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Chapter 2 2 Shifts in Supply and Demand Influence Price 3 Economists Love Competitive Markets Demand Coal Qd = f (Pc-, Psb+, Pcm-, Y, T+/-, Pol+/-, #buy+) Ceteris Paribus hold constant everything but P & Q P D Q 4 World Coal Use by Sector World % Coal Use By Sector 2007 r, 2.4% i, 20.4% tr, 0.1% fdsk, 0.1% ag, 0.5% c, 0.6% el&ht, 75.9% 5 Economists Love Competitive Markets Demand Coal Qd = f (Pc, Psb, Pcm, Y, Tech, Policy, +/- +/- +/+ Ceteris Paribus hold constant everything but P & Q P D #buy) + 6 Supply Suppliers Qs = f(Pc, Pf, Psm, Pby,T, Pcy, #sel) + P - - + + +/S Q + 7 Sum Up Where are coal reserves Conversions E1 in unit 1, (u1) or E2 in unit 2 (u2) conversion is units of 1 per unit of 2 (u1/u2) E1 E 2 E2 u1 u2 E1 u1 u2 E1u 2 u1 8 Sum Up Qualitative create D and S hold all variables but P&Q constant started to look behind supply P S D Q 9 Behind Supply for firm to maximize profits = P*Q – TC = P*Q – FC – VC(Q) competitive firms take price as given f.o.c. /Q = P - TC/Q = P - VC(Q)/ Q = 0 MC ↑ 2.o.c 2/Q2 = - TC2/Q2 = - MC(Q)/ Q<0 MC(Q)/ Q>0 operate where price equals marginal variable cost short run supply equals marginal cost curve 10 Typical Competitive Firm Cost Short Run Supply P S P P MC1 MC2 AVC1 AVC2 Psr D Q1 Q Si = MCi above AVC Market is horizontal sum Q2 Q Q1 +Q2 Q 11 Where They Cross Determines P & Q Supply = Demand P S Pe D Qe Q Model Building Blocks 12 Out of Equilibrium P S Price too high PH PL Price too low D Qs Qd Q 13 Shift in D Change in Qs – movement along S P S Pe' Pe Pe" D"(decrease)← Qe" Qe Qe' D D' (increase)→ Q 14 Shift in S Movements along the D curve P S"(decrease)← S Pe" Pe S' (increase)→ Pe' D Qe" Qe Qe' Q 15 More than one Change Coal Mine Productivity Per Miner Increases S P →S' Pe Pe' D Q Q' Q PQ 16 1.Chinese Coal Mine Productivity 2. Plus Cheaper Sequestration P↓Q↑ P ↑ Q↑ S S P Pe' Pe Pe Pe' D Q Q' D Q Q Q' →D' Q 17 Supply and Demand Building Blocks-Two markets Coal Natural Gas if all market - general equilibrium 18 Two Markets Qdo = a + bPo +cPg + dY Qso = e + fPo + gPG + hCost Cost Exogenous Y Exogenous Qdg = i + jPo + kPg + lY Qsg = m + nPo + oPG + pCost Qdo = Qso Qdg = Qsg 6 endogenous variables, 6 equations 19 Supply and Demand Building Blocks Dynamic -Two time periods Time 1 Time 2 n time periods 20 Trade Models- Two Areas in World P S2 S1 S1 S1+S2 D1 Q1 D2 D1+D2 Q2 Qw 21 Market Power Seller P S D Q 22 Market Power Buyer P S D Q 23 Quantitative Models Chapter 2&3 Buyers Qd = f(Pc, Psb, Pcm, T, Pot, Pcy, #buy) Suppliers Qs = f(Pc, Pf, Psm, Pby,T, Pcy, #sel) Functions – with numbers often start with qualitative model to get intuition 24 Quantitative S-D Example Qd =99 - 2Pc + 1Psb - 2Pcm + 0.1Y Qs =30 + 1Pc – 1Pk - 0.2Pl - 0.4Pnr Pc = price of coal Psb = price of substitute to coal (natural gas) =1 Pcm = a complement to coal =10 Y = income = 200 Pk = price of capital = 20 Pl = price of labor = 40 Pnr = price of other natural resources used in production of coal = 10 25 Qd = 99 - 2Pcd + Psb - 2Pcm + 0.1Y Qs =30 + 1Pcs – 1Pk - 0.2Pl - 0.4Pnr Qd = 99 - 2Pcd + 1 - 2*10+0.1*200 = 100 -2Pcd Qs = 30 + Pcs – 1*31 - 0.2*30 - 0.4*10 = -11 + 1Pcs 26 Model02.xls: Worksheet S&D 27 Inverse Demand Qd = = 100 -2Pcd Qs = -11 + 1Pcs Sometimes want price as function of quantity invert Qd = 100 -2Pcd solve demand for Pcd 2Pcd = 100 – Qd → Pcd = 50 – (1/2)Qd invert Qs = -11 + 1Pcs solve supply for Pcs Pcs = 11 + Qs 28 Graph and Forecast Pd = 50 – (1/2)Qd Ps = 11 + Qs P Forecast P & Q 60 Pd = Ps 50 – (1/2)Q = 11 + Q 40 50-11 = Q+(1/2)Q P = 37 39 = (3/2)Q 20 Q = (2/3)39 = 26 Pd = 50– (1/2)26 = 37 Ps = 11 + 26 = 37 S D 50 Q = 26 100 Q 29 Is Equilibrium Stable? Price above Equilibrium Pd = 50 – (1/2)Qd Qd=100 – 2Pd P 60 Ps = 11 + Qs Qs = -11 + Ps What if P = 40 P = 40 Qd=100 – 2*40 = 20 Qs = -11 + 40 = 29 Excess quantity supplied P↓ 20 S D Qd = 20 Qs = 29 100 Q 30 Quantitative Need numbers for ceterus paribus values Substitute in to Qd and Qs Qd = f(Pd) is demand Qs = f(Ps) is supply Solve for P and Q Sometimes inverse is easier or more useful Solve for price as a function of quantity Pd = f-1(Qd) is inverse demand Pd = f-1(Qd) is inverse demand We graph the inverses 31 General Equilibrium Model (1) Think about but not to be tested Markets for all products all factors or production consumers buy m final goods: a,b,c,…. at prices pb, pc, pd,…. their demand for final goods: db, dc, dd,…. consumers own and sell n factors of production: qt, qp, qk, …. at prices pt, pp, pk, … their supply/of n factors: st, sp, sk,… m + n unknown prices 32 General Equilibrium Model (2) Think about but not to be tested in real world things are priced in money $/liter, etc in simplest G.E. model no money pick a numeraire good its price is one m + n - 1 unknown prices equilibrium in household sector stpt + sppp+ skpk + …. = da + dbpb+ dcpc + …. income = expenditure If holds for each household, holds for market 33 General Equilibrium Model (3) Think about but not to be tested producers buy n factors of production demand: dt, dp, dk, producers produce m end use goods s demand: st, sp, sk, m commodities and n factors there are m+n unknowns quantities m + n - 1 unknown prices total: 2m + 2n - 1 unknowns 34 General Equilibrium Model (4) Think about but not to be tested Consumer Demand for goods (m-1 independent) da = da(pt, pp, pk, …, pb, pc, pd,…) Supply of factors (n) st = st(pt, pp, pk, …, pb, pc, pd,…) Producers Demand for factors (n) dt = dt(pt, pp, pk, …, pb, pc, pd,…) Supply of goods m sa = sa(pt, pp, pk, …, pb, pc, pd,…) 35 Last Time - Sum Up Qualitative create D and S hold all variables but P&Q constant P S D Q 36 Models for Policy What if Government Sets Maximum Price of 30 Shortages P Likely to be black market 60 Could to subsidize What would subsidy cost? Ps = 51 To get suppliers to produce 40 P = 30 Need Ps=11+40 =51 Cost (51-30)(40)=840 Qs = 19 S D Qd = 40 100 Q 37 What happens with following policies? P 60 S Pmax Pmin D 100 Controls Non-binding Q 38 Demand Price Elasticity Q responsiveness to price P P2 P1 Dlr D1 may change over time Qlr Q2 Q1 Q 39 Back to 1973 Oil Market OPEC Supply Shocks 73&79 S79 P S73 P79 Dlr D1 Q82 Q79 Q 40 Elasticity Definition How much quantity responds to price d = % change quantity % change in price If d = –0.5 price goes up by 100%, quantity demanded falls by % change quantity = % change in price* d = 100%*-0.5 = 50% 41 Let’s Develop Formal Definition d = d = = % change quantity % change in price Qd *100 Qd Pd *100 Pd Q2-Q1 Q1 P2 - P1 P1 42 Suppose We Have Price Increase P $2.00/g $3.00/g Q 500 106 g/d 400 106 g/d Qd d = Qd Pd Pd (400 106 g/d – 500 106 g/d) 500 106 g/d ($3.00 g – $2.00 g) $2.00 /g = -0.20/0.5 = -0.4 (no units) 43 Lets Go Back to Lower Price P $2.00/g $3.00/g Q 500 106 g/d 400 106 g/d Q2 – Q1 d = Q1 P2-P1 P1 (500 – 400) 400 = (1/4) = (2– 3) -(1/3) 3 = -(1/4)(3/1) = - 3/4 = - 0.75 44 Sum Up Computing Arc Elasticities d = d = = % change quantity % change in price Qd Qd Pd Pd Q2-Q1 (Q1+ Q2)/2 P2 - P1 (P1+ P2)/2 45 Sum Up Elasticity = Responsiveness to Price x = % change quantity % change in X Q could be quantity demanded Q could be quantity supplied X could be Price X could be income X could be the price of a substitute (cross price elasticity) X could be any other variable that influences Q Q likely more responsive in long run than short run 46 More Convenient for Elasticity Qs and Qd responsiveness to other variables x = x = % change quantity % change in Q Q = Q X X X Q X Take limit as X→0 x = Q X X Q 47 Where do they come from? Estimate whole function market data Qd = f(Pd, Y, Ps, Pc, . . ., etc. ) εp = Q P P Q Function Forms linear: Q = a – bP εp= -b(P/Q) 48 P Linear Function a/b =- a/2b |Elastic| > 1 D |Unit Elastic| = 1 =-1 |Inelastic| (1,0) =0 a Q a/2 Q = a - bP p = -b(P/Q) Graph: P = 0 then Q = a -b*0 = a = -b(0/a) = 0 Q = 0 = a-bP then P = a/b = -b(a/b)/0 = - P = (a/b)/2 then Q = a - b(a/b)/2) = a - a/2 = a/2 = -b(a/b)/2/(a/2) = -1 49 Demand Price Elasticities and Revenues How Does Price Change Revenue TR = PQ = PQ(P) TR/P = Q + (Q/P)*P =Q(1+ (Q/P)*(P/Q)) = Q(1+εp) Sign of TR/P = sign (1+εp) TR/P < 0 when (1+εp)<0 subtract -1 from both sides (elastic) εp<-1 Raising price lowers revenue Lowering price raises revenue 50 Demand Price Elasticities and Revenues TR/P < 0 when εp<-1 elastic P and TR opposite direction P TR P TR TR/P = 0 when (1+εp)=0 εp= -1 unitary elasticity TR/P > 0 when (1+εp)>0 0> εp> -1 P TR? P TR? 51 Demand Price Elasticities and Revenues TR/P < 0 when εp<-1 elastic P and TR opposite direction P TR P TR TR/P = 0 when (1+εp)=0 εp= -1 unitary elasticity TR/P > 0 when (1+εp)>0 0> εp> -1 P TR? P TR? 52 Demand Price Elasticities and Revenues TR/P < 0 when εp<-1 elastic P and TR opposite direction P TR P TR TR/P = 0 when (1+εp)=0 εp= -1 unitary elasticity TR/P > 0 when (1+εp)>0 0> εp> -1 P TR? P TR? 53 Elasticities and Revenues Intuition d = % change quantity % change in price Revenue = P*Q P TR Q TR so what happens to TR? depends on whether P or Q effect larger elastic -2 1 unitary elastic -1 1 inelastic -1 2 54 Elasticities and Revenues Intuition d = % change quantity % change in price Revenue = P*Q P TR Q TR so what happens to TR? depends on whether P or Q effect larger elastic -2 1 unitary elastic -1 1 inelastic -1 2 55 Where do they come from? Other Function Forms multiplicative: Q = aP-b Linearize ln Q = ln a -blnP Another way to write lnQ = -b = εp = Q P lnP P Q εp = -baP-b-1P/Q = -baP-b/aP-b = -b 56 Be Able to Compute for Other Functional Forms Other functional forms ln(Q) = a – bP + cY Q = a – bln(P) + c lnY ln(Q) = a - blnP + clnP2 + dlnY Q = a + bP +cY +dPY 57 Good for Back of the Envelope Forecasting Q Q x Q X X X Q X Q X X New Q = Q+Q = Q(1+Q/Q) X X Q 58 Sum Up Price Elasticity P = % change quantity % change in P Three ways to compute Q2-Q1 p = Q1+ Q2)/2 P2 - P1 (P1+ P2)/2 = Q P P Q = lnQ lnP 59 Price elasticity and revenue elastic P↑ → TR ↓ and P ↓ → TR ↑ elastic P↑ → TR ↑ and P ↓ → TR ↓ P 60 L9 - More on D&S Responsiveness Elasticities S1 S2 D2 D1 Q 61 Direct Purchases 62 Direct Purchases vs Cradle to Grave 63 Last Time Quantitative S-D Example Qd =99 - 2Pc + 1Psb - 2Pcm + 0.1Y Qs =30 + 1Pc – 1Pk - 0.2Pl - 0.4Pnr Pc = price of coal Psb = 1 Pcm = 10 Y = 200 Pk = 20 Pl = 40 Pnr = 10 Qd = 99 - 2Pcd + 1 - 2*10+0.1*200 = 100 -2Pcd Qs = 30 + Pcs – 1*31 - 0.2*30 - 0.4*10 = -11 + 1Pcs Invert, Shift, Solve for equilibrium, Price controls 64 Last Time Price Elasticity How responsive Qs or Qd is to price flatter is more responsive P = % change quantity % change in P P S1 Slr Dlr D1 Q Q 65 Last Time Price Elasticity P = % change quantity % change in P Three ways to compute Q2-Q1 p = Q1+ Q2)/2 P2 - P1 (P1+ P2)/2 = Q P P Q = lnQ lnP 66 Last Time Price elasticity and revenue elastic P↑ → TR ↓ and P ↓ → TR ↑ inelastic P↑ → TR ↑ and P ↓ → TR ↓ 67 Elasticities to Forecast Oil Price $40 to $60, p = -0.2, Q = 80 mb/d Q Q p P P 0 . 