In the figure below, the point d indicates the location of the consumer that is under competition. Transportation costs are given by td.

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1 UC Berkeley Economcs 11 Sprng 006 Prof. Joseph Farrell / GSI: Jenny Shanefelter Problem Set # - Suggested Solutons. 1.. In ths problem, we are extendng the usual Hotellng lne so that now t runs from [-a, 1+a]. Frms are located at 0 and 1. Unless there s a large prce dfference, consumers n the tals buy from the closest frm. We wll consder only the case n whch the consumers n the tals buy from the closer frm, so the frms compete only for the consumers n the (0,1) range. Note that for large enough cost dfference, however, there s a dscontnuty of demand the frm wth the lower cost (prce plus transportaton costs) wll take the whole market. In the fgure below, the pont d ndcates the locaton of the consumer that s under competton. Transportaton costs are gven by td a 0 d 1 1+a We can wrte out the total cost to the consumer n the mddle range: If a consumer n the mddle purchases from Frm 1, pays: p 0 +td If a consumer n the mddle purchases from Frm, pays: p 1 +t(1-d) Consumers n [-a,0] wll buy from Frm 0 unless there s a very large prce dfference,.e., that p 0 plus transportaton costs s greater than p 1 plus (hgher) transportaton costs. Consumers n [1,1+a] wll smlarly buy from Frm 1. To fnd the consumer who s ndfferent between the two frms: p 0 +td=p 1 +t(1-d) Solvng for d: d=(1/) + (p 1 p 0 /t) Now thnk about maxmzng profts of Frm 0. π= (p 0 -c)*a + (p 0 -c)*[(1/)+( p 1 p 0 )/t] π= a p 0 ca + (p 0 -c)/ + (p 0 p 1 /t) (p 0^/t) - (c p 1 /t) + (c p 0 /t) dπ/d p 0 = a + ½ + p 1 /t p 0 /t + c/t = 0 ta + t + p 1 p 0 + c =0 p 0 = ta + t + p 1 +c By symmetry, p 1 = ta + t + p 0 + c and, p 0 = p 1 =p

2 Substtutng and addng together, 4p = 4ta + t + p + c p = ta + t + c Note that the prce s now hgher by the amount ta the transportaton cost to cover the extra dstance of a on ether end. Prce s ncreasng n both a and t a greater number of captve consumers and hgher transportaton costs (or equvalently, more dfferentaton), mply a hgher prce. As n class, another way to solve once we have d=(1/) + (p 1 p 0 )/(t) s to use resdual demand elastcty and the Lerner equaton. Snce the slope of demand (for the frm at 0, say) s -1/(t), and n symmetrc equlbrum the quantty for each frm s now a + (1/), the elastcty of resdual demand s p/[t(a+1)]. So the Lerner equaton tells us that (p c)/p = t(a+1)/p, or p = c + (a+1)t. Note that the case a=0 confrms the calculatons we dd n class.. Snce consumers pay more under ths framework, and relatve to the case dscussed n class, the addtonal prce at exceeds the addtonal transportaton cost, whch s at most at, t follows that consumers have greater ncentve to credbly gnore dfferentaton. A central purchasng agency would thus be more attractve n ths set-up.. a. Ths s a surprsngly hgh elastcty, especally for a durable good lke cars. In 1955, 45% change n output only led to a 6% change n prce. (Arc) Elastcty = 45/6 = 7.5. b. Output ncreased a lot (45%), whle prces fell only a lttle bt (6%). So, f profts fell whle output ncreased and prces fell a small amount, ths mples that frms were prcng close to margnal cost. If not, you would expect a large change n output accompaned by a small change n prce to ncrease profts. Specfcally, f p s the normal prce and Q the normal quantty, we have: (0.94p c)*(1.45q) < (p c)q. Dvdng both sdes by Q and dong some arthmetc yelds 1.363p 1.45c < p c, or 0.45c > 0.363p, or c > 0.8p. c. If MC, AC < p and prce s close to margnal cost, then there are not sgnfcant economes of scale n the ndustry. Scale economes are ncreasng n the rato of AC/MC. d. The key dfference s that cars are a durable good. You would expect that a durable good would be more senstve to prce changes n nearby years. Ignorng 1955, the trend of automoble output gven n Car Wars s surprsngly steady. It looks a bt more lke what you would expect for a non-durable good, such as arlne travel. You would expect that f prces were low (and output was hgh) n one year that the output n the followng year would be abnormally low (snce everyone thnkng about buyng a car n 1956 was lured nto buyng a car n 1955 nstead). For arlne travel, you mght expect that a temporary decrease n prce would (mostly) expand rather than shft demand for that year. 3. p = A Q, N frms n the market frm-specfc unt costs, c c = (1/N)Σc

