ࡱ> TV Sq` xbjbjqPqP 4::h5 VVVV222<d /////GNX0~111122 2G G G G G G G$IhcLv.G92>722>7>7.GVV11gG|9|9|9>7@VV121G|9>7G|9|9Ch2D1L0 @Iٝ/~7$DET}G0G4DLZ8L DL2D2dK4|9I56)222.G.Gl9222G>7>7>7>7d//F^8VVVVVV NAME: Problem Set #1 Micro II Spring Term 2010 Suggested Solutions About 100 million pounds of jelly beans are consumed in the United States each year, and the price has been about $0.50 per pound. However, jelly bean producers feel that their incomes are too low and have convinced the government that price supports are in order. The government will therefore buy up as many jelly beans as necessary to keep the price at $1 per pound. However, government economists are worried about the impact of this program because they have no estimates of the elasticities of jelly bean demand or supply. Could the program cost the government more than $50 million per year? Under what conditions? Could it cost less than $50 million per year? Under what conditions? Illustrate with a diagram. If the quantities demanded and supplied are very responsive to price changes, then a government program that doubles the price of jelly beans could easily cost more than $50 million. In this case, the change in price will cause a large change in quantity supplied, and a large change in quantity demanded. In Figure 9.5.a.i, the cost of the program is (QS-QD)*$1. Given QS-QD is larger than 50 million, then the government will pay more than 50 million dollars. If instead supply and demand were relatively price inelastic, then the change in price would result in very small changes in quantity supplied and quantity demanded and (QS-QD) would be less than $50 million, as illustrated in figure 9.5.a.ii. Could this program cost consumers (in terms of lost consumer surplus) more than $50 million per year? Under what conditions? Could it cost consumers less than $50 million per year? Under what conditions? Again, use a diagram to illustrate. When the demand curve is perfectly inelastic, the loss in consumer surplus is $50 million, equal to ($0.5)(100 million pounds). This represents the highest possible loss in consumer surplus. If the demand curve has any elasticity at all, the loss in consumer surplus would be less then $50 million. In Figure 9.5.b, the loss in consumer surplus is area A plus area B if the demand curve is D and only area A if the demand curve is D.  EMBED Word.Picture.6  Figure 9.5.a.i  EMBED Word.Picture.6  Figure 9.5.a.ii  EMBED Word.Picture.6  Figure 9.5.b In 1998, Americans smoked 23.5 billion packs of cigarettes. They paid an average retail price of $2 per pack. Given that the elasticity of supply is 0.5 and the elasticity of demand is -0.4, derive linear demand and supply curves for cigarettes. Let the demand curve be of the general form Q=a+bP and the supply curve be of the general form Q=c+dP, where a, b, c, and d are the constants that you have to find from the information given above. To begin, recall the formula for the price elasticity of demand  EMBED Equation  You are given information about the value of the elasticity, P, and Q, which means that you can solve for the slope, which is b in the above formula for the demand curve.  