20 ( 60 40 ) 0 .1 40 Q = 80*(-0.1) = - 8 mb/d New Q = 80-8 = (1-0.1)80 = 72 million b/d 68 PQ and Q P P D Q 69 Compute Price Increase P = $2.50 Ep = -0.08 Q = -0.10 spread over 8 weeks = -0.0125 Q P = Q/Q = -0.0125 = 0.208 P εp -0.08 P = 0.208*2.5 = 0.521 P new = P + P = $2.50 + $0.521 = $3.021 70 Compute Price Increase P = $1.70 Ep = -0.08 Q = -0.10 Q P = Q/Q = -0.10 = 1.25 P εp -0.08 P = 1.25*1.70 = 2.125 P new = P + P = $1.70 + $2.125 = $3.825 71 Sum Up Defined arc and point elasticities Uses of Demand Price Elasticity Relationship of Revenue, Price and Elasticity Simple Forecasting 1. ΔQ=εpΔP Q P 2. ΔP = εp ΔQ P Q 72 L7- More on Elasticities |Elastic| > 1 P a/b =- a/2b D =-1 a/2 |Unit Elastic| = 1 |Inelastic| (1,0) =0 a Q 73 Last Time Defined demand price elasticities Arc Q2-Q1 (Q1+ Q2)/2 P2 - P1 (P1+ P2)/2 Point (∂Q/∂P)(P/Q) = ∂lnQ/∂lnP Relationship of Revenue, Price and Elasticity 74 Last Time Demand Elasticities and Revenue εp = LnQ = Q P LnP P Q Elastic < -1 TR/P= 1+εp <0 P TR P TR Unit Elastic = -1 TR/P= 1+εp = 0 P TR P TR Inelastic (-1,0) TR/P=1+εp>0 P TR? P TR ? 75 Last Time Simple Forecasting 1. ΔQ=εpΔP Q P 2. ΔP = εp ΔQ P Q 76 Forecast Using Income Elasticity Q y Q Y Y a. If εy China = 0.8, Q = 3 billion tons per year (almost half of world's total) income grows at historical rate of about 9% what is Qchina? Q Y Y Y Q = 0.8*0.09*3 = 0.216 billion tonnes 77 Price Change to Offset Coal Growth Let εy China = 0.8, Y/Y=0.09 εp = -0.5 What P/P do we need to choke off coal growth Q p Q P p P 0.5 P P P p P Q y Q y Y Y Y y Y 0.8 * 0.09 P P 0.144 78 Cross Price Elasticities of Demand Px = % change quantity % change in another good X price If Q and X are substitutes what is sign Px? coal and natural gas for electricity natural gas and electricity for heating If Q and X are complements what is sign Px? coal and boiler gasoline and automobile + - Q 79 Cross Price p Q Px Iooty, Queiroz, and Roppa (2007) P Cross price elasticity of ethanol with respect to gasoline substitute or complement? εpg=+0.6 QEth = 100 Pg = $2 per gallon Increases to $3 Q/Q = εPg (P/P) = 0.6*(1/2) = 0.3 Q new = Q(1+ Q/Q) = 100(1+0.3) = 130 What might happen to ethanol price? What might happen to sugar price? x 80 Create Function from Elasticity Q = PY Can add more variables εp= -0.80 ε y= 1.40 P =$1.15 Q = 8.00 Y = 5.40 = -0.8 = 1.4 Q = PY = (1.15-0.85.41.4) = Q/(PY) = 8/(1.15-0.85.41.4) = 0.84 Q = 0.84P-0.8Y1.4 81 Elasticities to create demand equations linear (Q = a + bP + cY) around the following values. Price Elasticity εp= -0.80 Income Elasticity ε y= 1.40 Price per gallon =$1.15 Consumption millions of barrels per day = 8.00 Income in trillions of U.S. dollars = 5.40 82 Create Demand from Elasticities Q = a + bP + cY P =$1.15 Q = 8.00 Y = 5.40 εp= -0.80 εy= 1.40 p = (dQ/dP)(P/Q) = b*(P/Q) -0.8 = b(1.15/8) b = -0.8*8/1.15=-5.57 y = (dQ/dY)Y/Q = c*(Y/Q) 1.4=c(5.4/8) c = 1.4*8/5.4 = 2.07 a = Q - bP - cY = 8 - (-5.57)*1.15 - 2.07*5.4 = 3.2 Q = 3.2 - 5.57Pd + 2.07Yd Could add another variable X Need values ε and X 83 Another Way With Constant Elasticity of Demand Example: Q1= P1 Q2 =P2 Q2/Q1 = (P2/P1) if P growing at 2 then P2/P1 = 1.02P1/P1= 1.02 Q2/Q1 = (1.02) only need ratio prices or growth rates no units to forecast Q2 multiply Q1 by your forecast of Q2/Q1 only works exactly for constant elasticity functions 84 Forecasts with Elasticity-2 ^ Q2 = (0.98Ps1)b2(1.01Ppl)b3(1.02Pal) b4(1.015Y1)b5 Q1 Ps1b2 Pplb3 Palb4 Y1b5 = 0.98b21.01b31.02b41.015b5 = 0.98-0.826571.010.1991.02 0.437141.015 1.10167 = 1.0447 Q1 = 111.9 ^ Q2 = 1.0447*Q1 = 1.0447*111.9= 116.9 85 Sum Up: Why are Demand Elasticities important? Why are they important? Forecast P→Q Q→P Y→Q Pother → Q P,Y, etc. Q ΔQ/Q = εp(ΔP/P) ΔP/P = (ΔQ/Q)/εp ΔQ/Q = εy(ΔY/Y) ΔQ/Q = εo(ΔPo/Po) ΔQ1/Q1+ ΔQ2/Q2 Policy analysis P to offset Y increase p Effect of carbon tax Create demand from elasticities linear and log P P Y y Y Chapter 3 87 Elasticities so Far 1. Measure of responsiveness 2. Where do they come from? a. compute from market data P Q 6 80 4 100 (Q2-Q1) = (Q1+ Q2)/2 P2 - P1 (P1+ P2)/2 problem if other variables change beside P 88 Elasticities so Far 2. Where do they come from? b. Estimate whole function market data Qd = f(Pd, Y, Ps, Pc, . . ., etc. ) εp = LnQ = Q P LnP P Q Function Forms linear: Q = a – bP εp= -b(P/Q) multiplicative: Q = aP-b εp= -b mixed: ln(Q) = a – bP εp= -bP mixed: Q = a – bln(P) εp= -b/Q Other 89 Demand Elasticities εx = LnQ = Q X LnX X Q |Elastic| > 1 |Unit Elastic| = 1 |Inelastic| (1,0) 3. Uses of elasticity price to revenue (P*Q) forecasting PQ QP YQ PcrossQ policy: price increase to offset income growth 90 Price Change to Offset Coal Growth Let εy China = 0.8, Y/Y=0.09 εp = -0.5 What P/P do we need to choke off coal growth Q p Q P p P 0.5 P P P p P Q y Q y Y Y Y y Y 0.8 * 0.09 P P 0.144 91 YQ and Q Y? P Po D Q D' 92 Studies of Oil Price on U.S. Macro Economy GDP Po Po GDP studies seem to suggest around 0.05 smaller than in 1970s and 1980s asymmetric affect when prices up not down mechanism GDP (K, L, O, etc.) - less oil GDP Po more inflation, tighter monetary policy r up, GDP down Po income transfer to OPEC 93 Elasticity Approximation - Linear Q = 20-4P P = 3 Q= 8 εp = -4*3/8 = -1.5 P2= 4 Use elasticity dQ/Q = εp*dP/P = -1.5*1/3 = -0.5 Qnew = Q(1+dQ/Q) = 4 With function Qnew = 20 - 4*4 = 8(1-0.5) = 4 94 Elasticity Approximation - Log Q = 10P-1 P=2 Q=5 εp= -1 P2 = 2.1 Use elasticity: dQ/Q = εp*dP/P = -1*0.1/2 = -0.05 Qnew = Q(1+dQ/Q)= 5*(1-0.05) = 4.75 With function Qnew= 10(2.1)-1 = 4.76 Approximation gets worse the larger the price change Q 95 Cross Price p Q Px P Iooty, Queiroz, and Roppa (2007) Cross price elasticity of ethanol with respect to gasoline substitute or complement? εpg=+0.6 QEth = 100 Pg = $2 per gallon Increases to $3 Q/Q = εPg (P/P) = 0.6*(1/2) = 0.3 Q new = Q(1+ Q/Q) = 100(1+0.3) = 130 What might happen to ethanol price? What might happen to sugar price? x 96 Create Function from Elasticity Q = PY Can add more variables εp= -0.80 ε y= 1.40 P =$1.15 Q = 8.00 Y = 5.40 = -0.8 = 1.4 Q = PY = (1.15-0.85.41.4) = Q/(PY) = 8/(1.15-0.85.41.4) = 0.84 Q = 0.84P-0.8Y1.4 97 Elasticities to create demand equations linear (Q = a + bP + cY) around the following values. Price Elasticity εp= -0.80 Income Elasticity ε y= 1.40 Price per gallon =$1.15 Consumption millions of barrels per day = 8.00 Income in trillions of U.S. dollars = 5.40 98 Create Demand from Elasticities Q = a + bP + cY P =$1.15 Q = 8.00 Y = 5.40 εp= -0.80 εy= 1.40 p = (dQ/dP)(P/Q) = b*(P/Q) -0.8 = b(1.15/8) b = -0.8*8/1.15=-5.57 y = (dY/dQ)Y/Q = c*(Y/Q) 1.4=c(5.4/8) c = 1.4*8/5.4 = 2.07 a = Q - bP - cY = 8 - (-5.57)*1.15 - 2.07*5.4 = 3.2 Q = 3.2 - 5.57Pd + 2.07Yd Could add another variable X Need values ε and X 99 Another Way With Constant Elasticity of Demand Example: Q1= P1 Q2 =P2 Q2/Q1 = (P2/P1) if P growing at 2 then P2/P1 = 1.02P1/P1= 1.02 Q2/Q1 = (1.02) only need ratio prices or growth rates no units to forecast Q2 multiply Q1 by your forecast of Q2/Q1 only works exactly for constant elasticity functions 100 Forecasts with Elasticity-2 ^ Q2 = (0.98Ps1)b2(1.01Ppl)b3(1.02Pal) b4(1.015Y1)b5 Q1 Ps1b2 Pplb3 Palb4 Y1b5 = 0.98b21.01b31.02b41.015b5 = 0.98-0.826571.010.1991.02 0.437141.015 1.10167 = 1.0447 Q1 = 111.9 ^ Q2 = 1.0447*Q1 = 1.0447*111.9= 116.9 101 Another Way With Constant Elasticity of Demand Example: Q1= P1 Q2 =P2 Q2/Q1 = (P2/P1) if P growing at 2 then P2/P1 = 1.02P1/P1= 1.02 Q2/Q1 = (1.02) only need ratio prices or growth rates no units to forecast Q2 multiply Q1 by your forecast of Q2/Q1 only works exactly for constant elasticity functions 102 Forecasts with Elasticity ^ Q2 = (0.98Ps1)b2(1.01Ppl)b3(1.02Pal) b4(1.015Y1)b5 Q1 Ps1b2 Pplb3 Palb4 Y1b5 = 0.98b21.01b31.02b41.015b5 = 0.98-0.826571.010.1991.02 0.437141.015 1.10167 = 1.0447 Q1 = 111.9 ^ Q2 = 1.0447*Q1 = 1.0447*111.9= 116.9 103 Tax Qualitative tax affect on price, quantity, government revenue or cost Ps+t P incidence Ps social welfare of tax Unit Ps + t = Pd Pe add to supply Ps = Pd-t Pd subtract from demand Pd-t Q Qe 104 Tax Supplier What happens to P and Q Ps+t Ps + t = Pd add to supply P Ps Pd’ Pe Ps’ Pd Qe’ Qe Q 105 Tax Government Revenues Ps+t Ps P Pd’ t Pe Ps’ Pd Qe’ Qe Q 106 Coal Ad Valorem Tax 50 % of Price ←(1+t%)Ps P Ps Pd' Pe Ps' tax Pd Qe' Qe Q Ad Valorem 50% of Ps (1+0.5)Ps = Pd 107 Tax Demander Subtract from Pd: What happens to P and Q Ps = Pd -t Ps P Pd’ Pe Ps’ Qe’ Qe Pd Pd-t Q 108 Coal Ad Valorem Tax 50 % of Buyer Price Ad Valorem 50% of Pd (1-0.5)Pd = Ps P Ps ←(1-t%)Pd Pd' Pe Ps' tax Pd Qe' Qe Q 109 Quantitative Model Tax supplier Qd = 30 -2Pd P 15 Qs = -3 + Ps 13 Solve for equilibrium 11 30 -2P=-3+P 7 P = 11, Q = 8 Add tax of 6 to supply price Invert demand and supply 3 Pd = 15 - 0.5Qd Ps = 3 + Qs Pd=15 - 0.5Qd = Ps+t = 3 + Qs + 6 Solve Q = 4, Pd=13, Ps = 7 Ps+6 Ps Pd 4 8 Q 30 110 Government Revenues Supplier tax of 6 Q=4 t=6 t*Q = 6*4=24 P Ps+6 15 Ps 13 11 7 Pd 3 4 6 Q 30 111 Who Pays the Tax Depends on Shape of Demand and Supply Ps+t Perfectly Elastic Supply Ps+t P P Ps Pd' Pd' Ps Pe Ps'= Pe Ps' Pd Qe' Qe share the tax Q Pd Qe' Qe consumer pays Q 112 Incidence of Tax Depends Shape of Supply and Demand (Practice Four Extreme Cases) P P Ps+t Pd-t Ps D P Q Ps+t S Pd Q Ps P Ps+t Ps Pd Q Q 113 Incidence Depends on Elasticity Inputs: d = -0.5 s = 1 dPd = εs dPs εd dPd+|dPs| = t dPd-dPs = t dPd=t+dPs But also dPd = εs/εddPs = (1/-0.5)dPs (1/-0.5)dPs-dPs = 4.50 -3dPs=4.50 dPs =-1.50 dPd =4.5 + (-1.5)=3 t = 4.5 114 Social Welfare Effects: Behind the Supply Curve Perfect competitors take P from market = PQ – TC(Q) pick Q to maximize f.o.c /Q = P – TC(Q)/Q = 0 P – MC = 0 P = MC 2.o.c 2/Q2 = – TC(Q)2/Q2 < 0 TC(Q)2/Q2 > 0 MC slopes up - increasing marginal cost 115 Social Welfare - Producer Surplus P Ps Pe Pd Qe Q Price Set by Marginal Producer and Consumer Ricardian Rent 116 Social Welfare - Consumer Surplus P Ps=MC Pe Pd=Marginal Benefit Qe Q 117 Social Welfare Effects: Behind the Supply Curve Perfect competitors take P from market = PQ – TC(Q) pick Q to maximize f.o.c /Q = P – TC(Q)/Q = 0 P – MC = 0 P = MC 2.o.c 2/Q2 = – TC(Q)2/Q2 < 0 TC(Q)2/Q2 > 0 MC slopes up - increasing marginal cost 118 Social Welfare P 15 Qd = 30 -2Pd Qs = -3+ Ps 11 Invert demand Pd = 15 - 0.5Qd Ps = 3 + Qs Solve for Equilibrium 3 P = 11, Q = 8 Consumer Surplus = ½(15-11)*8 =16 Ps Pd 8 Q 30 Producer Surplus = (1/2)(11-3)*8= 32 119 Ps+6 Welfare Cost of a Tax P DWL = 15 Pd = 15 - 0.5Qd Ps = 3 + Qs Pd=13 t=6 Pe=11 Invert demand Ps=7 Ps + t = Pd 3 + Qs + 6 = 15 - 0.5Qd 3 1.5Q = 6 Q=4 Ps= 7 Tax revenues = Pd= 13 6*4 = 24 (1/2)(13-7)*(8-4) Ps Pd 4 8 CS=16 Q 30 PS=32 120 Welfare Cost of a Subsidy P 15 DWL = Pd = 15 - 0.5Qd (1/2)(10.4-8)*(13.4-9.8)= Ps=13.4 Ps = 3 + Qs Ps-3.6 Ps Qe=8, Pe=11 11 sb= 3.6 Pd= 9.8 Ps - sb = Pd Pd 3 + Qs - 3.6 = 15 - 0.5Qd 3 1.5Q = 15.6 8 10.4 Q 30 Q = 10.4 Ps= 13.4 CS=16 PS=32 Subsidy Cost = Pd= 9.8 3.6*10.4= CS'=? PS'=? 