3 Q = Σq a. Frst, wthout loss of generalty, look at the proft functon for frm 1: π I = pq 1 c 1 q 1 π I = (A-Q) q 1 c 1 q 1 = A q 1 q 1 Σ q c 1 q 1 Maxmze wth respect to q 1. dπ I /d q 1 = A q 1 - Σ q + q 1 c 1 = 0 q 1 = A Q c 1 That s Frm 1 s quantty. Now add up q 1 + q + + q n Σ q = NA NQ - Σc Q = NA NQ - Σ c Q/N = A Q (1/N)Σc = A Q - c Q = N(A c ) Then plug ths back nto the prce equaton: p = A Q = A N(A c ) p = A + Nc Notce (not strctly part of answerng the queston, but worth notcng) that as N gets bg, ths says prce wll be near cost; for N=1 t gves well known monopoly prce formula. Notce also that near cost n that last sentence means near c. The extremely alert student wll already be askng how frms wth notceably above-average unt costs can survve see part (c) below! b. Average cost of producton = total cost/q Frm s total cost of producton = TC = c q Industry total cost of producton =TC = Σ TC Σ TC = Σc (A-Q-c ) = Σ(Ac Qc c ) TC = ΣAc ΣQc Σc ) Recall, var[c] = 1 N (c c ) Expandng, we fnd that

4 var[c] = 1 N c c c + Nc Rearrangng and substtutng c = N var[c] + Nc c = Nc : Substtute ths nto above total cost equaton: TC = A c Q c N var[c] + Nc Agan substtutng c = Nc and dvdng by Q, to express as average cost: TC ANc QNc N var[c]+ Nc ANc N var[c] + Nc = = Nc Q Q Q Recall that Q = N(A c ) (from part a.) and substtute: TC Q = (ANc N var[c] + Nc ) N(A c ) Nc Cancellng out, rearrangng and combnng: TC Q = Ac c var[c](n +1) ( N + 1)var[ c] = c A c A c By the way, we can also say the followng (not asked n the queston) about average producton cost n Cournot equlbrum wth nonlnear demand. Average cost of producton s ( q / Q ) c = sc where s s frm s market share. Rememberng that each frm s resdual demand elastcty n Cournot s the market demand elastcty dvded by ts share, we have ( p c )/ p= s / ε Multplyng through by p and by s and addng up for all frms gves us s ( p c ) = p s / ε. Snce the shares add up to 1, and rememberng the defnton of the Herfndahl ndex H, ths s

5 p sc = ph / ε so the average cost of producton s equal to p(1 H / ε ). Ths would gve an alternatve route to answer the queston here, though not necessarly a smpler one gven that both p and market demand elastcty are functons of all the c s. c. If frms shut down, t wll be the hgh cost frms that do so. So, the theoretcal average cost of producton s hgher than the actual average cost of producton, gven that frms can decde not to produce. Frm wll ext the market f frm AC (=c ) exceeds MR (=p). Ext f: p = A + Nc A c < c (c c )N + c > A ( N + 1) c > Nc + A, whch says that the frm s unt cost exceeds a weghted average of all (ncludng ts own) cost draws together wth the demand ntercept (note that the demand ntercept s the cost level above whch even a monopoly would ext). A mnute s thnkng about how averages work should tell you that ths means the same as sayng that the frm s unt cost exceeds a weghted average of other frms cost draws and the demand ntercept. d. Undfferentated Bertrand competton assgns all producton to the lowest cost frm, so the average cost of producton wll be lower than under Cournot (gven varyng c s). Bertrand prcng gves the second-lowest cost as prce. If the cost of the second-lowest frm s below the Cournot prce then the Bertrand prce wll be lower. It s temptng to say that f the second-lowest cost level s above the Cournot prce then the Bertrand prce would be hgher. But actually the only way that could happen (snce then only the lowest-cost frm would be producng n Cournot equlbrum) s f the monopoly prce, for the lowest-cost frm, s below the second-lowest cost: what we descrbed n class as a drastc cost dfference. And n that case, of course, the lowest-cost frm pcks ts monopoly prce and other frms stay out, both n Bertrand and n Cournot.