EMBED Equation  To find the constant a, substitute for Q, P, and b into the above formula so that 23.5=a-4.7*2 and a=32.9. The equation for demand is therefore Q=32.9-4.7P. To find the supply curve, recall the formula for the elasticity of supply and follow the same method as above:  EMBED Equation  To find the constant c, substitute for Q, P, and d into the above formula so that 23.5=c+5.875*2 and c=11.75. The equation for supply is therefore Q=11.75+5.875P. A new tax was added of $0.15 in 2002. What will this increase do to the market-clearing price and quantity? The tax of 15 cents will shift the supply curve up by 15 cents. To find the new supply curve, first rewrite the equation for the supply curve as a function of Q instead of P:  EMBED Equation  The new supply curve is now  EMBED Equation  To equate the new supply with the equation for demand, first rewrite demand as a function of Q instead of P:  EMBED Equation  Now equate supply and demand and solve for the equilibrium quantity:  EMBED Equation  Plugging the equilibrium quantity into the equation for demand gives a market price of $2.11. How much of the tax will consumers pay? What part will producers pay? Since the price went up by 11 cents, consumers pay 11 of the 15 cents or 73% of the tax, and producers will pay the remaining 27% or 4 cents. 3. Suppose the market for widgets can be described by the following equations: Demand: P = 10 - Q Supply: P = Q - 4 where P is the price in dollars per unit and Q is the quantity in thousands of units. a. What is the equilibrium price and quantity? To find the equilibrium price and quantity, equate supply and demand and solve for QEQ: 10 - Q = Q - 4, or QEQ = 7. Substitute QEQ into either the demand equation or the supply equation to obtain PEQ. PEQ = 10 - 7 = 3, or PEQ = 7 - 4 = 3. b. Suppose the government imposes a tax of $1 per unit to reduce widget consumption and raise government revenues. What will the new equilibrium quantity be? What price will the buyer pay? What amount per unit will the seller receive? With the imposition of a $1.00 tax per unit, the demand curve for widgets shifts inward. At each price, the consumer wishes to buy less. Algebraically, the new demand function is: P = 9 - Q. The new equilibrium quantity is found in the same way as in (2a): 9 - Q = Q - 4, or Q* = 6.5. To determine the price the buyer pays,  EMBED Equation , substitute Q* into the demand equation:  EMBED Equation  = 10 - 6.5 = $3.50. To determine the price the seller receives,  EMBED Equation , substitute Q* into the supply equation:  EMBED Equation  = 6.5 - 4 = $2.50. Note that you could also shift the supply curve up as a result of tax imposition (P = Q 3). You should get the same answer. c. Suppose the government has a change of heart about the importance of widgets to the happiness of the American public. The tax is removed and a subsidy of $1 per unit is granted to widget producers. What will the equilibrium quantity be? What price will the buyer pay? What amount per unit (including the subsidy) will the seller receive? What will be the total cost to the government? The original supply curve for widgets was P = Q - 4. With a subsidy of $1.00 to widget producers, the supply curve for widgets shifts outward. Remember that the supply curve for a firm is its marginal cost curve. With a subsidy, the marginal cost curve shifts down by the amount of the subsidy. The new supply function is: P = Q - 5. To obtain the new equilibrium quantity, set the new supply curve equal to the demand curve: Q - 5 = 10 - Q, or Q = 7.5. The buyer pays P = $2.50, and the seller receives that price plus the subsidy, i.e., $3.50. With quantity of 7,500 and a subsidy of $1.00, the total cost of the subsidy to the government will be $7,500. 