121 Calculating the Deadweight Loss in Practice Supply Elasticity (eS) 1.2 Demand Elasticity (eD) 0.3 Initial Gasoline Price per Gallon Initial Gallons Supplied / Demanded Per-unit Tax $2.40 4,500,000 $0.32 2 0.32 0.3 *1.2 D W 0.5 (2.4) * 4, 500, 000 $38, 400 2.40 1.2 0.3 122 Sum Up P 15 Pd=13 Ps Pe=11 Ps=7 Pd 3 4 8 Q 30 123 DWL and Elasticity Ps+t P Ps+t P S Ps D Pd Q Q More losses the more elastic are demand and 124 U.S. Tariff on Brazilian Ethanol (2008) P U.S. Ethanol Market Sus tariff revenues Pw + tariff Pw Dus Q1Q2Q3 Q4 Q 125 Brazil Ethanol 2008 Exports of Q4-Q3 P SBrazil Pw DBrazil Q3 Q4 Q 126 MR = MC P MC Pm P(Q) MR = MC ATC ACm Qm MR Q Monopoly profit = TR – TC = Pm*Qm – AC*Qm = Pm*Qm – o Qm MC dQm 2.o.c. Is slope MR< slope of MC? 127 MR = MC P MR = MC Pm P(Q) ACm Qm MR ATC MC Q Monopoly profit = TR – TC = Pm*Qm – AC*Qm = Pm*Qm – o Qm MC dQm 2.o.c. Is slope MR< slope of MC? 128 Social Optimum P = MC P Pd ATCo Pso Qso Losses Q ATC MC Pd = MC Choices: regulate or government own P = MC collect losses some other way P = ATC Chapter 4 130 tax revenues Incidence of Tax Ps+t P Ps t Pd Pd Pe Ps Ps Pd Q’ Qe Q 131 Incidence of Tax Depends on Demand Shape Two Extreme Cases P Perfectly inelastic D Ps+t D P Perfectly elastic D Ps+t Ps Ps Pd Ps P=Ps Pe=Pd D Ps Qe= Q’ Q Q’ Qe Producer Pays Q 132 Incidence of Tax Depends on Supply Shape Two Extreme Cases Perfectly elastic S P Pd Ps+t Pe=Ps Ps Perfectly inelastic S S P Pe=Pd Pd Ps Q’ Qe Consumer Pays Q Qe= Q’ Pd-t Q 133 Incidence of Tax – Depends on Elasticity Depends on elasticity d = -0.5 s = 1 t = 1.5 dPd = εs = 1 dPs εd -0.5 Ps+t P Ps Pd Pe Ps Pd Q' Qe Q 134 Incidence of Tax – Depends on Elasticity dPd = εs = 1 dPs εd -0.5 Ps+t P Ps Pd (1) dPd = (1/-0.5)dPs = -2dPs Pe Ps dPd-dPs = t dPd>0 dPs<0 Pd (2) dPd-dPs = 1.5 Two equations two unknowns Q' Qe Q 135 Solve Two Equations for dPs, dPd Ps+t P Ps (1) dPd = -2dPs Pd (2) dPd-dPs = 1.5 Pe Substitute (1) into (2) Ps -dPs-dPs = 1.5 -3dPs = 1.5 dPs = -0.5 dPd = -2Ps = -2(-0.5) = 1 dPd=1 dPs=-0.5 Pd Q' Qe Q 136 Social Welfare Qd = 30 -2Pd Qs = -3 + Ps P = 11, Q = 8 Invert demand Pd = 15 - 0.5Qd Ps = 3 + Qs P 15 Ps 11 Pd 3 8 Consumer Surplus = ½(15-11)*8 =16 Q 30 137 Behind the Supply Curve Perfect competitors take P from market = PQ – TC(Q) pick Q to maximize f.o.c /Q = P – TC(Q)/Q = 0 P – MC = 0 P = MC 2.o.c 2/Q2 = – TC(Q)2/Q2 < 0 TC(Q)2/Q2 > 0 MC slopes up- increasing marginal cost 138 S = MC in competitive market P S = MC 139 S = MC in competitive market P S = MC Pe Producer Surplus 140 Social Welfare Qd = 30 -2Pd Qs = -3 + Ps P = 12, Q = 6 Invert demand Pd = 15 - 0.5Qd Ps = 3 + Qs P Consumer Surplus = ½(15-11)*8 =16 15 Ps 11 Pd 3 8 Q 30 Producer Surplus = (1/2)(11-3)*8 = 32 141 Social Welfare - Consumer Surplus P Ps=MC Pe Pd=Marginal Benefit Qe Q 142 Social Welfare - Producer Surplus P Ps Pe Pd Qe Q Price Set by Marginal Producer and Consumer Hotelling Rent 143 Welfare loss from an ad valorem tax Ps(1+t%) P Ps Pd Q'QePd(Q)dQ Q'QePs(Q)dQ Pe Ps Pd Q' Qe Q 144 Government Revenues Ps(1+t%) P Ps Pd Pe Ps Pd Q' Qe Q 145 Change in Consumer and Producer Surplus Ps(1+t%) P Ps Pd Pe Ps Pd Q' Qe Q 146 Welfare loss from unit subsidy tax P Ps Ps Ps-sb Q'QePs(Q)dQ Q'QePd(Q)dQ Pe Pd Pd Qe Q' Q 147 What if You Export Your Product? Ps(1+t%) P gain Ps Pd Pe loss Ps Pd Q' Qe Q 148 What if your Demand is Perfectly Elastic Ps(1+t%) P Ps Pd'=Pe Pd Ps' Q' Qe Q 149 Tariff is a Tax on Imports Small Consumer and Producer Crude Price determined on world markets S P Pw D Qs Qd Q 150 Tariff on Crude Imports Add tariff t S P Pw+t Pw D Qs Qd Qs' Qd' Q 151 Tariff on Crude Imports Add tariff t S P Pw+t Pw D Qs Qd Qs' Qd' Q 152 Welfare loss from unit subsidy tax benefit to producer cost to government P Ps Ps-sb Ps' Pe Pd' benefit to consumer Pd Qe Q' Q DWL loss 153 Welfare one wrinkle Price increase/decrease what happens to consumer welfare P e.g. price increase – buy less loss in consumer surplus P2 P1 But two effects use less so lose utility but less real income D Q 154 Look at real income with two goods X1 P1X1 + P2X2=Y Example 2X1 + 4X2=100 Graph X1 = 100/2 – (4/2)X2 X1 = 50 – 2X2 Raise P1 to 4 X1 = 100/4 – (4/4)X2 X1 = 25 – X2 50 25 Budget 25 X2 155 Sum Up Tariff from Last Time P U.S. Ethanol Market Sus Loss in consumer surplus from tariff Pw + tariff Pw Dus Q1Q2Q3 Q4 Q 156 Sum Up Tariff from Last Time P U.S. Ethanol Market Sus Gain in domestic producer surplus from Pw + tariff tariff Pw Dus Q1Q2Q3 Q4 Q 157 Sum Up Tariff from Last Time P U.S. Ethanol Market Sus tariff revenues social loss Pw + tariff Pw Dus Q1Q2Q3 Q4 Q 158 Electricity - Decreasing Cost Industry P D ATC Q Natural Monopoly 159 MR = MC P MC Pm P(Q) ACm Qm MR ATC Q MR = MC Monopoly profit = TR – TC = Pm*Qm – AC*Qm = Pm*Qm – o Qm MC dQm 2.o.c. Is slope MR< slope of MC? 160 Example (small village) – Monopoly Solution P is US cents per kWh Q is kWh per year P demand is 75 P = 75 - 4Q total cost curve in cents is TC = 19Q - 0.25Q2 45.132 AC = TC/Q = 19 – 0.25Q 19 MC = TC/Q = 19 –0.50Q MR = 75 - 8Q =MC=19 –0.50Q Q = 7.467 P = 75 – 4(7.467)=45.132 Monopoly Profits 7.467 D 18.75 Q 161 Example (small village) – Monopoly Solution P is US cents per kWh = 45.132 Q is kWh per year = 7,467 P TC = 19Q - 0.25Q2 75 = PQ – TC =45.132*7.467 -19(7.467) + 0.25(7.467)2= 209.1 units 45.132 Units 19 PQ = cents/kWh*kWh = cents TC must be measured in cents What if TC measured in $ TC$*100¢ $ Monopoly Profits 7.467 D 18.75 Q 162 Social Optimum – Maximize Welfare P D P1 Q1 Q MC welfare (W) = sum of consumer plus producer surplus CS = area below demand and above price PS =area above marginal cost and below price Q Q W = 0 Pd(X)dX - PQ + PQ - 0 MC(X)dX 163 What is Q 0 MCdQ P MC ` Q ∫0QMCdQ=TVC 164 What is Social Loss with Natural Monopoly Decreasing Average Cost = Natural Monopoly Monopoly MR = MC Optimum P = MC P Social Loss Pm P(Q) Po Qm Qo MR market failure Q ATC MC 165 Lets Examine the Optimum P Pd ATCo Pso Qso Losses Q ATC MC Pd = MC Choices: regulate or government own P = MC collect losses some other way P = ATC 166 Example piqi < expenses + s(RB) i=1 pi qi 0.08 i=2 0.05 s = 10.5% ci oi RB 1,966,667 0.02 0.03 750,000 799,999 0.01 0.02 0.08*1,966,667 + 0.05*799,999 < (0.02+0.03)*1,966,667 + (0.02+0.01)*799,999 +0.105*750,000 197,333.31 ? 122,333.32 + 78,750= 201,083.32 < 201,083.32 Rates would be approved R7 Examples Discounting (Annual Compounding) B dollars, interest rate r, in t years, annual compounding B=10, r=0.1, t=20, then B/(1+r)t =10/(1+0.1)20 = $1.486 B=10, r=0.2, t=20, then B/(1+r)t =10/(1+0.2)20 = $0.261 B=10, r=0.2, t=40, then B/(1+r)t =10/(1+0.2)40 = 0.007 B=20, r=0.2, t=40, then B/(1+r)t = 20/(1+0.2)40 = 0.014 B=20, r=0.0, t=40, then B/(1+r)t = ? R8 Compounding More than Once a Year Compounding twice a year (r annual rate) one half year A (1+r/2) after a year A(1+r/2)(1+r/2) after a year and a half A (1+r/2)3 after t years or 2t half years A (1+r/2)2t Example A = 20, r = 8%, t = 10 20(1+0.08/2)2*10 = $43.82 Compare to compounding annually 20(1+0.08) 10 = $43.18 R9 Compounding p times a year compounding p times a year A(1+r/p)tp A = 100, t = 50, r = 10% p=4 13956.39 p = 10 14477.28 p = 365 14831.16 continuous compounding p goes to = ertA = 14841.32 R11 Discounting with Compounding p Times a Year B dollars in t years at interest rate 10% is worth ? today A(1+r/p)tp=B A = B/(1+r/p)tp B = 100, t = 50, r = 10% p=4 0.717 p = 10 0.691 p = 365 0.674 continuous compounding p goes to B = ertA A = B/ert = Be-rt = 100e-rt A = 100e-0.10*50 = $0.674 R14 Value a Stream of Income D1 dollars at the end of 1 year D2 at the end of 2 years NPV = D1 + D2 (1+r) (1+r)2 Example D1 = 50, D2 = 51, r = 0.10 NPV = 50 + 51 = $89.256 (1.1) (1.1)2 Could have changing interest rates NPV = D1 (1+r1) + D2 (1+r2)2 R20 Internal r (IRR) Invest Equipment costing 100 now year 0 Yields income after 1 and 2 years of 60 59 Flow of income is -100 60 59 NPV of flow of income is -100 + 60 + 59 (1+r) (1+r)2 solve for the r that makes NPV = 0 R21 Internal r (IRR) Solve for r that Makes NPV 0 -100 + 60 + 59 = 0 (1+r) (1+r)2 Alternatively rearrange 100 = 60 + 59 (1+r) (1+r)2 Find r that makes price of asset (100) = DCF of income flow Solve: 100(1+r)2 = 60(1+r) + 59 100(1+2r+r2) = 60 +60r +59 100r2 +140r - 19 =0 R22 Using Quadratic Formula 100r2 +140r - 19 =0 Quadratic formula ar2 +br + c =0 -b (b2 - 4ac)0.5 = -140 (1402 - 4*100*140)0.5 2a 2*100 = 0.125 = 1.525 Excel alternative - Put stream of income in A1 to A3 -100 60 59 =irr(a1.a3,guess) = irr(a1.a3, 0.05) = 12.5% seems to always take + root R24 Internal r (IRR) Power Plant Power plant costing 200 now year 0 two years to build Stream of income -100 -100 30 65 65 25 65 65 65 -20 What is the NPV or DCF of this power plant? -100 -100 + 30 + 65 + 65 + 25 + 65 + . . . – 20 (1+r) (1+r)2 (1+r)3 (1+r)4 (1+r)5 (1+r)6 (1+r)9 solve for the r that makes above sum zero = irr(addresses, guess) = 14.4% to see other excel functions >insert >function 177 Fully Distributed Cost (FDC) Lump Sum to Each Consumer Class residential customers (L) CL = 1200 + 20QL industrial (H) CH = 1000 + 10QH if produce both CLH = 1500 + 20 QL + 10QH but CL + CH = 2200 + 20QL + 10QH sub-additive How to allocate 1500? one group not subsidize another > CQLQH 178 Marginal Cost Pricing for Low Voltage P 80 PL = 80 – 2QL CL = 1200 + 20QL CLH = 1500 + 20 QL + 10QH MCL = 20 20 PL = MCL 80 – 2QL = 20 80-20 = 2QL QL = 30 PL = 20 Consumer surplus 0.5(80-20)30 = 900 PL MCL 30 40 QL Standalone Fixed 1200 179 You Do Marginal Cost Pricing for High Voltage CLH = 1500 + 20 QL + 10QH PH = 100 – 3QH MCH= PH PH MCH QH = QH PH = Consumer surplus Standalone fixed 180 What is Maximum We Should Charge H 1. Charge less than stand alone 2. Charge less than consumer surplus What is maximum we can charge H? PH = 100 – 3QH CLH = 1500 + 20QL + 10QH P CH = 1000 + 10QH Stand fixed cost = Consumer surplus = PH MCH QL 181 Pricing Across Time - Peak load pricing one simple case – quantity independent of price in other period peak shifting more complicated problem P ck+co Dpk Dopk Q ck 182 CS peak Peak load pricing Social optimum Ppk = ck + co Popk = co P Qpk co+ck Qopk Qopk' Qpk' CS offpeak co Q 183 Numerical Example Peak Load Pricing No peak switching Qpk = 50 - 5Ppk Qopk = 8 - 2Popk ck = 3 co = 2 Ppk = ck + co Popk = co Ppk = 10 - (1/5)Qpk Popk = 4 - (1/2)Qopk 184 Solve for Qpk and Qopk Social optimum Ppk = 10 - (1/5)Qpk = ck + co = 3 + 2 = 5 10 - (1/5)Qpk = 5 Qpk = 25 P Qpk Popk = 4 - (1/2)Qopk = co = 2 Ppk 4 - (1/2)Qopk = 2 Qopk = 4 Popk Co+Ck=5 Qopk 4 25 Co= 2 Q 185 Often Charge One Price If Charge One Price: P=5 Social Loss P Qpk co + ck=5 P=5 Qopk Qopk' Qpk' co = 2 Q 186 If charge one price: P = 2 Social Loss P Qpk co+ck=5 Qopk co=2 P=2 Qopk Qpk Qpk' also not covering capital cost Q 187 If charge one price: P=2.5 See if you can figure out losses Losses in Off Peak Losses in Peak P Qpk P=2.5 Co+Ck=5 Qopk Co= 2 Qopk Qpk Qpk' Qopk' Q Peak Load Price if Losses Greater than Metering Cost 188 Fully Distributed Cost (FDC) Lump Sum to Each Consumer Class residential customers (L) CL = 1200 + 20QL industrial (H) CH = 1000 + 10QH if produce both CLH = 1500 + 20 QL + 10QH but CL + CH = 2200 + 20QL + 10QH sub-additive > CLH 189 Fully Distributed Cost (FDC) Lump Sum to Each Consumer Class residential customers (L) CL = 1200 + 20QL industrial (H) CH = 1000 + 10QH if produce both CLH = 1500 + 20 QL + 10QH but CL + CH = 2200 + 20QL + 10QH sub-additive How to allocate 1500? one group not subsidize another > CQLQH 190 You Do Marginal Cost Pricing for High Voltage CH = 1000 + 10QH CLH = 1500 + 20QL + 10QH PH = 100 – 3QH MCH= QH = PH PH MCH QH PH = Consumer surplus Standalone fixed 191 You Do Marginal Cost Pricing for High Voltage PH CH = 1000 + 10QH 100 CLH = 1500 + 20 QL + 10QH PH = 100 – 3QH MCH= 10 10 100 – 3QH=10 QH =30 PH MCH 30 33.33 QH PH = 10 Consumer surplus 0.5(100-10)*30 =1350 Standalone fixed 1000 192 Pricing Across Time - Peak load pricing one simple case – quantity independent of price in other period peak shifting more complicated problem P ck+co Dpk Dopk Q ck 193 Two Curves Shift - Gasoline Market Oil Prices Up P ↑ Q↓ Income Up P ↑ Q↑ ←S' P S S Pe' Pe Pe' Pe D Q' Q D Q Q Q' →D' Q 194 Incidence of Subsidy – on Supply P Ps Ps-sb Ps' Pe Pd' Pd Qe Q' Q 195 Incidence of Subsidy – on Demand P Ps Ps' Pe Pd' Pd+sb Pd Qe Q' Q 196 Pmax P S Pmax not binding Pmax D Qd Qs Q 197 MR = MC P MR = MC Pm P(Q) ACm Qm MR Q ATC MC Monopoly profit = TR – TC = Pm*Qm – AC*Qm 2.o.c. Is slope MR< slope of MC? 198 Marginal Cost Pricing PQL = 100 – 2QL PQL = 70 – 4QL PL 100 PQL PQL = 100 – 2QL = MCL = 20 20 100-20 = 2QL 50 40 QL = 40 PQH = 70 – 4QH = MCL = 30 70-30 = 4QH QH = 10 Haven't allocated fixed costs of 1700 MCL QL 199 Marginal Cost Pricing PQL = 100 – 2QL 100 CQLQH = 1700 + 20 QL + 30QH CQL = 1400 + 20QL PL PQL 20 Consumer surplus 0.5(100-20)40 = 1600 MCL 40 50 QL 200 Pricing Across Time - Peak load pricing one simple case – quantity independent of price in other period peak shifting more complicated problem 201 Example (small village): P is US cents per kilowatt hours Q is measured in kilowatt hours per year demand and total cost curve are P = 75 - 4Q TC = 19Q - 0.25Q2 AC = TC/Q = 19 – 0.25Q TC/Q = MC = 19 –0.50Q MR = 75 - 8Q = MC = 19 –0.50Q => Q = 7.467 P = 75 – 4(7.467)=45.132 = PQ – TC=45.132*7.467* -19(7.467) + 0.25(7.467)2 = 209.1 202 Example (small village): P cents per kilowatt hours Q kilowatt hours per year P P = 75 - 4Q 75 MR = 75 – 8Q AC = 19 – 0.25Q 45.132 MC = 19 – 0.50Q Q = 7.467 19 P = 45.132 Monopoly Profits D = 209.1 units? 