Test, ECON 6090-9, Summe 0 Instuctions: Answe all questions as completely as possible. If you cannot solve the poblem, explaining how you would solve the poblem may ean you some points. Point totals ae in paentheses.. Shot answe questions a Thee ae two cigaette manufactues in an industy. They both choose advetising levels simultaneously. The payo matix fo thei choices is shown below: Fim Advetise Don t Advetise Fim Advetise 7 ; 7 6 ; Don t Advetise ; 6 4; 4 A ( points) Find the Nash Equilibium to this game. The Nash equilibium is that Fim chooses Advetise and Fim chooses Advetise. B ( points) The fedeal govenment in the late 960s and ealy 970s passed legislation that pohibited cigaette companies fom many foms of advetising. Based on this game, explain how the govenment s legislation can incease the po t to both ms. By pohibiting advetisement, it is as if the govenment is causing the ms to cedibly commit to choosing Don t Advetise, and they will both eceive 4 if they choose Don t Advetise as opposed to the 7 they eceive fo choosing Advetise. b ( points) Explain why the typical shape of the shot-un aveage total cost function is U-shaped. The shot-un ATC is U-shaped because fo low quantities of poduction thee is a high aveage xed cost and fo high quantities of poduction thee is a high maginal cost (due to diminishing maginal etuns). Thus, the shot-un ATC begins at a high level and then declines as AFC declines, but then tuns upwad as maginal cost begins to incease. c ( points) Explain the shutdown ule fo a m efeencing speci c cost and evenue tems. Thee ae basically states fo the m: continue opeation (if making nonnegative economic po t), go out of business (if making an economic loss and does not expect to make a po t in the futue), and shutdown (cuently making an economic loss but expects to make nonnegative economic po t in the futue). If a m is cuently making an economic loss but expects to make a nonnegative economic po t in the futue, then it must decide whethe to continue opeation o to shutdown. A m will continue opeation if it can cove its total vaiable costs and will shutdown othewise. Thus, the shutdown ule is if T R < T V C then shutdown, and if T R T V C continue to opeate. The eason is that the m will incu its xed costs egadless of this decision, but if T R T V C then the m can take some of its evenue in excess of total vaiable cost and apply it to the xed costs. d ( points) Conside the following pictue.
P S P MC ATC MR D Maket Q Fim q Is this pefectly competitive maket in long un equilibium? Explain why o why not. No, this pefectly competitive maket is not in long un equilibium. The epesentative m is making positive economic po ts, which should attact othe ms to the maket. The pefectly competitive maket would be in long un equilibium if the m was making zeo economic po t.. ( points) Conside a simultaneous quantity choice game between ms. Each m chooses a quantity, q and q espectively. The invese maket demand function is given by P (Q) = 06 0Q, whee Q = q + q. Fim has total cost function T C (q ) = 6q + 000 and Fim has total cost function T C (q ) = 6q + 000. a (0 points) Find the best esponse functions fo Fims and. To nd the best esponse functions we need to take the patial deivative of each m s po t function, set the esult each to zeo, and solve fo each m s quantity. Fo Fim we have: = (06 0q 0q ) q 6q 000 @ @q = 06 0q 0q 6 0 = 00 0q 0q Fo Fim we have: 0q = 00 0q q = 00 0q 0 q = 0 q = (06 0q 0q ) q 6q 000 @ @q = 06 0q 0q 6 0 = 00 0q 0q 0q = 00 0q q = 00 0q 0 q = 0 q
Technically the best esponse functions fo Fim and Fim ae q = Max 0; 0 Max 0; 0 q because the m s will not poduce a quantity less than zeo. b (0 points) Find the Nash equilibium to this game. Fo this we need to solve the best esponse functions simultaneously, so we have: q = 0 q q = 0 q Substituting the st equation into the second we have: q = 0 0 q 0 q = 0 4q = 00 0 + q q = 0 q = 0 Now using Fim s best esponse function and that q = 0 we have: So the Nash equilibium is q = q = 0. q = 0 0 q = 0 c ( points) Find the total maket quantity (Q), maket pice, and po t fo each m. q q and q = The total maket quantity is Q = q +q = 0+0 = 00. The maket pice is P (Q) = 06 000 = 06. Each m s po t is: = P q T C (q ) = 06 0 (6 0 + 000) = 00 00 = 4000 and = P q T C (q ) = 06 0 (6 0 + 000) = 4000. (0 points) Conside the constant elasticity of substitution poduction function: q (K; L) = (L + K ) Let =, and let = be the pice of capital and w = be the pice of labo. a (0 points) Find the maginal ate of technical substitution.