4. A particular metal is traded in a highly competitive world market at a world price of $9 per ounce. Unlimited quantities are available for import into the United States at this price. The supply of this metal from domestic U.S. mines and mills can be represented by the equation QS = 2/3P, where QS is U.S. output in million ounces and P is the domestic price. The demand for the metal in the United States is QD = 40 - 2P, where QD is the domestic demand in million ounces. In recent years, the U.S. industry has been protected by a tariff of $9 per ounce. Under pressure from other foreign governments, the United States plans to reduce this tariff to zero. Threatened by this change, the U.S. industry is seeking a Voluntary Restraint Agreement that would limit imports into the United States to 8 million ounces per year. Under the $9 tariff, what was the U.S. domestic price of the metal? With a $9 tariff, the price of the imported metal on U.S. markets would be $18, the tariff plus the world price of $9. To determine the domestic equilibrium price, equate domestic supply and domestic demand:  EMBED Equation P = 40 - 2P, or P = $15. The equilibrium quantity is found by substituting a price of $15 into either the demand or supply equations:  EMBED Equation  and  EMBED Equation  The equilibrium quantity is 10 million ounces. Because the domestic price of $15 is less than the world price plus the tariff, $18, there will be no imports. If the United States eliminates the tariff and the Voluntary Restraint Agreement is approved, what will be the U.S. domestic price of the metal? With the Voluntary Restraint Agreement, the difference between domestic supply and domestic demand would be limited to 8 million ounces, i.e. QD - QS = 8. To determine the domestic price of the metal, set QD - QS = 8 and solve for P:  EMBED Equation , or P = $12. At a price of $12, QD = 16 and QS = 8; the difference of 8 million ounces will be supplied by imports. 5. Among the tax proposals regularly considered by Congress is an additional tax on distilled liquors. The tax would not apply to beer. The price elasticity of supply of liquor is 4.0, and the price elasticity of demand is -0.2. The cross-elasticity of demand for beer with respect to the price of liquor is 0.1. a. If the new tax is imposed, who will bear the greater burden, liquor suppliers or liquor consumers? Why? Section 9.6 in the text provides a formula for the pass-through fraction, i.e., the fraction of the tax borne by the consumer. This fraction is  EMBED Equation , where ES is the own-price elasticity of supply and ED is the own-price elasticity of demand. Substituting for ES and ED, the pass-through fraction is  EMBED Equation  Therefore, 95 percent of the tax is passed through to the consumers because supply is relatively elastic and demand is relatively inelastic. b. Assuming that beer supply is infinitely elastic, how will the new tax affect the beer market? With an increase in the price of liquor (from the large pass-through of the liquor tax), some consumers will substitute away from liquor to beer, shifting the demand curve for beer outward. With an infinitely elastic supply for beer (a perfectly flat supply curve), there will be no change in the equilibrium price of beer. 6. The domestic supply and demand curves for hula beans are as follows: Supply: P = 50 + Q Demand: P = 200 - 2Q where P is the price in cents per pound and Q is the quantity in millions of pounds. The U.S. is a small producer in the world hula bean market, where the current price (which will not be affected by anything we do) is 60 cents per pound. Congress is considering a tariff of 40 cents per pound. Find the domestic price of hula beans that will result if the tariff is imposed. Also compute the dollar gain or loss to domestic consumers, domestic producers, and government revenue from the tariff. To analyze the influence of a tariff on the domestic hula