7.467 18.75 Q 203 Social Optimum P D P1 Q1 Q MC welfare (W) = sum of consumer plus producer surplus CS = area below demand and above price PS =area above marginal cost and below price W = 0QPd(Q)dQ - PQ + PQ - 0QMCdQ 204 Maximize CS = 0QPd(Q)dQ - 0QPs(Q)dQ maximizing the area between D and MC f.o.c. W = 0QPd(Q)dQ - 0QMCdQ = 0 Q Q Q = Pd(Q) – MC = 0 2.o.c. 2CS = Pd(Q) – MC < 0 Q2 Q Q Pd(Q)< MC Q Q Slope of inverse demand less than slope of MC 205 Last Time Quiz - Cost Curves Sunk costs are part of total costs TC = FC+VC P MC = TC/ TC ATC = ATC/Q FC=FCsun + FCnosunk FCsunk Q Chapter 5 207 Generating Costs D' D 208 Price Regulation Transportation and Distribution 1. Rate of Return (piqi < expenses + s(RB) U.S. 2. Price Cap (RPI-X) prices can to up no more than (RPI) rate of inflation - (X) rate of productivity change CPI07= 115 and CPI08 = 123 RPI=(123-115)/115 = 0.07 productivity change some measure of output/input (O/I) (O/I)07=0.21 and (O/I) 08=0.22 209 Price Regulation Transportation and Distribution X = (0.22-0.21)/0.21 = 0.048 RPI-X = 0.07 - 0.048 = 0.022 P07 = $0.10 P08<(1+0.022)*0.10 = $0.102 popular in UK 3. Light Handed New Zealand 4. Yardstick Scandinavia 210 Wholesale Market Q1=19 One sided bidding hour ahead, day ahead get bids – put in order one-sided P P1 Q1 Q 211 Wholesale Market System Marginal Price = SMP Two sided bidding again get bids – put in order P $0.07 P1 Q1 Q 212 SMP = System Marginal Price + Capacity Charge Total capacity charge loss of load probability (LOLP) times value of the lost load (VOLL) Example: 5% probability of a 10 kWh short fall. loss of output from a 10 kWh shortfall ~ $15 LOLP*VOLL = (0.05*$15)/ = $0.75 Dividing this over all kilowatts consumed (100 kWh) CC = $0.75/100 = $0.0075 Power Pool Price = PPP PPP = SMP + CC = 0.07 + 0.0075 = $0.0775 213 How to allocate power at capacity Role of price signals Gaming the system S P Dpk Dopk Qopk Q 214 Last time: SMP = System Marginal Price + Capacity Charge Total capacity charge loss of load probability (LOLP) times value of the lost load (VOLL) Example: 5% probability of a 10 kWh short fall. loss of output from a 10 kWh shortfall ~ $15 LOLP*VOLL = (0.05*$15)/ = $0.75 Dividing this over all kilowatts consumed (100 kWh) CC = $0.75/100 = $0.0075 PPP = SMP + CC = 0.07 + 0.0075 = $0.0775 215 1990 - 1999 Demand up Supply down S' P S Pe D Qs' Qe Imports Qd' Q D' Chapter 6 217 Typical Competitive Firm Cost Short Run Supply P S P P MC1 MC2 AVC1 AVC2 Psr D Q1 Q Q2 Si = MCi above AVC Market is horizontal sum Q Q1 +Q2 Q 218 Last Time Reviewed- Long Run Supply With Entry and Exit srMCi = Ssr P Slr D1 D2 Increasing Cost Industry D3 D Q 219 Long Run Supply With Entry and Exit srMCi = Ssr P Slr D1 D2 Increasing Cost Industry D3 D Q 220 Long Run Supply With Entry and Exit srMCi = Ssr P Slr D1 D2 Constant Cost Industry D3 D Q 221 Long Run Supply With Entry and Exit srMCi = Ssr P Slr D1 D2 Constant Cost Industry D3 D Q 222 Inelastic Supply and Demand P S' S S'' P S' S S'' P1 D D Q Q 223 Multiplant Monopoly Marginal Cost – 2 countries TC1 = 10 + Q1 + (1/2)Q12 TC2 = 20 + 2Q2 + Q22 MC1 = TC1/ Q1 = 1 + Q1 MC2 = TC2/ Q2 = 2 + 2Q2 224 MC for Monopolist- Horizontal Sum Firm 1 MC1= 1 + 1Q1 MC Firm 2 MC2 = 2 + 2Q2 MC2 MC1 MC= 1 + 1Q 0<Q<1 Q = Q1+Q2 Q >1 MC1+2 2 1 Q1 Given MC sum the Q Q2 225 MC Above Kink Firm 1 MC1= 1 + 1Q1 MC Firm 2 MC2 = 2 + 2Q2 Q = Q1+Q2, Q>1 MC1 MC1+2 2 1 Q1 Given MC sum the Q Q1 = -1 + MC1 Q2 = -1 + (1/2)MC2 Q1 + Q2 = -2 + (3/2)MC Q = -2 + (3/2)MC MC = 4/3 + 2/3Q 226 Now Add Demand What Should Monopolist Do? P=75-0.5Q P MC1 MR = MC 52.9=Pm MR= 75-Q=4/3+2/3*Q Q=44.2 P=75-0.5Q = 75-0.5*44.2 = 52.9 MC1+2 MCm MC=MR = 75-44.2=30.8 Q1 = -1 + MC 2 D 1 = -1 + 30.8 = 29.8 1 Qm MRQ Q2 = -1 + (1/2)MC =44.2 =-1 + (1/5)30.8 = 14.4 227 Sum Up Competitive Market Short Run Supply Competitive Market P = MC above AVC P P P MC1 MC2 Q1 ΣMCi MC3 Q2 Q3 MCi=fi(Qi) Invert Qi = fi-1(MCi) Horizontal Sum Q1+Q2+Q3= f1-1(MC1)+ f2-1(MC2) + f3-1(MC3) Set Q = Q1+Q2+Q3 and MCi=MCj Q 228 Competitive Market Long Run Supply With Entry and Exit Increasing, Constant, Decreasing Cost Industry P Slri Slrc Slrd Q 229 2 Order Conditions MR – MC = 0 P MR – MC <0 MC1 Q Q Pm MR = 75-Q MC = 4/3+2/3*Q MR = -1 MC1+2 MCm Q D MC = 2/3 1 Q Qm MRQ MR – MC = -1 – (2/3) < 0 Q Q 230 Individual Producer's Profits Profits 1 = P*Q1 - TC1 = P*Q1 - 10 - Q1 - (1/2)Q1 = 52.9*29.8 - 10 – 29.8 - (1/2)29.82=1751 2 = P*Q2 – TC2 = P*Q2 – 20 - 2Q2 – Q22 = 52.9*14.4 - 20 - 2*14.4 – (14.4)2=824 231 Competitive Model (P=MC) P MC1 Q1 Ppc Q2 MC1+2 D Qpc Q Be able to solve for Ppc, Qpc, Q1, Q2, 1, 1 232 Market Failure from Monopoly redistributed from consumers to monopolist P Pm Ppc Efficiency Distribution Social Losses? MC D Qm Qpc Q MR 233 Sources of Cost - economic model if competitive market supply = marginal cost fit function to data P Q 234 Where to Get Demand Qd = f(Pd, Y, Ps, Pc, . . ., etc. ) Collect data on Qd, Pd - all variables that change Fit a function using statistical techniques Simplified Two Variable Illustration Qt = 1 + 2Pt + et (truth) P R.V. = et ~ 0, s2 et êt Q 235 MRP = Factor Demand PEEo = Po MRP = Po marginal revenue product must slope down Po Can compute demand if know Eo PEEo Po1 Po2 O1 O2 Q 236 Abdel Reviewed Competitive Short Run Supply P P S P MC1 MC2 AVC1 AVC2 Psr D Q1 Q Q2 Si = MCi above AVC Market is horizontal sum Q Q1 +Q2 Q 237 Competitive Long Run Supply With Entry and Exit MC = S sr i sr P Slr Slr Slr D D' Q 238 Market Failure from Monopoly redistributed from consumers to monopolist P Pm Ppc Efficiency Distribution Social Losses? MC D Qm Qpc Q MR 239 Supplier Oil Price and Transport Cost Demand and Supplier Separated by Transport Cost tr P S+ tr2 S+ tr1 S Price lower the farther from the market Ps1 Ps2 D Q2 Q1 Q 240 Location - Supplier Price and Arbitrage S2 >$69? <$69? $1 D $70 $1 S1 $69 241 Supplier Price and Arbitrage S2 >$68? <$68? $2 D $70 $1 S1 $69 Prices can only differ by transport and transaction cost 242 Income Redistribution If Producer Exports all of Product Income distribution before tax Consumer Surplus P MC Pm Producer Surplus D Qm MR Q 243 Sum Up: Horizontal Sum MC Competitive Supply, MC for multi-plant Monopoly P MC 244 Sum Up: Factor Demand = MRP = PQMPE Horizontal sum from individual to Market P D Q 245 Sum Up: Add Demand to MC What Should Monopolist Do? P MC1 Put together Demand and MC Pm MR = MC MC1+2 MCm 2 1 D 1 Qm MRQ 246 Next Horizontal Difference: Dominant Firm's Demand World Demand Supply of fringe Qo = Qw – Qs call on OPEC horizontal difference Qs P Qw MRL Q MR 247 What should OPEC Do? Add MC – Case 1 QsMCf Pf MCo P Po Fringe? 2 places Qw Qo Q MR 248 What should OPEC Do? Add MC – Case 2 MCf P MCo Po Qw Qo MR Q 249 What should OPEC Do? Add MC – Case 3 MCf P MCo Po Qw Qo MR Q 250 More on Price and Elasticity P = MC (1-1/|p|) One other implication What if p inelastic = -1/2 Then | p| = 1/2 Formula say P = MC = MC = -MC (1-1/(1/2)) (1 - 2) Whoops - negative price? conclusion monopolist not in inelastic range of demand 251 Numerical Example - 2 Country OPEC Costs OPEC MC1 = 2 + Q1 MC2 = 2 + 2Q2 Qs P MC World Demand Qw = 30 - 0.5P Supply fringe Qf = -10 + P MRU Qw MRL Q 252 Numerical Example - 2 Country OPEC OPEC MC Marginal costs MC1 = 2 + Q1 MC2 = 2 + 2Q2 P Horizontal sum Q Invert (let MC1 = MC2 = MC) Q1 = -2 + MC Q2 = -1 +(1/2)MC 2 add Q's, Q1+Q2 = Q = -3 + (3/2)MC invert back => MC = 2 + (2/3)Q MC Q 253 OPEC Demand - Find Kink MC fringe or supply fringe is MCf = P = 10 + Q => Qf = -10 + P P Inverse Demand World P = 60 - 2Qw => Qw = 30 - 0.5P 10 Kink P Qf = 0 = - 10 + P => P = 10 Kink Q 25 World demand = Q = 30 - 0.5(10) = 25 Qs Qw Q 254 OPEC Demand Above kink P > 10 and Q < 25 Qw - Qf Qo = 30 - 0.5P - (-10 + P) = 40 -1.5P Below Kink P<10, Q > 25 Qo = Qw = 30 - 0.5P Qs P 10 Qw 25 Q 255 Marginal Revenue Above kink P > 10 or Q < 25: Qw - Qf Qo = 40 -1.5P Invert P P = 40/1.5 -2/3Q MR = 40/1.5 - 4/3Q Below Kink Qo = 30 - 0.5P Invert P = 60 -2Q MR = 60 - 4Q Qs Qw Q 256 Solution - 3 choices Try above the kink MR = MC MR = 40/1.5 - 4/3Q P MC = 2 + (2/3)Q 40/1.5 - 4/3Q = 2 + (2/3)Q 74/3 = (6/3)Q 6Q = 74 10 Q = 12.333 MRU less than 25 P = 40/1.5-2/3(12.333) = 18.444 Qs MC Qw 25 MRL Q 257 Income Distribution Affect in Monopoly Market Consumer Surplus P MC Pm Producer Surplus D Qm MR Q 258 Tax in Monopoly Market: Global Changes New Pm' and Qm' New CS , Producer G tax revenue, New PS MC+t P Pm' Pm MC Qm'Qm MR Q Same Effect - unit tax Tax Consumer MR-t = MC Tax Producer MR = MC + t 259 Income Distribution in Monopoly Market Assume Producer Exports all of Product Income distribution before tax Consumer Surplus P MC Pm Producer Surplus D Qm MR Q 260 Tax in Monopoly Market: Global Changes New Pm' and Qm' New CS , Producer G tax revenue, New PS MC+t P Pm' Pm MC Qm'Qm MR Q Same Effect - unit tax Tax Consumer MR-t = MC Tax Producer MR = MC + t 261 261 261 Tax in Monopoly Market: Global Changes New Pm' and Qm' New CS , Producer G tax revenue, New PS MC+t P Pm' MC Pm Qm'Qm MR Q 262 Tax in Monopoly Market Tax Producer Government Consumer Country Loss MC+t P Pm' Pm MC tax revenues to producer government transfer to producer government loss Qm'Qm MR Q 263 Tax in Monopoly Market Effect on Producers Producer Losses MC+t P Pm' Pm MC Qm'Qm MR Q Transfer to Producer Government 264 Tax in Monopoly Market Net Effect on Producer Country MC+t P Pm' Pm MC Qm'Qm MR Q 1. Producer DW Losses 2. Tax revenues from consumer country Change in Producer Country Welfare = 2-1 265 Numerical Example- P&Q Before Tax P = 50 -2Q MC = 1 + 3Q MR = 50-4Q MR = MC 50 – 4Q = 1 + 3Q 7Q = 49 Q=7 P = 50 – 2*7 = 36 MC = 1 + 3*7=22 P 50 MC 36 = Pm 22 1 D 7= Qm MR Q 266 Numerical Example- CS & PS P Before Tax Consumer Surplus 50 =0.5(50-36)7=49 Producer Surplus 36 = Pm =(36-22)*7+0.5*(22-1)*7 22 =171.5 1 MC D 7= Qm MR Q 267 Tax in Monopoly Market Producer Tax of 7 P P = 50 -2Q MC = 1 + 3Q 50 MR = 50-4Q MR = MC+7 38 36 50 – 4Q = 1 + 3Q+7 387 7Q = 42 Q=6 19 P = 50 – 2*6 = 38 1 MC = 1 + 3*6=19 MC+t MC 6 MR Q 268 Welfare Effects Producer Tax of 7 Before CS = 49 Now consumer surplus P CS = 0.5(50-38)*6= 36 50 Change in consumer surplus 49 - 36 = 13 38 36 387 Tax from Consumer transfer to producer Gov 19 (38-36)*6=12 1 Consumer DWL = 13-12 = 1 MC+t MC 6 MR Q 269 Welfare Effects Producer Tax of 7 Before PS = 171.5 Now Producer Surplus PS = (31-19)*6 + 0.5*(19-1)*6 = 126 Change in PS 171.5-126 = 45.5 Tax from Producer (7-2)*6 = 30 DWL Producer 45.5-30 = 15.5 P MC+t 50 MC 38 36 387 19 1 6 MR Q 270 Welfare Effects Producer Tax of 7 Net Effect for Producing Country DWL from producers = 15.5 Tax revenue gain from consumer country = 12 Net effect = Loss of 15.5 - 12 271 Total Global Welfare Effects Before CS = 49 PS = 171.5 Total= 204 P Now CS = 36 PS + TR = 126 + 42 = 168 MC Pm Total Losses 220.5-204 = 16.5 D Qt Qm MR Q 272 Do for Tax by Consumer Government Demand P= 50 - 2Q MC = 1 + 3Q Subtract tax from demand P - t = 50 -2Q - 7 = 34 - 2Q Create MR from new demand MR = 43 - 4Q Set MR = MC and solve See file ch06-Monopoly Tax to check problems for producer tax consumer tax both taxes 273 Consumer Tax in Monopoly Market Consumer Government Adds a Tax P MC Pt P-T Qt MR MRt P Q 274 What should OPEC Do? Add MC – Case 1 QfMCf Pf MCo P Po Qw Qf Qo Fringe? 