Fist nd the maginal poduct of labo: Now nd the maginal poduct of capital: @q @L = (L + K ) L @q @L = (L + K ) L @q @K = (L + K ) K @q @K = (L + K ) K The maginal ate of technical substitution is simply the negative of the atio of those two maginal poducts: Substituting in fo we have: MP L MP K (L + K ) L (L + K ) K L K L K K L = = b (0 points)find the cost-minimizing bundle of inputs if the m wishes to poduce 0 units. You could set it up as an optimization poblem, but we know that at the optimal bundle (inteio) that: w so Now we can nd L in tems of K: L K = w. L = w K. 4
Putting this back into the poduction function and substituting q = 00 we have: q = w K + K q = w K + K q = w K + q = w K + q K = w + Substituting in we nd: K = K = 0 = = + 0 + =(=) K = 0 + K = 0 = 0 = 4 You could simplify if you want but no need to. Now just plug K into the equation fo L: w L = K 8 L = 4 L = 4 L = 4 L = 4 64 9 (=) = L = 4 9 = So the cost minimizing bundle, if the solution is inteio, is K = 4 and L =, which leads to a T C = 4 + = 600. If a cone solution existed it would involve using all capital o all labo, and the poduction function would be eithe q (K; L) = L o q (K; L) = K. If the m uses only L it needs 0 units, and this costs 960. If the m uses only capital it needs 0 units, and this costs 600. Neithe of those is less expensive than using (K; L) = (4; ). 4. (0 points) Bates Gill is the sole develope of undewate opeating systems. His m is potected by consideable baies to enty. Gill faces the following invese demand function fo his undewate opeating systems: P (Q) = 40 Q
His total cost function is: T C = 4Q + 00; 000 a ( points) Find Gill s maginal evenue function solely as a function of quantity. Maginal evenue is simply the deivative of T R with espect to Q: T R = (40 Q) Q @T R @Q = 40 4Q MR = 40 4Q b (0 points) Find Gill s po t-maximizing - Quantity Just set MR = MC. We know that MC = @T C @Q : MC = 8Q So that: MR = MC 40 4Q = 8Q 40 = Q = Q - Pice The pice is given by: P (Q) = 40 Q P (Q) = 40 P (Q) = 700 - Po t at the po t-maximizing pice and quantity Po t is simply T R T C at Q =, so: = (40 Q) Q 4Q + 00000 = (40 ) 4 + 00000 = 6400 7900 = 9600 Adding the govenment The govenment ealizes that Gill is a monopolist and that consideable deadweight loss is being ceated in the undewate opeating systems maket. Use the functions above to answe these questions. a ( points) Daw a pictue (it does not have to be dawn to scale) that illustates 6
- Gill s po t-maximizing pice and quantity - The quantity and the pice that would be chaged if the maket wee to opeate e ciently (with no deadweight loss) - The deadweight loss due to this monopoly (shade it in using a pencil o pen o make) Pice Qty. The po t-maximizing pice and quantity ae given by the ed dashed lines. The e cient pice and quantity ae given by the puple (lowe pice, highe quantity) dashed lines. The geen line is the invese demand function, the bown line is the maginal evenue function, and the black line is the maginal cost function. Deadweight loss is the tiangle outlined by the invese demand function (geen line) between the ed dashed line and the intesection of invese demand and MC, the MC function (black line) between the ed dashed line and the intesection of the MC and invese demand function, and the ed dashed line between the invese demand function and MC. b ( points) Fo the completely e cient outcome (no deadweight loss), nd the - quantity The completely e cient outcome would be whee P = MC. So: 40 Q = 8Q 40 = 0Q 6 = Q - pice To nd the pice simply sub in Q = 6 into P (Q): and we should nd that P = 78. P (6) = 40 6 (Note that these should be actual numbes based on the invese demand and cost function given at the beginning of this poblem.) c ( points) Suppose that Gill had not yet enteed into this maket, but was meely planning to ente into the maket. If the govenment wee to tell Gill ahead of time that they would egulate his pice at the e cient level, would Gill ente this maket? Explain. 7
Gill s po t in this maket at the e cient quantity of Q = 6 is: = (40 6) 6 4 6 + 00000 = 748 8664 = 76 Because Gill would be eaning a negative po t he should NOT ente the maket if he will be egulated. 8