bean market, start by solving for domestic equilibrium price and quantity. First, equate supply and demand to determine equilibrium quantity: 50 + Q = 200 - 2Q, or QEQ = 50. Thus, the equilibrium quantity is 50 million pounds. Substituting QEQ equals 50 into either the supply or demand equation to determine price, we find: PS = 50 + 50 = 100 and PD = 200 - (2)(50) = 100 cents The equilibrium price P is $1 (100 cents). However, the world market price is 60 cents. At this price, the domestic quantity supplied is 60 = 50 - QS, or QS = 10, and similarly, domestic demand at the world price is 60 = 200 - 2QD, or QD = 70. Imports are equal to the difference between domestic demand and supply, or 60 million pounds. If Congress imposes a tariff of 40 cents, the effective price of imports increases to $1. At $1, domestic producers satisfy domestic demand and imports fall to zero. As shown in Figure 9.12, consumer surplus before the imposition of the tariff is equal to area a+b+c, or (0.5)(200 - 60)(70) = 4,900 million cents or $49 million. After the tariff, the price rises to $1.00 and consumer surplus falls to area a, or (0.5)(200 - 100)(50) = $25 million, a loss of $24 million. Producer surplus will increase by area b, or (100-60)(10)+(.5)(100-60)(50-10)=$12 million. Finally, because domestic production is equal to domestic demand at $1, no hula beans are imported and the government receives no revenue. The difference between the loss of consumer surplus and the increase in producer surplus is deadweight loss, which in this case is equal to $12 million. See Figure 9.12.  EMBED Word.Picture.6  Figure 9.12 The following table shows the demand curve facing a monopolist who produces at a constant marginal cost of $10: PriceQuantity2702422141861581210912614316018 Write an equation for the firms marginal revenue curve. (Hint: It may help to write the equation for the demand curve first.) P/Q = -3/2 = Slope of demand curve or AR P = a  1.5Q, At Q = 0, P = 27 Therefore, a = 27 P = 27  1.5Q TR = 27Q  1.5Q2 MR = 27  3Q What are the firm s profit maximizing output and price? What is its profit? MC = MR ( 27 3Q = 10 ( Q = 5.67 and P = 18.5 Profits = 27 (5.667) 1.5 (5.667)2 - 10 (5.667) = 48.17 What would the equilibrium price and quantity be in a competitive market? P = MR = MC ( 27 1.5Q = 10 ( Q = 11.33 and P = 10 (Equal to MC) What would the social gain be if this monopolist were forced to produce and price at the competitive equilibrium? (In other words, make Q so that P=MC.) Who would gain and lose as a result? 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FАIٝАIٝOle [PIC  \dft Word 6:NormalMichael CaputoMichael Caputo'@v@v@֗@Microsoft Word 6.0.12dX   @ .  & META ^PICT #n)CompObjKObjInfo"$Timesw@ [wdw0-!ETimesw@ R[wdw0-!P !D PSymbolw@ [wdw0-!=Timesw@ S[wdw0-!P !Q-  &PSymbolw@ [wdw0-!D *Timesw@ T[wdw0-!Q 1PSymbolw@ [wdw0-!D*Timesw@ U[wdw0-!P2 ) 9Timesw@ [wdw0-!.; & ')@dxpr MTHU @Grphbj @"@ currentpoint " +E +P( D + =( P(Q"  ( *D)Q(*D)P" )(;./MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2048 div 960 3 -1 roll exch div scale currentpoint translate 64 39 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 885 406 moveto 310 0 rlineto stroke 1259 406 moveto 558 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (E) 6 505 sh (P) 932 264 sh (Q) 902 798 sh (Q) 1524 264 sh (P) 1543 798 sh 224 ns (P) 241 601 sh (D) 277 333 sh 384 /Symbol f1 (=) 567 505 sh 384 /Symbol f1 (D) 1289 264 sh (D) 1308 798 sh 384 /Times-Roman f1 (.) 1855 505 sh end MTsave restore dSMATHG  E PD =PQDQDP.m FMathType Equation1ELO EquationG E PD =PQDQDP.Ole10Native%KOle10FmtProgID  _1015669883S8* FАIٝPIٝOle  EquationdhT h 0  > .  & PSymbolw@ M[wdw0-!-Timesw@ _[wdw0PIC ')dMETA hPICT (,fCompObjK-!0!.!4PSymbolw@ N[wdw0-!=Timesw@ `[wdw0-!2 ,!23%!.1!54- $ 9PSymbolw@ O[wdw0-!D =Timesw@ a[wdw0-!Q DPSymbolw@ P[wdw0-!D=Timesw@ b[wdw0-!PE < LPSymbolw@ Q[wdw0-!D*Timesw@ c[wdw0-!Q* PSymbolw@ R[wdw0-!D;Timesw@ d[wdw0-!P; ..PSymbolw@ S[wdw0-!=2!-2 Timesw@ e[wdw0-!02'!.2-!420!23*=!.*I!5*L!2;D.<.QPSymbolw@ T[wdw0-! ,6! 86! ,R! 8R!=2[!-2eTimesw@ f[wdw0-!42k!.2q!72tPSymbolw@ U[wdw0-!=2}Timesw@ g[wdw0-!b2Timesw@ V[wdw0-!.2 & 'f>dxpr MTHU>Grphbj >"> currentpoint " +-)0).)4) =( ,2(%23) .)5" $( =D)Q(=D)P" <(*D)Q(;D)P".(2=) -)0).)4(*=23) .)5(;D2".<(,6 *  (,R *  (2[=) -)4).)7) =) b)./MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4640 div 1984 3 -1 roll exch div scale currentpoint translate 64 34 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1101 406 moveto 701 0 rlineto stroke 1866 406 moveto 558 0 rlineto stroke 0 1473 moveto 558 0 rlineto stroke 1871 1473 moveto 701 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (-) -4 505 sh (=) 783 505 sh (=) 666 1572 sh (-) 980 1572 sh (\346) 1674 1390 sh (\350) 1674 1799 sh (\366) 2582 1390 sh (\370) 2582 1799 sh (=) 2862 1572 sh (-) 3176 1572 sh (=) 3954 1572 sh 384 /Times-Roman f1 (0) 206 505 sh (4) 494 505 sh (2) 1355 264 sh (23) 1122 798 sh (5) 1602 798 sh (0) 1190 1572 sh (4) 1478 1572 sh (23) 1892 1331 sh (5) 2372 1331 sh (2) 2125 1865 sh (4) 3386 1572 sh (7) 3674 1572 sh 384 /Times-Roman f1 (.) 398 505 sh (.) 1506 798 sh (.) 1382 1572 sh (.) 2276 1331 sh (.) 3578 1572 sh (.) 4449 1572 sh 384 /Symbol f1 (D) 1896 264 sh (D) 1915 798 sh (D) 30 1331 sh (D) 49 1865 sh 384 /Times-Italic f1 (Q) 2131 264 sh (P) 2150 798 sh (Q) 265 1331 sh (P) 284 1865 sh (b) 4263 1572 sh end MTsave restore dMATH_ -0.4=223.5DQDPDQDP=-0.423.52()=-4.7=b.m FMathType Equation1ELO Equation -0.4=223.5DQDPDQDP=-0.423.52()=-4.7=b. EquationObjInfo+-Ole10Native.Ole10FmtProgID  _10156698823 FPIٝPIٝOle PIC 02dMETA  PICT 15 d ,   r  a .  & Timesw@= [wdw0-!ETimesw@ [wdw0-!P !S PSymbolw@= [wdw0-!=      !"#$%&'()*+,-/2348:;<=>?@ABCDEFGHIJKLMNOPQRSUVWXYZ[\]^_`abcdefghijklmnoqtuy{|}~Timesw@ [wdw0-!P !Q-$PSymbolw@= [wdw0-!D (Timesw@ [wdw0-!Q 0PSymbolw@= [wdw0-!D)Timesw@ [wdw0-!P0'8Timesw@= [wdw0-!04!.4!54 PSymbolw@ [wdw0-!=4Timesw@= [wdw0-!2,%!23=!.=)!5=,002PSymbolw@ [wdw0-!D,6Timesw@= [wdw0-!Q,=PSymbolw@ [wdw0-!D=6Timesw@= [wdw0-!P=>050EPSymbolw@ [wdw0-!DMTimesw@= [wdw0-!QM PSymbolw@ [wdw0-!D^Timesw@= [wdw0-!P^ QQPSymbolw@ [wdw0-!=UTimesw@= [wdw0-!0U !.U&!5U)!23M6!.MB!5ME!2^=Q5QJPSymbolw@ [wdw0-! O/! [/! OK! [K!=UTTimesw@= [wdw0-!5U]!.Uc!875UfPSymbolw@ [wdw0-!=U{Timesw@= [wdw0-!dUTimesw@ [wdw0-!.U & ' adxpr- MTHU-aGrphbj a"a currentpoint " +E +P( S +=( P(Q"( (D)Q()D)P"'(40).)5) =(,%2(=23) .)5"0(,6D)Q(=6D)P"05(MD)Q(^D)P"Q(U=) 0).)5(M623) .)5(^=2"Q5(O/ *  (OK *  (UT=) 5).)875)=) d)./MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4576 div 3104 3 -1 roll exch div scale currentpoint translate 64 58 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 835 406 moveto 310 0 rlineto stroke 1209 406 moveto 558 0 rlineto stroke 873 1507 moveto 701 0 rlineto stroke 1638 1507 moveto 558 0 rlineto stroke 0 2574 moveto 558 0 rlineto stroke 1643 2574 moveto 701 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (E) 6 505 sh (P) 882 264 sh (Q) 852 798 sh (Q) 1474 264 sh (P) 1493 798 sh (Q) 1903 1365 sh (P) 1922 1899 sh (Q) 265 2432 sh (P) 284 2966 sh (d) 4201 2673 sh 224 ns (P) 241 601 sh (S) 269 333 sh 384 /Symbol f1 (=) 517 505 sh (=) 555 1606 sh (=) 666 2673 sh (\346) 1446 2491 sh (\350) 1446 2900 sh (\366) 2354 2491 sh (\370) 2354 2900 sh (=) 2634 2673 sh (=) 3888 2673 sh 384 /Symbol f1 (D) 1239 264 sh (D) 1258 798 sh (D) 1668 1365 sh (D) 1687 1899 sh (D) 30 2432 sh (D) 49 2966 sh 384 /Times-Roman f1 (0) -9 1606 sh (5) 279 1606 sh (2) 1127 1365 sh (23) 894 1899 sh (5) 1374 1899 sh (0) 975 2673 sh (5) 1263 2673 sh (23) 1664 2432 sh (5) 2144 2432 sh (2) 1897 2966 sh (5) 2940 2673 sh (875) 3228 2673 sh 384 /Times-Roman f1 (.) 