2 places Qf Qd-Qf Q MR 275 What should OPEC Do? Add MC – Case 2 MCf P MCo Po Qw Qo MR Q 276 What should OPEC Do? Add MC – Case 3 MCf P MCo Po Qw Qo MR Q 277 Quiz - left of Kind Qf MCo P Po 2. Price is the same for OPEC and the fringe Qw Qf Qo Q MR N.B. 1. read price off of OPEC demand not world demand 278 Qf = 0 Quiz OPEC at Kink MCf P MCo Po Qw Qo MR Q 279 Quiz - OPEC to right of kink Qf = 0 MCf P MCo Po Qw Qo MR Q 280 Quiz Key -Graphically there are two ways to show economic profits Profits = producer surplus = area below price and above marginal cost Profits = (P-ATC)*Q 281 Numerical Example - OPEC Optimum Pick to right or left Pick Left MR = MC Left of kink Q< 123 P P = 92 - (2/3)Q MR = 92 - (4/3)Q OPEC Marginal Cost 10 MC = 2 + (2/3)Qo MR = MC 92 - (4/3)Qo= 2 + (2/3)Qo 90 = (6/3)Qo 45 Qo = 45 MC Qs Qw 123MR Q 282 What Else do We Know About the Market Qo = 45 P = 92 - (2/3)Q = 92 - (2/3)45 = 62 Qf = -10 + P = -10 + 62 = 52 MC Qs P 62 10 Qw 45 52 123MR Q 283 OPEC Quotas MC2 MC = 2 + (2/3)Q = 2 + (2/3)45 = 32 OPEC Quotas MC1 = 2 + Q1 32 = 2 + Q1 Q1 = 30 MC2 = 2 + 2Q2 32 = 2 + 2Q2 Q2 = 15 MC1 MC P 62 32 Qw 30 15 45 MR Q 284 Industry Profits OPEC π= P*Q - TC = P*Q - ATC*Q = P*Q - 0QMC(Q)dQ Solve = 62*45 - 045(2 + (2/3)Q)dQ = 2790 - (2Q + (1/3)Q2)|045 = 2790 -[2*45 + (1/3)452 - 2*0 + (1/3)02] = 2025 Individual OPEC countries or the fringe Qi = P*Q - MC (Q )dQ 285 Adjust for Technical Change and Depletion depletion curve AC learning curve Cumulative Q Chapter 7 287 Price control versus Quantity control P D3 P D1 D3 D1 D2 D2 Pc QcapacityQ Qcontract shortage Q 288 Opportunism - quasi rent rent MC P ATC AVC P1 P2 Q quasi-rent Chapter 8 290 Policy - Negative Externalities on Supply? Ssoc Spv P Psoc Ppv Dpv Qsoc Qpv Q 291 Numerical example Qd = 90 - Pd Qs = 2Ps Externality X = 9 Solve for equilibrium Qd = 90 - Pd = Qs = 2Ps Drop subscripts and solve P = 30 Q = 90-30 = 2*30 = 60 292 Numerical example Externality = 9 Invert Qd and Qs Pd = 90 - Qd Ps = Qs/2 Add negative externality to Ps 30 Pd = Ps + X drop subscripts 90 - Q = Q/2 + 9 Q = 54, P = 90 - 54 = 36 Ps = Qs/2 Pd = 90 - Qd 60 293 Numerical example- Social Costs Pd = Ps + X = 30 + 9 = 39 Welfare loss 0.5(39-30)(54-60)=27 units? = units of PQ price in $/ton quantities - millions of tons P*Q = $ * millions tons ton = millions $ 39 9 Ps = Qs/2 30 Pd = 90 - Qd 54 60 294 Social Loss - positive externality on supply Spv Ssoc P Dpv Q 295 MB = MC MB,MC MC MB Xo X 296 Model Two Pollution - Optimal Level $ Benefits Costs MB of MC of Pollution Pollution G E A B C D X Pollution 297 Polluter has Property Rights? What are Social Losses? $ MB,MC MC of Pollution MB of Pollution F G E A B C D X Pollution 298 One Who Suffers has Property Rights? You Show Social Losses? $ MB,MC MB of Pollution MC of Pollution G F E A B C D X Pollution Coase’s Law 299 No Transaction Costs Suppose Dow has property rights $ MB,MC Dow MB of Pollution Exxon MC of Pollution G F Most benefit E A B C D X Pollution 300 $ Benefits Costs Distribution Affects Polluter Has Property Rights MB of Pollution MC of Pollution social loss F G E A B D C polluter benefits from pollution Q Pollution 301 $ Benefits Costs Command and Control You can only emit C MB of Pollution MC of Pollution F at C no social loss G E A B Pollution at C C Q Pollution D Polluter clean up cost 302 What Happens if Pollution Tax = T1 $ Benefits Costs MB of MC of Pollution Pollution F G E T1 A B C pollution D Q Pollution 303 What Happens if Pollution Tax = T2 $ Benefits Costs MB of MC of Pollution Pollution F T2 G E A B C D Q Pollution 304 $ Benefits Costs Optimal Pollution Tax MB of Pollution MC of Pollution F G E A B C pollution taxes T3 D Q Pollution 305 $ Benefits Costs Polluter Had Property Rights Redistribution Affects With Tax MB of Pollution MC of Pollution F G E society gains back tax abatement A B C polluter losses from tax D Q Pollution 306 Distribution Affects Sufferer Had Property Rights $ Benefits Costs H I MB of Pollution MC of Pollution social loss F G E A B C polluter benefits D Q Pollution 307 Distribution Affects With Tax $ Benefits Costs MB of I Pollution H MC of Pollution F G E J A B tax C polluter gains JIG-AEJB D fix Q Pollution $ Benefits Costs H 308 Issue Marketable Permit of AC MB of I Pollution MC of Pollution F G E P1 A B C J polluters will want to buy AJ D P Price will go to AE Q Pollution $ 309 BenefitsPolluter Had Rights Subsidize Clean Up Costs MB of MC of Pollution Pollution H F G E Sb A C D Q Pollution Total Subsidy 310 Distribution Affects $ from Subsidy Polluter Had Property Rights Benefits Costs H MB of Pollution MC of Pollution social loss F E G A B C Total Polluter Benefits AHGC+GKD K D Total Subsidy Q Pollution 311 Which Policy Does Polluter Prefer 312 of Abatement CD Model 3 Optimal level Optimal Level – Pollution AC $ Benefits Costs MB of Pollution MC of Pollution F G E A B C D Q Pollution 313 Model 3 Abatement over to firms of CD MC P2 MC1 A1 ’ A1 Price of permits MC2 P3 A2 ’ needed abatement CD What happens at P2? P3? A2 Chapter 9 315 Public Good Quantitative Separate Players MC = 6 MB1 = 30 - 3A1 MB2 = 20 - 2A2 MC= MB1 6 = 30 - 3A1 3A1 = 30 - 6 = 24 A1 = 24/3 = 8 MC = MB2 6 = 20 - 2A2 2A2 = 14 A2 = 7 MC MB MB1=30-3A1 MC=6 A1o=8 A1 MC MB MB2= 20 - 2A2 MC=6 A2o =7 A2 316 Public Good Quantitative Gaming the System MC MB Non-excludeable A1 wants A2 to produce? 7 A1 will produce 1 A2 wants A1 to produce? MC 8 MB A2 will produce 0 Each will want to free ride MB1=30-3A1 MC=6 A1o=8 A1 MB1+MB2 MB2 A2o=7 Aso MC A2 317 Public Good Quantitative Social Optimum Since non-rivalrous benefits MB1 + MB2 MC = 6 MB1 = 30 - 3A1 MB2 = 20 - 2A2 MB = 50 - 5A MB = MC 50-5A = 6 5A = 44 A = 44/5 = 8.8 MC MB MB1=30-3A1 MC=6 A1o=8 MC MB A1 MB1+MB2 MB2 A2o=7 Aso=8.8 A2 MC 318 Value of life Occupation increases the probability of dying by 1/1000 = 0.001 Salaries are 5,000 higher in this occupation How are they valuing their lives Die lose = V Don't die from work accident loss = 0 0.001V + 0.999*0 = 5000 V = 5,000,000 319 Conservation – levelized costs 75-watt incandescent bulb (75/1000 = 0.075 kilowatts) lasts 600 hours buy packs of two $1.40 more than 90% of energy lost to heat 20-watt (20/1000 = 0.020 kilowatts) compact fluorescent bulb same amount of light lasts around 8,400 hours costs around $14.50 320 Conservation – levelized costs Suppose lights will run 1200 hours per year electricity costs $0.10 per kilowatt-hour interest rate is 12% compounded once a month operating costs/hour for incandescent bulb (oi) = kilowatts per bulb X costs per kilowatt hour = (0.075)*0.10 = $0.0075 per hour operating costs of/hour for fluorescent = (0.020)*0.10 = $0.0020/hr 321 Levelized Capital Costs for each Bulb a bit harder to compute. X monthly output of light (1200/12) lasts for n years K is initial capital costs, let $ equal levelized cost K= $X + $X + .... $X (1+r/12) (1+r/12)2 (1+r/12)n*12 Then K = $X i=1n*12 (1/(1+r/12) i) Solving for $ = (K/X)/i=1n*12 (1/(1+r/12)i) 322 Levelized Cost Fluorescent & Incandescent Package of incandescents costs K = $1.40 n=1 year, X = 100 hours per month $i = i=1n*12 (1/(1+r/12) i ) = (1.40/100)/11.255 = $0.0012 capital costs per unit of light lower than operating for incandescent compact fluorescent cost K =$12.00 n=7 years, X = 100 hours per month $f = (12/100)/Σi=112*7(1/(1+0.1/12)i) = (12/100)/56.648 = $0.0021 compact fluorescent operating costs lower than capital costs 323 Total unit Cost Fluorescent & Incandescent Adding capital and operating costs total incandescent costs $i + oi = $0.0012 + $0.0075 = $0.0087 total compact fluorescent costs $f + of = $0.0021 + $0.0020 = $0.0041 Total Formula unit cost = kilowatts*Pe + $i = (K/X)/i=1n*12 (1/(1+r/12)i) 324 Market Power Seller power MC P P Competition Monoply MR = MC MC D Competition D Q Q Buyer Power 325 Graph the Decision Process MRP = P PL & MRPL PL-1 PL-2 D= MRPL L1 L2 L 326 Marginal Factor Cost from Supply Market Power of Buyer = PEE(L) – PL(L)*L L = PEEL – (PL + dPLL)= 0 dL MRP - MFC =0 Example: L = -10 + 2PL supply P = 5 + 0.5L 327 Numerical Example of Marginal Factor Cost to Monopsonist TC = PL L PL = 5 + 0.5L TC = PL L = (5 + 0.5L)L = 5L + 0.5L2 MFC= TCL= 5 + 2*0.5L = 5 + L Chapter 10 329 Factor Demand Sum Up P ln g 1 P ln g 2 M R P ln g LNG 330 Monopsony Outcome 331 Bilateral Monopoly Assume Both Want the Same Quantity 332 Bilateral Monopoly Assume Both Want the Same Quantity 333 Reservation Prices 334 Graph the Decision Process MRP = P PL & MRPL PL-1 PL-2 D= MRPL L1 L2 L 335 MRP = D (Marginal Benefit) Need Buyer Marginal Cost MFCL PL & MRPL SL =Seller Marginal production cost PL-ms D= MRPL Lms L 336 Numerical Example – Monopsony Market Sell Electricity PE = $10 per megawatt MFCL Produce electricity from LNG (let Lng = L) = 20+4L 2 E = 8L – 2L Buy LNG supply PL & L = -10 + 0.5PL MRPL PL PL = 20 + 2L = 20+2L Maximize profits PE*E - PLL P L-ms 2 = 10(8L – 2L ) – (20+2L)L D = MRPL L = 80 – 10*4L - 20 – 4L = 0 =80 – 10*4L 80 – 10*4L = 20 + 4L L L 337 Numerical Example – Monopsony Market MRP = MFC 80 – 10*4L = 20 + 4L 44L = 60 L = 60/44 = 1.36 PL= 20 + 2L = 20 + 2*1.36 = 22.72 E = 8L – 2L2 = 8*1.36 – 2*1.362 = 8.156 MFCL = 20+4L PL & MRPL PL = 20+2L PL-ms =22.72 D = MRPL =80 – 10*4L Lms= = 8.156 L Chapter 11 339 Duopoly theory – Cournot model Two Players Choose quantity to maximize profits given the other firms output Inverse demand function demand P = 100 - 0.5(q1 + q2) C1 = 5q1, C2 = 0.5q22 Profit functions 1 = (100 - 0.5(q1 + q2))q1- 5q1 2 = (100 - 0.5(q1 + q2))q2 - 0.5q22 340 Duopoly theory – Cournot model First order conditions Firm 1 1/q1=(100 - 0.5(q1 + q2)) - 0.5q1 - 5 = 0 rearranged to = 95 - q1 - 0.5q2 = 0 reaction function q1 = 95 - 0.5q2 Firm 2 1/q2 = (100 - 0.5(q1 + q2)) - 0.5q2 - q2 rearranged to = 100- q2 - 0.5q1 - 2q2 reaction function q2 = 100/2 - (0.5/2)q1 = 50 - 0.25q1 341 342 Equilibrium Solve Where Reaction Functions Cross second equation into the first. q1 = 95 - 0.5(50 - 0.25q1) = 95 – 25 + 0.125q1 = 70 + 