w * 2 w * 1 D 1 D 2 n * 1 n * 2 employment n But the project shifts the demand for labour from D 1 to D 2 more jobs are created (n * 2 n * 1) workersmovefrom lower-paid to higher-paid jobs and there is no involuntar y unemployment

27 Week 3, Lecture 5 AGSM 2006 Page 27 Labour (cont) Suppose there is a minimum wage w for labour set by the IRT Then the employment level n 1 will be less than the competitive level, n * 1 wage w $/unit S w w2 SP w 1 * w SP 1 D D 1 D 2 n 1 n * 1 n 2 employment n The projects shifts out demand for labour from D 1 to D 2 If the additional workers who receive jobs value leisure at w SP then w1 SP is their shadow wag e Newemployedworkers may have ahigher value of leisure w SP 2 than w SP 1 this higher average value w SP 2 should be used for them The average social cost (shadow price) is lower than the market price w Why? Because there is unemployment at minimum wage w 1,

28 Week 3, Lecture 5 AGSM 2006 Page 28 Labour (cont) $/unit w S w SP 2 w SP 1 D D 1 D 2 n 1 n 2 employment n The chang e in the wage bill = w (n 2 n 1 ); in FA it was the rectangle (brown + green) The chang e in the social cost = w 1 SP + w2 SP 2 (n 2 n 1 ): brown area The opportunity cost of getting a job is less than w,which is reflected in the supply cur ve w SP = 1 2 (w SP 1 + w SP 2 )

29 Week 3, Lecture 5 AGSM 2006 Page Example 5: Foreign exchang e [C&B Ch 8; FP Ch 92; DoF 38] $/unit D S of expor ts e 2 e 1 D 1 D 2 for impor ts q 1 q 2 traded goods q

30 Week 3, Lecture 5 AGSM 2006 Page 30 Foreign Exchang e (cont) In the figure: the vertical axis shows the real price of traded goods = the inverse of the exchang e rate the supplyand demand for foreign currency isinitially inbalance at q 1, e 1 if the demand for imports by Australians goes up by D, the real price of traded goods will cost more the $A will devalue in terms of foreign currency asthe real price of imports rises (and e rises) Australian expor ters will gain more revenue in $A terms, to encourage additional expor ts an upwards sloping supply cur ve S

31 Week 3, Lecture 5 AGSM 2006 Page 31 Foreign Exchang e (cont) e 1 +t $/unit B S of expor ts e 1 A t D gross of tax D net of tax for impor ts q 1 traded goods q Atariff (tax) of t is nowimposed on imports A represents the equilibrium value of expor ts B is the tariff-inclusive value of imports measured in terms of foreign currency, the value of imports and expor ts is equal

32 Week 3, Lecture 5 AGSM 2006 Page 32 Foreign Exchang e (cont) Project nowincreases our supply offoreign exchang e S via additional expor ts Let S = 1 e 1 +t $/unit B S of expor ts S =1 S e 1 A C D tariff inclusive D tariff exclusive forimpor ts a 1 a q 1 traded goods q this situation is similar to the tax example (p5-12)

33 Week 3, Lecture 5 AGSM 2006 Page 33 Foreign Exchang e (cont) increased expor ts facilitates increased imports valued at e 1 +t (area base 1 a) increased newexpor ts displaces traditional expor ts a valued at net-tax price of e 1 shadowprice = a (post-tax price of traded goods) +(1 a)(pre-tax price of traded goods) shadowexchang e rate will exceed market exchang e rate, since expor ts are under-valued by the market exchang e rate The Gregor y Thesis or Dutch Disease

34 Week 3, Lecture 5 AGSM 2006 Page 34 Summar y of Lecture 5 This lecture introduced the use of market prices suitably adjusted to become shadow prices which accuratelyreflect the oppor tunity cost of the goods and services used bythe project, whether produced in response to the project s demand (incremental) orbid away from existing uses (displaced) in CBA studies Remember: No price chang e no displacement How toadjust market prices for taxes (whichare transfers, by and large) How toadjust market prices for price chang es caused by the project When input prices rise, FAoverstates the cost When output prices fall, FA understates the benefit How toadjust market prices for regulated prices, suchas minimum wages with unemployment among the workers the project will hire Shadow wag es <