183 1606 sh (.) 1278 1899 sh (.) 1167 2673 sh (.) 2048 2432 sh (.) 3132 2673 sh (.) 4405 2673 sh end MTsave restore dMATH% E PS =PQDQDP0.5=223.5DQDPDQDP=0.523.52()=5.875=d.۠ FMathType Equation1ELO EquationCompObj.KObjInfo460Ole10Native71Ole10FmtProgID 5  E PS =PQDQDP0.5=223.5DQDPDQDP=0.523.52()=5.875=d. 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J!75 M!5?!.E!875H > YPSymbol GP؟ww wf-!+]Times P؟ww wf-!.f!15iPSymbol HP؟ww wf-!=wTimes P؟ww wf-!0!.!17Times IP؟ww wf-!QTimes P؟ww wf-!SPSymbol JP؟ww wf-!-Times P؟ww wf-!1!.!85!. & 'dxpr MTHU Grphbj " currentpoint " +P) =( Q + S (5).)875" (5-( >11) .)75(?5).)875" >(]+) .)15)=) 0).)17) Q +S (-)1).)85) ./MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 6336 div 896 3 -1 roll exch div scale currentpoint translate 64 39 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 655 406 moveto 892 0 rlineto stroke 1940 406 moveto 892 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (P) 6 505 sh (Q) 895 264 sh (Q) 4741 505 sh 224 ns (S) 1152 360 sh (S) 4998 601 sh 384 /Symbol f1 (=) 337 505 sh (-) 1639 505 sh (+) 2925 505 sh (=) 3765 505 sh (-) 5230 505 sh 384 /Times-Roman f1 (5) 675 798 sh (875) 963 798 sh (11) 1945 264 sh (75) 2425 264 sh (5) 1960 798 sh (875) 2248 798 sh (15) 3297 505 sh (0) 4074 505 sh (17) 4362 505 sh (1) 5489 505 sh (85) 5777 505 sh 384 /Times-Roman f1 (.) 867 798 sh (.) 2329 264 sh (.) 2152 798 sh (.) 3201 505 sh (.) 4266 505 sh (.) 5681 505 sh (.) 6161 505 sh end MTsave restore dMATH P=Q S 5.875-11.755.875+.15=0.17Q S -1.85.۠ FMathType Equation1ELO Equation P=Q S 5.875-11.755.875+.15=0.17Q S -1.85. EquationOle10NativeIOle10FmtProgID  _1015672511AN FPIٝPIٝOle drH r    .  & Times PP؟ww w'f-!Q Times P؟ww w'f-!D PSymbolPIC KMdMETA :PICT LPCompObjK QP؟ww w'f-!= Times P؟ww w'f-!32 !. (!9 +PSymbol RP؟ww w'f-!- 3Times P؟ww w'f-!4 <!. B!7 ETimes SP؟ww w'f-!P LPSymbol P؟ww w'f-! VTimes TP؟ww w'f-!P ePSymbol P؟ww w'f-!= oTimes UP؟ww w'f-!7 yPSymbol P؟ww w'f-!- Times VP؟ww w'f-!. !21 Times P؟ww w'f-!Q Times WP؟ww w'f-!D Times P؟ww w'f-!. & 'dxpr MTHUGrphbj " currentpoint " + Q +D ( =) 32) .)9)-) 4).)7)P) )P) =) 7)-) .)21) Q +D ( ./MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5440 div 448 3 -1 roll exch div scale currentpoint translate 64 59 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (Q) -15 261 sh (P) 2383 261 sh (P) 3185 261 sh (Q) 4820 261 sh 224 ns (D) 250 357 sh (D) 5085 357 sh 384 /Symbol f1 (=) 540 261 sh (-) 1590 261 sh (\336) 2701 261 sh (=) 3516 261 sh (-) 4091 261 sh 384 /Times-Roman f1 (32) 842 261 sh (9) 1322 261 sh (4) 1887 261 sh (7) 2175 261 sh (7) 3827 261 sh (21) 4462 261 sh 384 /Times-Roman f1 (.) 1226 261 sh (.) 2079 261 sh (.) 4366 261 sh (.) 5271 261 sh end MTsave restore dnMATHb Q D =32.9-4.7PP=7-.21Q D .m FMathType Equation1ELO Equationb Q D =32.9-4.7PP=7-.21Q D .ObjInfoOQOle10NativeRfOle10FmtProgID  _1005649588 WFPIٝPIٝ EquationL8Ladxpr MTHUGrphbj " currentpoint " + 0).)17) Q) -)1).)85)=) 7)-).)21) Q) )Q) =) 23) .)12) Ole PIC TVLPICT aCompObjUY W   !#&)+,-./0134568;>@ABCDEFHIJKMPSUVWXYZ[]^_`behjklmnoqrtwz|}~./MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5856 div 448 3 -1 roll exch div scale currentpoint translate 64 56 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (0) -9 264 sh (17) 279 264 sh (1) 1270 264 sh (85) 1558 264 sh (7) 2337 264 sh (21) 2972 264 sh (23) 4825 264 sh (12) 5305 264 sh 384 /Times-Roman f1 (.) 