0.125q1 q1 - 0.125q1 = 70 q1(1-0.125) = q1*0.875 = 70 q1 = 70/(0.875) = 80 Then q2 = 50 - 0.25*80 = 30 Price from P= 100 - 0.5(80 + 30) = 45 343 What If Out of Equilibrium q1 = 95 - 0.5q2 q2 = 50 - 0.25q1 344 Profits 1 = Pq1 - C1 = 45*(80) - 5*80 = 3200 2 = Pq2 - C2 = 45*(30) - 0.5*302 = 900 345 Competitive Model P = MC P = 100 - 0.5(q1 + q2)C1 = 5q1, C2 = 0.5q22 MC1 = 5 MC2 = q2 P M C1 D M C2 q1 q2 MC q 346 Competitive Model P = MC P = 5 = 100 - 0.5(q1+q2) q1+ q2 = 190 MC2 = P = 5 = q2 q2 = 5 q1 = 190- 5 = 185 1 = 185*5 - 5(185) = 0 no Ricardian rents normal rate of return 2 = 5*5 - 0.5*(52)= 12.5 Ricardian rents 347 What if equal n-opolist P = a –bnqi TC = c + dqi If act as competitors P = MC a –bnqi = d => qi = (a-d) P = a –bn(a-d) = d bn bn If act as duopolist i = (a-b[(n-1)qj+qi])qi – c – dqi = 0 i = aqi-b(n-1)qjqi+qi2 – c – dqi = 0 348 What if equal n-opolist i/qi = a – b(n-1)qj+2bqi – d = 0 a – b(n+1)qi – d = 0 qi = (a-d) 2b(n+1) P= (a-b[a-d/2b(N+1)_= 0 349 What if sold gas on a monopoly market? D P MC q MR= MC 100 - (q1+q2) = 5 q1+ q2 = 95 P = 100 -0.5(95) = 52.5 350 How much does each player produce? MC2 = 5 = q2 q2 = 5 q1 = 95-q2 = 95-5 = 90 1 = 90*52.5- 5(52.5) = 4462.5 2 = 5*52.5 - 0.5*(52)= 250 Monopoly rents 351 Perfectly Price Discriminating Monopolist P D MC q 352 Stackleberg solution q1 one firm more information or more dominant optimizes given the other firm’s reaction function In the above, suppose 1 is the dominant firm 1 = (100 - 0.5(q1 + q2))q1 -5q1 but knows that firm 2’s reaction function is q2 = 50 - 0.25q1 1 = (100 - 0.5(q1 + (50 - 0.25q1))q1 - 5q1 1 = 100q1 - 0.5q12 - 25q1 + 0.125q12 - 5q1 1/q1= 100 – q1 - 25 + 0.25q1 – 5 = 0 0.75q1 = 70 => q1 = 70/0.75 = 93 1/3 353 Stackleberg solution q2 q2’s reaction function is the same as before q2 = 50 - 0.25q1 = 50 - 0.25*93.33 = 26.67 Stackleberg Cournot PC Monopoly q1 93.33 80 185 90 q2 26.67 30 5 5 P= 40.00 45 5 52.5 profit 1= 3266.67 3200 0 4275 profit 2=711.11 900 12.5 250 What If Both354 Try to be Leader Firm 1 produces q1 = 93 1/3 expecting q2 = 26.67 Firm 2 maximizes 2 = (100 - 0.5((95 - 0.5q2) + q2))q2 - 0.5q22 = 52.5q2 - 0.75q22 2/q2 = 52.5 - 1.50q2 = 0 q2 = 35 expecting q1 = 95-0.5(35) = 77.5 P = 100 – 0.5*(93.33+77.5) = 15 1 = 933 2 = -88 Not a stable equilibrium 355 Bilateral Monopoly Model P b MC c1 MRP X Xc Quantity agreed upon – Xc = 1 c1 reservation price of seller b reservation price of buyer Price between c1 and b 356 Add a Second Supplier with a Reservation (c2) P MC b c2 c1 MRP Xc X c1 <c2 < b possible rents, b-c1 divided between all players 1. 1 + 2 +3 = b - c1 357 Possible Rents at P If 1 sells 1 = p-c1 = rent supplier 1 If 2 sells 2 = p-c2 = rent supplier 2 3 = b-p = rent buyer Find core no coalition can block 358 Core no coalition can block 1. 1 + 2 +3 = b - c1 Core = 5. i > 0 6. 1 + 2 > 0 7. 1 + 3 > b - c1 8. 2 + 3 > b - c2 359 Core no coalition can block 1. 1 + 2 +3 = b - c1 Core = 2. i > 0 3. 1 + 2 > 0 4. 1 + 3 > b - c1 5. 2 + 3 > b - c2 1 & 3 => 6. 2 = 0 (insight #1) 5 & 6 => 7. 3 > b - c2 Substituting 6 into 1 8. 1 + 0 + 3 = b - c1 Rearranging 8 9. 1 = b - c1 - 3 Using 2 and 7 10. 0 < 1 < b - c1 - (b - c2) 11. 0 < 1 < c2 - c1 (insight #2) 360 Case 2: c1 < c2 best that Firm 1 can do is difference between its costs and rival 2 if Firm 1 charges slightly lower price will get all sales Redo for one seller and two buyers 361 Limit Pricing Model Chapter 12 363 first order conditions (foc) International Energy Workshop collected forecasts P 73 79 86 364 Graph - Two Period Model 2 periods – now and next year Q = 10 – 2.5P + 0.1Y Res = 50 no income growth r = 0.2 no costs Y = 500 Q = 10 -0.5P + 0.1(500) = 60 - 0.5 Q Inverted Demand P = 200 – 2Q 365 Demand Now Two Period Model 150 100 50 0 0 10 20 30 Qo=> <=Q1 40 50 366 Demand Now and Next Year Two Period Model 150 100 50 0 0 10 20 30 Qo=> <=Q1 40 50 367 Discount Next Year Two Period Model 150 100 50 0 0 10 20 30 Qo=> <=Q1 40 50 368 Mathematical Solution Basic Model Model Po = P1/(1+r) R = Qo + Q1 r = 0.2 Res = 50 Solution 120 – 2Qo = (120 – 2Q1)/(1+r) 120 – 2Qo = (120 – 2(50-Q1)/(1+0.2) Solve for Qo = 28.18 Q1 = 50 – 28.18 = 21.82 Po = 120-2*28.18 = 63.636 P1 = 120-2*21.82 = 76.364 369 Two Period Model with Income Growth 2 periods – now and next year Q = 10 – 2.5P + 0.1Y Res = 50 income growth 25% r = 0.2 no costs Y = (1+0.25)600 = 625 Q = 10 -0.5P + 0.1(625) = 72.5 - 0.5Q Inverted Demand P = 145 – 2Q Basic Model370 – Increase Income Green for More Money 150 100 50 0 0 8 17 25 33 42 50 371 Increasing Income Period 150 100 50 0 0 8 17 25 33 42 50 372 Discount P1 150 100 50 0 0 8 17 25 33 42 50 373 Two Period Model with Higher Interest Rate 2 periods – now and next year Q = 10 – 2.5P + 0.1Y Res = 50 income growth 25% r = 0.4 no costs Y = (1+0.25)500 = 625 Q = 10 -0.5P + 0.1(625) = 72.5 - 0.5Q Inverted Demand P = 145 – 2Q Raise 374 Interest Rate Green for More Interest 150 100 50 0 0 8 17 25 33 42 50 375 Discount Future More 150 100 50 0 0 8 17 25 33 42 50 376 Mathematical Solution Raise Interest Model Po = P1/(1+r) R = Qo + Q1 r = 0.4 Res = 50 Solution 120 – 2Qo = (145 – 2Q1)/(1+r) 120 – 2Qo = (145 – 2(50-Q1))/(1+0.2) Solve for Qo = 26.67 Q1 = 50 – 26.67 = 23.33 Po = 120-2*26.67 = 66.67 P1 = 1450-2*23.33 = 73.33 Model 4:377 Raise Reserves Green for More Reserves 150 100 50 0 0 8 17 25 33 42 50 378 Increase in Reserves 150 100 50 0 0 10 20 30 40 50 60 70 379 Add Demand Next Period Increase in Reserves 150 100 50 0 0 10 20 30 40 50 60 70 380 Discount Next Period Increase in Reserves 150 100 50 0 0 10 20 30 40 50 60 70 381 Mathematical Solution Raise Interest Model Po = P1/(1+r) R = Qo + Q1 r = 0.4 Res = 75 Solution 120 – 2Qo = (120– 2Q1)/(1+r) 120 – 2Qo = (120 – 2(75-Q1))/(1+0.2) Solve for Qo = 39.55 Q1 = 50 – 39.55 = 35.45 Po = 120-2*39.550 = 40.91 P1 = 120-2*35.45 = 49.09 382 Add Constant Costs to the Model = 20 Two Period Model 150 100 50 0 0 10 20 30 Qo=> <=Q1 40 50 383 Red = Marginal Cost 20 Two Period Model with Constant Marginal Costs 150 100 50 0 0 8 17 25 33 Qo=> <= Q1 42 50 384 Add Demand Next Period and Discount for Basic Model Two Period Model with Constant Costs 150 100 50 0 0 8 17 25 Qo=> <= Q1 33 42 50 385 P1 – MC! Two Period Model with Constant Costs 150 100 50 0 0 8 17 25 Qo=> <= Q1 33 42 50 386 Discount P1 - MC Two Period Model with Constant Costs 150 100 50 0 0 8 17 25 Qo=> <= Q1 33 42 50 387 Mathematical Solution Marginal Cost = 20 Model Po - MCo= (P1-MC1)/(1+r) R = Qo + Q1 r = 0.2 Res = 50 Solution 120 – 2Qo – 20 = (120 – 2Q1 – 20)/(1+r) 120 – 2Qo – 20 = (120 – 2(50-Q1) – 20 /(1+0.2) Solve for Qo = 27.27 Q1 = 50 – 27.27 = 22.73 Po = 120-2*27.27 = 65.45 P1 = 120-2*22.73 = 74.55 388 Costs a Function of Current Production MCi= a + bQi b > 0 = increasing cost industry b = 0 constant cost industry b< = decreasing cost industry MCo= a + bQo MC1= a + bQ1 Purple = Cost 389 Costs Increase with Production MCo Two Period Model Costs Increase with Current Production 100 P 50 MCo 0 1 2 3 4 5 Qo=> <=Q1 6 7 390 Add Po Po-MCo 150 100 P 50 0 0 8 17 25 33 Qo=> <=Q1 42 50 391 Po - MCo Two Period Model Costs Increase with Current Production 150 100 P Po 50 Po-MCo 0 MCo 0 8 17 25 33 Qo=> <=Q1 42 50 392 MC1 100 80 60 P 40 20 0 0 8 17 25 33 Qo=> <=Q1 42 50 393 PQ and MC1 Two Period Model Costs Increase with Current Production 150 100 P 50 P1 MC1 0 0 8 17 25 33 42 Qo=> <=Q1 50 394 P1 - MC1 Two Period Model Costs Increase with Current Production 150 100 P 50 P1 P1-MC1 0 MC1 0 8 17 25 33 42 50 Qo=> <=Q1 395 Put Two Sides Together Two Period Model Costs Increase with Current Production 150 P 100 Po 50 P1 Po-MCo P1-MC1 0 P1/(1+r) 0 8 17 25 33 42 Qo=> <=Q1 50 MCo MC1 396 Find Po-MCo=(P1-MC1)/(1+r) Two Period Model Costs Increase with Current Production 150 Po 100 P Po-MCo P1 50 P1-MC1 (P1-MC1)/(1+r) 0 P1/(1+r) 0 8 17 25 33 42 50 Qo=> <=Q1 MCo MC1 397 Quantitative NPV Consumer Surplus = Po dQo + P1 dQ1 - NPV = 0.5*(200-142.86)*28.57 + 0.5*(200-157.14)*21.43/(1.1) -7142.88 = 1233.74 398 Mathematical Solution MC = 10+1.5Qi Po – MCo = (P1-MC1) / (1+r) R = Qo +Q1 r = 0.2 MCo = 10 + 1.5 Qo MC1 = 10 + 1.5Q1 120-2Qo –(10 +1.5Qo) = 120-2Q1 – ( 10- 1.5Q1)/(1+0.2) 120-2Qo –(10 +1.5Qo) = (120-2Q1 – ( 10- 1.5 (50-Qo))) (1+0.2) Solution: Qo = 25.58 Q1 = 50 - 25.58 = 24.42 Po = 120 - 2*25.58 = 68.83 P1 = 120 - 2*24.42 = 71.17 Model: 399 T w o P eriod M odel C osts a F unction of C um ulative P roduction 150 125 M co 100 P 75 50 25 0 0 8 17 25 Q o=> <=Q 1 33 42 50 400 MCo = 20 + Qo T w o P eriod M odel C osts a F unction of C um ulative P roduction 150 Po 125 M co 100 P 75 50 25 0 0 8 17 25 Q o=>,Q 1=< 33 42 50 401 Po-MCo 120- 2Qo – ( 20 + Qo) T w o P eriod M odel C osts a F unction of C um ulative P roduction 150 Po 125 P o -M C o M co 100 P 75 50 25 0 0 8 17 25 Q o= > < = Q 1 33 42 50 402 Next Period MC1 = 20 + Qo T w o P eriod M odel C osts a F unction of C um ulative P roduction 150 P1 125 M co 100 P 75 50 25 0 0 8 17 25 Q o=> <=Q 1 33 42 50 403 T w o P eriod M odel C osts a F unction of C um ulative P roduction 150 P1 125 P 1-M C 1 M co 100 P 75 50 25 0 0 8 17 25 Q o=> <=Q 1 33 42 50 404 T w o P eriod M od el C osts a F u n ction of C u m u lative P rod u ction 150 Po P o -M C o P1 125 P 1 -M C 1 (P 1 -M C 1 )/(1 + r) M co 100 P 75 50 25 0 0 8 17 25 Q o= > < = Q 1 33 42 50 405 Income increases 200 150 Po P1 100 50 0 0 8 17 25 33 42 50 406 Change Interest Rate 200 150 Po P1 100 50 0 0 8 17 25 33 42 50 407 Non-Scarce resources 200 150 100 Po 50 P1 P 1/(1+ r) 0 -50 -100 -150 0 42 83 125 167 208 250 408 What is socially optimal use of resources? P S D Q instead of maximizing NPV profits maximize NPV of social welfare consumer + producer surplus 409 Social Welfare 200 150 Po P1 100 50 0 0 8 17 25 33 42 50 410 5. Model 3 + three cases for MC a. MC = constant = 20 b. MC = function of current production MCo = 2 + 0.2Qo MC1 = 1 + 0.2Q1 technical progress in 2 that lowers costs c. MC = function of cumulative production MCo = 2 + 0.2Qo MC1 = 2 + 0.2Qo 411 Model with Costs 250 200 Po P1 150 P o-M C 100 P 1-M C (P 1-M C )/(1+ R ) 50 MC 0 0 8 17 25 33 42 50 412 Po -MCo= P1 - MC1 (1+r) 200-2Qo -20= 206-2Q1 - 20 (1+0.05) substitute in the constraint 200-2Qo -20= 206-2(50-Qo) - 20 (1+0.05) Qo/Q1 = 25.12/24.88 Po/P1 = 149.76/156.24 NPV Net Rev = 6487.80 NPV Cons Surp =2196.864 413 Compare to case 3 Qo/Q1= 25.37/24.63 Po/P1= 149.26 /156.74 Reduce current consumption higher costs delays consumption 5b. MC = function of current production MCo = 20 + 0.2Qo MC1 = 10 + 0.2Q1 technical progress in 2 that lowers costs Model 6. Back Stop414 Fuel - Sweeney 1989 (LA) @10% Example (update) gasoline => $31.50 NG => methanol $45 per barrel coal => methanol $52 wood => methanol $73 compressed NG => $33 corn => ethanol $65 oil shale => oil $42 tar sands => oil $41 415 Back Stop Case 1: Po = 200 - 2Qo no Y grow r = 10% R = 50 MC = 0 416 Backstop @ 125 P1 = 125 Po = 125/(1.1)= 113.64 Q0 = 43.18 Q0 resource 43.18 Q0 bkstop 0.00 Q1 37.50 Q1 resource 6.82 Q1 bkstop 30.68 Resource price will gradually approach backstop price. 417 Backstop Analysis 200 Po 150 P1 P m ax 100 P m ax/(1+ r) 50 0 0 8 17 25 33 42 50 418 Shortage Case P = 200 - 2Q MR = 200 - 4Q 200-4Qo = 100 - 4Q1 (1+.1) 200-4Qo = 100 - 4(50-Qo) (1+.1) Qo Q1 P0 P1 26.19 23.81 147.62 152.38 419 Shortage Other cases - n periods Po = P1 + (1+r) P2 + (1+r)2 s.t. Q0 + Q1 + Q2 + P3 = (1+r)3 + Qn = R Example: No income growth Case 1: P = 200 - 2Q Q = 100 - 0.5P Maximum price = 200 Pn (1+r) 420 Monopoly 421 Price Control P = Pmax = P1 = $112.00 Po = $101.82 Q0 = 49.09 Q0 resource = 49.09 Q0 backstop = 0.00 Q1 = 44.00 at $101.82 but Q1 resource = 0.91 Then jumps to the backstop price P1 = $150 422 Compare to competitive case 423 Calculus of Variation Chiang pick a time path that optimizes a function 0T F(t, y(t),y’(t))dt y(0) = A Y(T) = Z know Z and T y might be oil production F could be discounted profits from the mine y’(t) is how production is changing 424 CV – Objective Function =PodQo-MCodQo+P1dQ1 - MC1dQ1 + (R-Qo -Q1) (1+r) (1+r) = R-Qo -Q1=0 Qo = P0 - MCo - = 0 Q1 = P1 - MC1 - = 0 (1+r) P0 -MCo= P1 - MC1 = (1+r) 425 Backstop 426 Backstop Quantitative Q = 60 – 0.5P Res = 60 P = 120 – 2Q r = 0.2 Backstop = $42 P1 = 42 Po = 42/1.2 = 35 Qo = 60 – 0.5*35 = 42.50 Q1 = 60 – 0.5*42 = 39.00 Q1 + Qo = 81.50 Qo + Q1 – Res = backstop consumption = 81.50 - 60 = 21.50 Chapter 13 428 Above Ground Costs - continuous Suppose Ro = 100 and = 0.10 (decline rate of ): Qo = 0.10*100 = 10 = Ro Q1 = e.10*t Ro = e-0.10*1 10 9.0484 Q2 = e.10*t Ro = e-0.10*2 10 8.1873 ... Q20 = e.10*t Ro = e-0.10*2 10 1.3534 ... Q100 = e.10*t Ro = e-.10*2 10 0.0005 429 Above Ground Costs - continuous (decline rate of ): Qo = Ro K = $Qte-rt dt = $Roe-t e-rt dt $ = (K/(Ro))/(oe(--r)tdt) = denominator = [(e(--r)t/(--r)]|o = [(e(--r)/(--r) - (e(--r)0/(--r)] = [(0)(-1)/(--r) = 1/(+r) Solving $ = (K/Ro)(+r)/ (K/Qo)(+r) K/Qo is referred to as capacity cost 430 Oil Costs Example: Decline rate 0.13, r = 0.10 $1 billion, R = 200 million $ = (1000/200)(0.13+0.10)/0.10 = $11.5 431 Nuclear Policy Hubbert: 1962 used logistics curves on US reserves Qt = Q (1+e(-(t-to)) Qt = cumulative production Q = total reserves that will ever be produced Q Chapter 14 434 Whole Blending Problem max s.t. = $0.08*X1 + $0.09*X2 0.4X1 + 0.57143X2 < 100,000 straight run 0.8X1 + 0.57143X2 < 140,000 cracked Graph in X1 X2 space constraint 1 constraint 2 X1 = 0 => X2 = 175,000 X1 = 0, X2 = 245,000 X2 = 0 => X1 = 250,000 X2 = 0, X1 = 175,000 435 Graph Constraints constraint 1 constraint 2 X1 = 0 => X2 = 175,000 X1 = 0, X2 = 245,000 X2 = 0 => X1 = 250,000 X2 = 0, X1 = 175,000 436 Objective Function = 0.08X1 + 0.09X2 X2 = /0.09 - (0.08/0.09)X1 Find highest line on constraint -slope dX2/dX1 = -0.8888 437 For this Shaped Constraint Set Always on Corners Check profits A, B, C (A) = 0.08X1+.09X2 = 0.08*(0) + 0.09(175,000) = 15750 (C) = 0.08X1+.09X2 = 0.08*(175,000) + 0.09(0) = 14000 (B) need to find what X1 and X2 are. 438 Solve simultaneously (1) (2) 0.4X1 + 0.57143X2 = 100,000 0.8X1 + 0.57143X2 = 140,000 Solve 1 for X1 (3) X1 = (100,000 - 0.57143X2)/0.4 Plug (3) into (2) (4) 0.8(100,000 - 0.57143X2)/0.4 + 0.57143X2 = 140,000 (5) X2=105,000 (6) X1 = (100,000 - 0.57143*105000)/0.4 = 100,000 439 Solve simultaneously X2 = 105,000 X1 = 100,000 = 0.08X1 + 0.09X2 (B) = 0.08(100000) +0.09(105000) = 17,450 (A) = 0.08*(0) + 0.09(175,000) = 15750 (C) = 0.08*(175,000) + 0.09(0) = 14000 440 How to Blend X2 = 105,000 straight run X1 = 100,000 0.40(100,000) + 0.57143(105,000) = 100,000 cracked 0.70(100,000) + 0.57143(105,000) = 140,000 u1 = 40,000 to grade 1 = 60,000 to grade 2 u2 = 80,000 to grade 1 = 60,000 to grade 2 441 Transport Problem five supply points for crude oil A, B, C, D, E available are 10, 20, 30, 80, 100 three refineries X, Y, Z crude oil requirements of 40, 80, 120 442 Transport Costs X Y Z A B C 7 10 5 3 2 0 8 13 11 D E 4 12 9 1 6 14 443 Math Formulation of Problem Objective: Minimize TTC = * = Cij*Aij. supply shipments jAij < Yi for all i refinery satisfy crude oil needs iAij = Xj for all j Set up in Excel Solver 444 Simple Example max s.t. = $0.08*X1 + $0.09*X2 0.4X1 + 0.57143X2 < 100,000 straight run 0.8X1 + 0.57143X2 < 140,000 cracked Blending Model Profit Function two processes grade 1 X1 = 2.5 min (u1 , u2/2) grade 2 X2 = 1.75 (u1,u2) u1 = straight run (100,000) u2 = cracked gasoline (140,000) 1 = $0.08/gal X1 2 = $0.09 / gal X2 = $0.08*X1 + $0.09*X2 What are technical constraints – u1 2.5 gallon of grade X1 requires 1 gallon of u1 gallon of u1 per gallon of X1 u1/X1 = 1/2.5 = 0.4 1.75 gallon of grade X2 requires 1 gallon of u1 gallon of u1 per gallon of X2 u1/X2 = 1/1.75 = 0.57 Total requirements of u2 for X1 and X2 2.5 gallon of grade X1 requires 2 gallon of u2 gallon of u2 per gallon of X1 u2/X1 = 2/2.5 = 0.8 1.75 gal of grade 2 requires 1 gallon of u2 gallon of u2 per gallon of X2 u2/X2 = 1/1.75 = 0.57 0.8X1 + 0.57X2 < 140,000 u2 constraint Whole problem max = $0.08*X1 + $0.09*X2 s.t. 0.4X1 + 0.57143X2 < 100,000 straight run 0.8X1 + 0.57143X2 < 140,000 cracked Graph in X1 X2 space Constraint 1 Constraint 2 X1 = 0 => X2 = 175,000 X1 = 0, X2 = 245,000 X2 = 0 => X1 = 250,000 X2 = 0, X1 = 175,000 Constraint Set 245,000 X2 A 2 175,000 1 B C 175,000 X1 250,000 = 0.08X1 + 0.09X2 X2 = /0.09 - (0.08/0.09)X1 Finding highest line that touches constraint set with slope dX2/dX1 = -0.8888 Will be one of points A, B, C Check the profit at each point (A) = 0.08X1+.09X2 = 0.08*(0) + 0.09(175,000) = 15,750 (C) = 0.08X1+.09X2 = 0.08*(175,000) + 0.09(0) = 14,000 (B) need to find what X1 and X2 are Solve simultaneously 0.4X1 + 0.57143X2 = 100,000 0.8X1 + 0.57143X2 = 140,000 If know matrix algebra 0.4 0.57143 X1 = 100,000 0.8 0.57143 X2 140,000 Invert and multiply -2.5 2.5 100,000 = 100,000 3.49999 -1.75 140,000 105,000 (C) = 0.08(100,000) +0.09(105,000) = 17,450 How much u1and u2 to blend to get X1 = 100,000 and X2 = 105,000 straight run 0.40(100,000) + 0.57143(105,000) = 100,000 u1 cracked 0.80(100,000) + 0.57143(105,000) = 140,000 u2 u1 = 40,000 to grade 1 60,000 to grade 2 u2 = 80,000 to grade 1 60,000 to grade 2 Transport Problem five supply points for crude oil A, B, C, D, E available are 10, 20, 30, 80, 100 three refineries X, Y, Z crude oil requirements of 40, 80, 120 Transport Costs X Y Z A B C 7 10 5 3 2 0 8 13 11 D E 4 12 9 1 6 14 Math Formulation of Problem Objective: Minimize TTC = * = Cij*Aij. supply shipments jAij < Yi for all i refinery satisfy crude oil needs iAij = Xj for all j Set up in Excel Solver Chapter 15 457 Cash Market Ignore transaction and storage costs Trader has a barrel of crude in transit Current spot price is St = $18 Trader is paid the spot price upon delivery at ST ST Value $18 Gain or Loss 17 -1 18 0 19 1 458 Trader Wants to Hedge Suppose FT = $18 to deliver oil at time T Sells one futures At time T the contract is worth FT – ST Futures Cash 18 - ST T ST Contract Sold $18 17 1 -1 18 0 0 19 -1 1 459 Hedging When Futures Price Different than Current Spot (Mia and Edwards 111-117) Heating Oil Distributor 1-Oct Cost of Crude 0.51 Spot price 0.54 Cost of carry/gal/month 0.008 Contract for Delivery Dec 15 420,000 at market price Delivery in 2.5 months If sell at current spot on Dec 15, profits would be (0.54 - 0.51 - 2.5*0.008)*420,000 = 4,200 Expecting price to rise but not certain If price < 0.54 less profits , > 0.54 more profits 460 Short with Zero Basis Risk Suppose Futures price is for Dec. 15 = $0.56 Basis = Cash Price – Minus futures Price = 0.54 – 0.56 = -$0.02 If products same –basis should go to zero at delivery But if using another product to hedge basis may not go to zero Suppose the basis stays constant You want to hedge to lock in profits 461 Long Hedge Distributor Short Crude Agreed to Deliver Crude Price Spot price Cost of carry/gal/month Deliver Crude March 15 Delivery in 1/1 2/15 0.55 0.535 0.008 420,000 1.5 month If buy at current spot, hold and sell at contract rate profits: (0.55 - 0.535 - 1.5*0.008 )*420,000 = $1,260 Could wait to buy in March for delivery Suspects price will be lower, more profit But price may be higher – doesn't want to take risk 462 Convenience Yield < Storage plus Interest Rate example r = 1%, = 1%, = 1%, St = 20. FT = Ste(r+ -)T = Ste(0.01 + 0.01 - 0.01)T T FT = $21.025 5 FT = $22.103 10 Further out is T the higher is FT Normal market - contango 463 Convenience Yield > Storage Plus Interest Rate (r+ -)< 0 => r+ < Example: r = 1%, = 1%, = 3%, St = 20. T FT = $19.025 5 FT = $18.097 10 Further out is T the lower is FT Backwardation or inverted market 464 Futures Markets in Contango (normal) and Backwardation (inverted) 2 What Determines Energy Future Prices Market 1 Market 2 S1 a S2 P1 PT PT b e c D1 Qb d Qc imports Q1 P2 f Qd D2 Qe Q2 exports 3 Optimal Hedge Ratio United Airlines will buy 500,000 gallons of jet fuel There is no futures market for jet fuel σs,jet = 0.028 σf,heating = 0.05 ρ = 0.9 s s , jet h 0 .5 s f , heating United Airlines should buy 250,000 gallons of heating oil at the futures market to hedge their risk 5 One small wrinkle to the spark spread An electricity contract is 736 mWh Gas contracts are in 10,000 MMBtu 8 MMBtu 1 mWh h 10 , 000 MMBtu 736 mWh h = 0.59 ≒ 0.6 lowest common denominator 3 gas contracts for every 5 electricity contracts 468 Hedging When Futures Price Different than Current Spot (Mia and Edwards 111-117) Heating Oil Distributor 1-Oct Cost of Crude 0.51 Spot price 0.54 Cost of carry/gal/month 0.008 Contract for Delivery Dec 15 420,000 at market price Delivery in 2.5 months If sell at current spot on Dec 15, profits would be (0.54 - 0.51 - 2.5*0.008)*420,000 = 4,200 Expecting price to rise but not certain If price < 0.54 less profits , > 0.54 more profits Chapter 16 470 Value of Call at Expiration 471 Value of Put at expiration 472 Value before expiration depends on following variables Increase Call Put 1. Underlying Asset Price Value K ST K ST 473 Value before expiration depends on following variables Increase Call Put 2. Exercise Price Value K ST K ST 474 Value before expiration depends on following variables Increase Call Put 5. Stock Risk Value K ST K ST 475 Single Period Binomial Pricing Model European Call know percentage rise or fall 476 Buy a stock and bond portfolio equivalent to C Let risk free rate = 6% Bond matures in one period Sell a bond 477 Buy a stock and bond portfolio equivalent to C Buy a Stock 478 After a Year If the stock price goes up you have 55-45 = $10 If the stock price goes down you have 45 – 45 =0 same portfolio as buying a call must be worth the same otherwise arbitrage Value of portfolio now $50 - 42.45 = $7.55 479 Solve for N anf Bt PortuT = N*U*St + R*Bt = cu = ST - K = 10 PortdT = N*D*St + R*Bt = cd.= 0 N = (cu - cd)/[(U - D)*St], Bt = [cu*D - cd*U]/[(U - D)*(-R)] N = (10 - 0)/[(1.1 - 0.9)*100] = 0.5, Bt = [(10*0.9) - (0*1.1)]/[(1.1 - 0.90)*(-1.06)] = -42.45. buy (+) half a stock sell (-) $42.45 worth of bonds Value of the portfolio is, as before, N*St + Bt = 0.5St - 42.45 = $50 - $42.45 = $7.55. 