183 264 sh (.) 1462 264 sh (.) 2876 264 sh (.) 5209 264 sh (.) 5689 264 sh 384 /Times-Italic f1 (Q) 658 264 sh (Q) 3330 264 sh (Q) 4149 264 sh 384 /Symbol f1 (-) 1011 264 sh (=) 2026 264 sh (-) 2601 264 sh (\336) 3686 264 sh (=) 4518 264 sh end MTsave restore dcMATHW7 0.17Q-1.85=7-.21QQ=23.12.poFMathType Equation 3.6+DNQEEquation.DSMT36W 0.17Q-1.85=7-.21QQ=23.12. EquationObjInfo Ole10NativeXZ [Ole10FmtProgID [ Equation Native sW 0.17Q-1.85=7-.21QQ=23.29.LWT|WK   .1  ` & & MathTypeP Centur_921259972^ FPIٝPIٝOle PIC ]`LMETA y Schoolbook- 2 @P Century Schoolbookv- 2  BCentury Schoolbookv- 2 06* & "System-  dPPNTCentury Schoolbook ,Century Schoolbook .+PdPPNTCentury Schoolbook +BdPICT _b CompObj"PObjInfoac$Ole10Native%DPPNTCentury Schoolbook ( *dPPNT"System FMathType Equation Equation Equation@ P B *e_921259971t\f FPIٝPIٝOle 'PIC eh(LMETA *LWT|WK   .1  ` & & MathTypeP Century Schoolbook- 2 @P Century Schoolbookv- 2  BCentury Schoolbookv- 2 06* & "System-  dPPNTCentury Schoolbook ,Century Schoolbook .+PdPPNTCentury Schoolbook +BdPPNTCentury Schoolbook ( *dPPNT"System FMathType Equation Equation PICT gj2 CompObj7PObjInfoik9Ole10Native:DEquation@ P B *eLWT|WF   _921259970n FPIٝPIٝOle <PIC mp=LMETA ?.1  ` & & MathTypeP Century Schoolbook- 2 @P Century Schoolbookv- 2 SCentury Schoolbookv- 2 06* & "System-  dPPNTCentuPICT orG CompObjLPObjInfoqsNOle10NativeODry Schoolbook ,Century Schoolbook .+PdPPNTCentury Schoolbook +SdPPNTCentury Schoolbook ( *dPPNT"System FMathType Equation Equation Equation@ P S *eLWT|WF   .1  ` & & MathTypeP Century Schoolbook- 2 @P _921259966|lv FPIٝPIٝOle QPIC uxRLMETA TCentury Schoolbookv- 2 SCentury Schoolbookv- 2 06* & "System-  dPPNTCentury Schoolbook ,Century Schoolbook .+PdPPNTCentury Schoolbook +SdPPNTCentury Schoolbook ( *dPPNT"PICT wz\ CompObjaPObjInfoy{cOle10NativedDSystem FMathType Equation Equation Equation@ P S *eL_921259942~ FPIٝPIٝOle fPIC }gLMETA i~   .1  ``&   & MathType-@ Century Schoolbook- 2 QV2 2 Q3 & "System-  @PICT pCompObjsPObjInfouOle10NativevD    "dPPNTCentury Schoolbook,Century Schoolbook .+ 2*3dPPNT"System FMathType Equation Equation Equation@ 23 LaserJe_921433109 FPIٝ@XIٝOle xPIC ydMETA {bdy Ty  -  o .  & Times' P؟ww w/ fY-!Q Times nP؟ww w/ fY-!D PSymbol' P؟ww w/ fY-!= Times oP؟ww w/ fY-!40 PSymbol' P؟ww w/ fY-!- ,Times pP؟ww w/ fY-!2 9PSymbol' P؟ww w/ fY-!( 5!) ?Times qP؟ww w/ fY-!15 FPSymbol' !P؟ww w/ fY-!( C!) RPSymbol rP؟ww w/ fY-!= XTimes' "P؟ww w/ fY-!10 a & 'podxpr MTHUoGrphbj o"o currentpoint " + Q ( D + =) 40)-) 2 (PICT pCompObjPObjInfoOle10NativeU 5() ))15 ( C()))=) 10/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3552 div 544 3 -1 roll exch div scale currentpoint translate 64 60 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (Q) -15 324 sh 224 ns (D) 284 152 sh 384 /Symbol f1 (=) 574 324 sh (-) 1354 324 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 1635 334 sh (\)) 1970 334 sh 384 1000 1190 /Symbol f3 (\() 2086 336 sh (\)) 2568 336 sh 384 /Symbol f1 (=) 2782 324 sh 384 /Times-Roman f1 (40) 888 324 sh (2) 1769 324 sh (15) 2189 324 sh (10) 3058 324 sh end MTsave restore d]MATHQ } Q D =40-2()15()=10 FMathType Equation1ELO EquationQ Q D =40-2()15()=10Ole10FmtProgID  _921433108 F@XIٝ@XIٝOle PIC d Equationd3 D3     Y .  & Times P؟ww w fv-!QTimes& P؟ww w fvMETA PICT CompObj\ObjInfo-!S PSymbol P؟ww w fv-!=Times& P؟ww w fv-!2 !!3!- &PSymbol P؟ww w fv-! ! ! '! 'Times& P؟ww w fv-!150PSymbol P؟ww w fv-!(-!);PSymbol& P؟ww w fv-!=BTimes P؟ww w fv-!10J!.U & 'Ydxpr MTHU YGrphbj Y"Y currentpoint " +Q ( S +=( !2*3"  (  *  ( ' *   (015 (-()) )=)10) .y/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2848 div 928 3 -1 roll exch div scale currentpoint translate 64 58 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 972 419 moveto 221 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 352 /Times-Italic f1 (Q) -14 518 sh 192 ns (S) 256 346 sh 352 /Symbol f1 (=) 489 518 sh (=) 2055 518 sh 384 ns (\346) 775 355 sh (\350) 775 726 sh (\366) 1203 355 sh (\370) 1203 726 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1075 /Symbol f3 (\() 1385 519 sh (\)) 1840 519 sh 352 /Times-Roman f1 (2) 994 277 sh (3) 999 811 sh (15) 1491 518 sh (10) 2317 518 sh 352 /Times-Roman f1 (.) 2669 518 sh end MTsave restore djMATH^   `Q `S `=23()15()=10. FMathType Equation1ELO Equation9qOle10NativebOle10FmtProgID  _921433107 F@XIٝ@XIٝOle ^  `Q `S `=23()15()=10. EquationdM 0M    Z .  & PIC dMETA 8PICT cCompObj\      +*"#$%&'()f=-./0123456789:;<>@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdeTimesw@M |[wdw0-!40PSymbolw@ [wdw0-!-Timesw@M }[wdw0-!2Timesw@ [wdw0-!P"PSymbolw@M ~[wdw0-!(!))PSymbolw@ [wdw0-!-/Timesw@M [wdw0-!2 9!39- 8 >Timesw@ [wdw0-!PAPSymbolw@M [wdw0-!=KTimesw@ [wdw0-!8T & 'cZdxpr MTHU ZGrphbj  !"%'()*+,-./012345689:;<=>?@ABCDEFGHIJKLMNOPRUXZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ Z"Z currentpoint " +40)-)2)P (()() )-( 92*3" 8(AP) =) 8/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2880 div 896 3 -1 roll exch div scale currentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1739 403 moveto 221 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 352 /Times-Roman f1 (40) 121 502 sh (2) 831 502 sh (2) 1761 261 sh (3) 1766 795 sh (8) 2625 502 sh 352 /Symbol f1 (-) 557 502 sh (-) 1455 502 sh (=) 2342 502 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1075 /Symbol f3 (\() -20 503 sh (\)) 1256 503 sh 352 /Times-Italic f1 (P) 1033 502 sh (P) 2029 502 sh end MTsave restore dOMATHC   `40-2P()-23P=8 FMathType Equation1ELO Equation9qObjInfoOle10NativeGOle10FmtProgID   _921433106d F@XIٝ@XIٝC  `40-2P()-23P=8 EquationL3"X3"Q   .1  &`Ole  PIC  LMETA HPICT 6m & MathType-@B Century Schoolbook- 2 ME 2 `E 2 TE Century Schoolbookv- 2 S 2 iMS 2 iODSymbol- 2 U- & "System-o  tu6- - -"(dPPNTCentury Schoolbook,Century Schoolbook .+ E(E)EdPPNTCentury Schoolbook ( S( S)DdPPNTSymbol, Symbol (-dPPNT"System FMathType Equation Equation EquationCompObjPObjInfoOle10Native _921433105 F@XIٝ@XIٝ E S E S -E D Dd2 48 l2 4    v .  & Ole #PIC $dMETA &PICT 7OTimes P؟ww wf-!4 !4PSymbol\ P؟ww wf-!- !-Times P؟ww wf-!0!.%!2(PSymbol\ P؟ww wf-!(!).-  1PSymbol P؟ww wf-!=6Times\ P؟ww wf-!4 E!4@!.F!2I @ OPSymbol P؟ww wf-!STimes\ P؟ww wf-!0]!.c!95f!.r & 'Ovdxpr MTHUvGrphbj v"v currentpoint " + 4(4) -) -)0).)2 (())" /(6=( E4(@4).)2" @(S) 0).)95) ./MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3776 div 992 3 -1 roll exch div scale currentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 403 moveto 1560 0 rlineto stroke 1986 403 moveto 530 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (4) 688 261 sh (4) 28 795 sh (0) 933 795 sh (2) 1221 795 sh (4) 2159 261 sh (4) 2014 795 sh (2) 2302 795 sh (0) 2928 502 sh (95) 3216 502 sh 384 /Symbol f1 (-) 301 795 sh (-) 723 795 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 582 805 sh (\)) 1422 805 sh 384 /Symbol f1 (=) 1668 502 sh (\273) 2623 502 sh 384 /Times-Roman f1 (.) 1125 795 sh (.) 2206 795 sh (.) 3120 502 sh (.) 3600 502 sh end MTsave restore djMATH^c 44--0.2()=44.20.95.st FMathType Equation1ELO Equation^ 44--0.2()=44.20.95. 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