480 What is Value of Your Portfolio? If risk neutral in the above example then 1.1 (p) + 0.9(1-p) = 1.06 1.1p - 0.9p = 1.06-0.9 p =0.16/0.20 = 0.8 value of call 0.8*(10) + 0.2(0) = $7.55 (1.06) Same value so can act as if risk neutral 481 P for general case 1.1 (p) + 0.9(1-p) = 1.06 (p)*USt + (1 - p)*DSt = (1 + r)*St = R*St Solving we get p = (R - D)/(U - D) 482 What is Value of Your Portfolio? What if two periods to maturity 0.8 (1.1)2100 S = 121, C = 21 0.8 (1.1)*100 0.2 100 (2)0.2*0.8 (1-.1)(1+.1)100 S = 99, C=0 0.2 (1-.1)*100 0.22(1-.1)(1-.1)100 S = 81, C= 0 Value today 0.8*0.8*21 + 2*(0.8*0.2)0 + 0.2*0.2(0) = $11.962 1.062 483 Finish - Value of 2 period American Call What if two periods to maturity 0.8 (1.1)2100 S = 121, C = 21 0.8(1.1)*100 0.2 100 (2)0.2*0.8(1-.1)(1+.1)100 S = 100, C=0 0.2(1-.1)*100 0.22(1-.1)(1-.1)100 S = 81, C= 0 Value today 0.8*0.8*21 + 2*(0.8*0.2)0 + 0.2*0.2(0) = $11.962 1.062 Chapter 17 Chapter 18 Chapter 19 487 Input Output Model - Leontief 0.05 basic materials (B) per B (tons) (B/B) 0.01 B per manufacturing (M) (tons) (B/M) 0.09 B into E (106 BTU) (E) (B/E) 0.00 M per B (M/B) 0.50 M per M (M/M) 0.01 M per E (M/E) 0.20 E per B (E/B) 0.07 E per M (E/M) 0.15 E per E (E/E) 488 Write in Equations: B = 0.05 B + 0.01M + 0.09E + 1 M= + 0.5M + 0.01E + 2 E = 0.20 B + 0.07M + 0.15E + 10 M all constants to the right and convert to matrices 1 0 0 0 1 0 0 0 1 I (I-A)x = d - 0.05 0.01 0.00 0.50 0.01 M =d 0.20 0.07 E A 0.09 0.15 B x d 489 Last Time I-O Regionalize – Consumption, Investment, Government, Net Exports a11 a12 a13 a14 C 0 0 0 a21 a22 a23 a24 0 I 0 0 ... 0 0 G 0 a91 a92 a93 a94 0 Sector 0 0X Sector (X-M) Region X Sector AU= a11C a12I a13G a14(X-M) a21C a22I a23G a24(X-M) ... a91C a92I a93G a94(X-M) Region X Sector 490 Last Time Keeping Track of Pollution (1) fij is the amount of pollutant i per unit of good j Total amount of pollution i is Pi = j(fijXj) Example 2 pollutants (P1,P2), 3 goods (X1, X2, X3) pollutant 1 pollutant 2 f1j f2j Xj X1 = coal 25 2 50 X2 = gas 14 3 40 X3 = oil 19 4 30 491 Last Time Keeping Track of Pollution (2) X1 = coal X2 = gas X3 = oil P = P1 P2 pollutant 1 f1j 25 14 19 pollutant 2 f2j Xj 2 50 3 40 4 30 F X P = F'X 492 Last Time Keeping Track of Pollution (2) f1j 25 14 19 P = F'X P = 25 14 19 2 3 4 f2j 2 3 4 Xj 50 40 30 50 40 30 = 25*50 + 14*40 + 19*30 = 2380 2*50 + 3*40 + 4*30 = 340 493 Last Time Units Sector output input energy (a) other (b) energy (a) 0.20 0.20 other (b) 0.30 0.10 a/a a/b b/a b/b a BTU b tons outputs = BTU and tons 494 If a and b both $ output input energy other energy 0.20 0.20 other 0.40 0.10 Sum 0.60 0.30 1- sum = value added 1-0.6=0.4 1-0.3=0.7 495 Three Sectors - $ E E 0.20 M 0.35 O 0.45 x = (I-A)-1d x = 81.45 75.56 129.98 M 0.30 0.10 0.15 O 0.25 0.15 0.40 d 10 20 30 496 Add Regulation - Change Coefficient E M O d E 0.30 0.30 0.25 10 M 0.35 0.10 0.20 20 O 0.50 0.25 0.40 30 x = (I-A)-1d xr = 158.27 136.86 238.92 xr-x = 158.27 - 81.45 = 76.82 136.86 - 75.56 = 61.31 238.92 - 129.98 = 108.94 sum = 247.06 497 How measure Above in dollars - $247.06 as % of GDP 247.06/10000*100 ~ 2.5% If E, M, O measured in tons E = 76.82 M = 61.31 O = 108.94 Value = Price'X = [1, 4, 3] 76.82 61.31 108.94 = 705.88 498 How Can the Rules be Written How much to clean up quantity Z=7 % of Pollution Z = αP How much you can pollute quantity Pa = 2 =P-Z %of pollution Pa = αP 499 More complicated - model control industry new control industry Z produce 0.05 lbs of pollutant/$ of energy current pollution = 0.05*X1 = 0.05*197.146=9.857 regulation remove 90% Z = 0.9*0.05*X1 But it takes resources to remove pollution $0.02 energy / lb removed $0.05 mineral / lb removed $0.12 mfg / lb removed 500 More complicated - model control industry New industry Z X1 = 0.2X1+0.2X2+0.25X3+ 0.15X4 + 0.02Z +10 X2 = 0.0X1+0.1X2+0.15X3+ 0.17X4 + 0.05Z+20 Fill in for X3 to X4 X3 = 0.3X1+0.3X2+0.30X3+ 0.10X4 + 0.12Z+100 X4 = 0.3X1+0.1X2+0.20X3+0.40X4 + 30 Add regulation Z = 0.9*0.05*X1 Notice: Old A plus extra row and extra column x includes Z x = Agx + d 501 Variables left, constants right Variables left, constants right X1 - 0.2X1-0.2X2-0.25X3- 0.15X4 - 0.02Z =10 X2 - 0.0X1-0.1X2-0.15X3- 0.17X4 - 0.05Z =20 X3 - 0.3X1-0.3X2-0.30X3- 0.10X4 - 0.12Z=100 Compute for X4 = 30 X4 - 0.3X1-0.1X2-0.20X3- 0.40X4 Z - 0.9*0.05*X1 =0 Let's write as [I-Ag]x = d 502 Write as [I-Ag]x = d 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 - 0.20 0.00 0.30 0.30 0.20 0.10 0.30 0.10 0.045 0.0 0.25 0.15 0.30 0.20 0.0 0.15 0.02 0.17 0.05 0.10 0.12 0.40 0.00 0.0 X1 10 X2 20 X3 = 100 0.0 X4 30 Z Solve:(I-Ag)x = d x = (I-Ag)-1d 0 503 Opportunity Cost of Pollution Regulation X1 X2 X3 X4 Z Before 197.146 128.245 321.918 277.253 0.000 After Opportunity Cost 199.323 2.177 129.801 1.556 325.414 3.496 279.766 2.513 8.970 8.970 pollution before = 0.05*X1 =0.05*197.146=9.8573 pollution after = 0.05*199.323 - 8.97 = 1.00 cost as percent of GDP = 504 Cradle to Grave x = Ax + d x1 = a11x1 + a12x2 + d1 x2 = a21x1 + a22x2 + d2 x = (I-A)-1d x1 = τ11d1 + τ12d2 x2 = τ21d1 + τ22d2 cradle to grave use of x1 to get 1 more d1 dx1/dd1 = τ11 cradle to grave use of xi to get 1 more dk= τik 505 Set up the Problem B= 0.05B/B*B + 0.01B/M*M + 0.09B/E*E + 1 506 Solve B = 0.05 B + 0.01M + 0.09E + 1 (1) M= (2) + 0.50M + 0.01E + 2 E = 0.20B + 0.07M + 0.15E + 10 Solve (1) - (3) simultaneously From equation 2, solve for M M - 0.50M = 0.01E + 2 M = (0.01E + 2)/(1-0.5) = 0.02E+4 Substitute M into equations 1 and 3 (3) 507 Solve Substitute for M in 1, 3 B = 0.05 B + 0.01(0.02E+4) + 0.09E + 1 (4) E = 0.20B + 0.07(0.02E+4) + 0.15E + 10 (5) Rearrange some terms and simplify B - 0.05 B - 0.0002E - 0.04 - 0.09E = 1 E - 0.20B - 0.0014E - 0.28 - 0.15E = 10 Further combine terms 0.95 B - 0.0902E = 1.04 - 0.20B + 0.8486E = 10.28 508 Solve Solve 2 equations with 2 unknowns 0.95 B - 0.0902E = 1.04 (1) -0.20B + 0.8486E = 10.28 From eq. (1) B = (0.0902E +1.04)/0.95 = 0.0940E + 1.095 Substitute B into eq. 0.968 (2) -0.20(0.0940E + 1.095) + 0.8486E = 10.28 Solve for E -0.0181E - 0.219+ 0.8436E = 10.28 0.8255E = 10.499 →E = 12.718 (2) 509 Solution Total E, M, B to support end-use demands of 10, 2, 1 E = 12.718 M= 0.02E+4 = 0.02*12.718 + 4 = 4.254 B = = 0.0940*E + 1.095 = 0.0940*12.718 + 1.095 = 2.244 510 Let's Rewrite Technical Coefficients as Input per unit Output Matrix From Slide 33 0.05 basic materials (B) per B (tons) (B/B) 0.01 B per manufacturing (M) (tons) (B/M) 0.09 B into E (106 BTU) (E) (B/E) Inputs down/outputs across B M E B 0.05 0.01 0.09 M 0.0 0.50 0.01 E 0.20 0.07 0.15 511 Cradle to Grave, Wells to Wheels A= 0.05 0.01 0.09 B = 2.30 tons 0.00 0.50 0.01 M = 4.25 tons 0.20 0.07 0.15 E = 12.66 X 106 BTU Why can't diagonal elements > 1? Cradle to grave use of energy to produce mfg Direct E*M = M = 0.07*4.25 = 0.30 512 Continue Solution A= 0.05 0.01 0.09 B = 2.30 tons 0.00 0.50 0.01 M = 4.25 tons 0.20 0.07 0.15 E = 12.66 X 106 BTU First Order Indirect (B) E*B M BM = 0.2*0.01*4.25 = 0.01 First Order Indirect (E) E*E*M = 0.15*0.07*4.25 = 0.04 E M Total = 0.30 + 0.01 + 0.04 = 0.350 513 Cradle to Grave, Wells to Wheels A= 0.05 0.01 0.09 B = 2.30 tons 0.00 0.50 0.01 M = 4.25 tons E = 12.66 X 106 BTU 0.20 0.07 0.15 Why can't diagonal elements > 1? Cradle to grave use of energy to produce mfg Direct: 0.07 per unit of M Total first order direct E*M = 0.07*4.25 = 0.30 M But first order indirect: need B to produce M which needs E need E to produce E which needs E 514 x = (I-A)-1d Take a Look Back Last Time Input Output A inputs from one industry to another k industries how much x to produce to get d solution for general case x = (I-A)-1d (kX1) (kXk)(kX1) 530-11f-19m.xlsx 515 Take a Look Back (b) When does solution exist When does solution make economic sense Disaggregate models (i regions, j products) aij = region i's share of total product j (A)(27X3) European Union (27) Fossil fuels (3) (Coal,Oil,Ngas) FF (3X3) i = 27, j = 3, FF_Region (27X3) a1,1 a1,2 a1,3 Coal 0 0 . . . 0 Oil 0 a27,1 a27,2 a27,3 0 0 Ngas 516 Take a Look Back (c) A times FF = FF_Reg a1,1 a1,2 a1,3 Coal 0 0 . . . 0 Oil 0 a27 a27,3 a27,3 0 0 Ngas FF_Reg = consumption of each fuel by region = a1,1 Coal a1,2Oil a1,3 Ngas a2,1 Coal a2,2Oil a2,3 Ngas … a27,1 Coal a27,2Oil a27,3 Ngas 517 Take a Look Back (d) fij is the amount of pollutant i per unit of good j i = 5 pollutants - O3, PM, CO, Nox, Sox j = 3 products electricity (E), metals (M), Pulp&Paper (PP) f11 f 21 F f 51 f12 f 22 f 52 f13 f 23 f 53 518 Take a Look Back (f) x = total output of the three products What do you want to know? I = total of each pollutant I = Fx E x M P P O3 PM I CO N ox S ox 519 Take a Look Back (g) Pollution by industry pij = total pollution i from good j p11 p 21 P p 51 P = F*X p12 p 22 p 52 p13 p 23 p 53 E X 0 0 0 M 0 0 0 P P 520 Direct Inputs from One Industry to Another A= 0.05 0.01 0.09 0.00 0.50 0.01 0.20 0.07 0.15 x = (I-A)-1d = 2.30 tons (B) 4.25 tons (M) 12.66 X 106 (E) Direct E into each industry 530-11f-19m.xlsx, IO!A20:A22 E*B = aEB*B E*M = E*E B M E = 0.20*2.3 = 0.46 = 0.07*4.25 = 0.30 = ?*? 521 Go to Excel - 530-11f-19m.xlsx You can change yellow, solution is in red 522 Solve In Excel 523 Multiply (I-A)-1d Highlight c5:c6 Type in =MMULT(A5:B6,C2:C3) Ctrl Shift Enter Should show { } if it’s a matrix Excel will not allow you to change single elements 524 Input Output in $ aij = $ of input i for 1 dollar of output j A = input output matrix of technology matrix B M E B .10 .40 .20 M .30 .20 .10 E .05 .25 .35 When B sells $1 of output B buys 0.10 +0.30 + 0.05 = 0.45 from other industries/$ The remainder 1-0.45 = 0.55 is called value added Value added for M = 1-0.40-0.20-0.25 = 0.15 Value added for E = ? 525 More complicated - model control industry new control industry Z produce 0.05 lbs of pollutant/$ of energy current pollution = 0.05*X1 = 0.05*197.146=9.857 regulation remove 90% Z = 0.9*0.05*X1=0.045X1 But it takes resources to remove pollution $0.02 energy / lb removed $0.05 mineral / lb removed $0.12 mfg / lb removed 526 More complicated - model control industry New industry Z X1 = 0.2X1+0.2X2+0.25X3+ 0.15X4 + 0.02Z +10 X2 = 0.0X1+0.1X2+0.15X3+ 0.17X4 + 0.05Z+20 Fill in for X3 to X4 X3 = 0.3X +0.3X +0.30X + 0.10X + 0.12Z+100 1 2 3 4 30 X4 = 0.3X1+0.1X2+0.20X3+0.40X4 + Add regulation Z = 0.45*X1 Notice: Old A plus extra row and extra column x includes Z x=A x+d 527 Variables left, constants right Variables left, constants right X1 - 0.2X1-0.2X2-0.25X3- 0.15X4 - 0.02Z =10 X2 - 0.0X1-0.1X2-0.15X3- 0.17X4 - 0.05Z =20 X3 - 0.3X1-0.3X2-0.30X3- 0.10X4 - 0.12Z=100 Compute for X4 = 30 X4 - 0.3X1-0.1X2-0.20X3- 0.40X4 Z - 0.9*0.05*X1 =0 Let's write as [I-Ag]x = d 528 Write as [I-Ag]x = d Z - 0.045*X1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 - 0.20 0.00 0.30 0.30 0.20 0.10 0.30 0.10 0.25 0.15 0.30 0.20 ? ? ? 0.045 0.0 0.0 Old A Ag Solve:(I-Ag)x = d (I-Ag)-1(I-Ag)x = (I-Ag)-1d x = (I-Ag)-1d =0 0.15 0.02 0.17 0.05 0.10 0.12 0.40 0.00 ? X1 10 X2 20 X3 = 100 ? X4 30 0.0 0.0 Z ? 0.0 529 Clean Up More Compliucated Model with Control Industry (I-Ag)x = d 0.8 -0.2 -0.0 0.9 -0.3 -0.3 -0.3 -0.1 -0.045 0 -0.25 -0.15 0.70 -0.20 0 -0.15 -0.17 -0.10 0.60 0 -0.02 -0.05 -0.12 -0 1 X1 = 10 X2 20 X3 100 X4 30 Z 0 (I-Ag) with augmented row and column 530 X-1 with pollution control industry x = (I-Ag) d 2.126 0.543 1.391 1.617 0.096 1.010 1.551 1.275 1.188 0.045 1.270 0.725 2.521 1.596 0.057 1.029 0.696 1.129 2.674 0.046 -0.245 10 = 199.323 -0.175 20 129.801 -0.394 100 325.414 -0.283 30 279.766 -1.011 0 8.970 pollution after = 0.05*199.323 - 8.970= 0.997 0.05*199.323 - .97 = 0.997 531 Opportunity Cost of Pollution Regulation X1 X2 X3 X4 Z Before 197.146 128.245 321.918 277.253 0.000 After Opportunity Cost 199.323 2.177 129.801 1.556 325.414 3.496 279.766 2.513 8.970 8.970 pollution before = .05*X1 =0.05*197.146=9.025 pollution after = 0.05*199.323 - 8.97 = 0.996 cost in extra output industry 2.177+1.556+3.496+2.513 = 9.742 sometimes as a percent of GDP = 9.742/GDP Chapter 20