Elements of Economic Analysis II Lecture VI: Industry Supply

Transcription

1 Elements of Economc Analyss II Lecture VI: Industry Supply Ka Hao Yang 10/12/2017 In the prevous lecture, we analyzed the frm s supply decson usng a set of smple graphcal analyses. In fact, the dscusson can be extended to a whole ndustry, whch s the man topc of ths lecture. 1 Industry Supply wth Entry Barrer We consder frst the case the there s an entry barrer so that there are only n frms operatng n the ndustry. 1.1 Short Run Supply wth Entry Barrer Recall that gven the short run cost functon c SR, a frm s proft maxmzaton problem s max y 0 py csr (y). We denote the soluton of ths short run problem as y SR (p), whch s exactly the unque soluton to the frst order condton p = MC SR (y). f p > AV C SR (p) MC SR (y SR (p)) s ncreasng, and s zero otherwse. We call y SR the short run supply functon of ths frm. Department of Economcs, Unversty of Chcago; e-mal: khyang@uchcago.edu 1

2 2 Instead of consderng only one frm, we can consder a whole ndustry. For each frm {1,..., n}, let c SR Let y SR be frm s short run cost functon and assume that t s strctly convex. denote frm s short run supply functon. The ndustry supply s then gven by: Y SR (p) = N =1 y SR (p). Recall that for each frm, the supply curve of frm s exactly the part of the margnal cost curve that les above the lowest pont of average varable cost curve. Under the assumpton that c SR s strctly convex, ths s smply the whole margnal cost curve snce margnal cost curve s always above the average cost curve. As such, the ndustry supply s smply the horzontal sum of the ndvdual supply curves. As depcted n Fgure 1. [Fgure 1.] Notce that gven any market prce p, the whole ndustry wll supply Y SR (p) unts. However, as dfferent frms may have dfferent cost functons, each frm may have dfferent profts. As llustrated n Fgure 2. Indeed, gven p, frm wll supply y SR soluton of [Fgure 2.] (p) unts of output, where y SR (p) s the unque p = MC SR (y). As such, dependng on the cost functon of frm, can ether make postve proft, negatve proft or zero proft under market prce p. Recall the the proft of a frm when producng y SR (p) unts gven prce p s smply y SR (p)(p AT C SR (y SR (p))). Thus, f MC SR ntersects wth p above the lowest pont of AT C SR, then frm wll be makng postve proft. Smlarly, f MC SR negatve proft. Fnally f p ntersects wth MC SR then frm wll be makng zero proft. ntersects wth p below AT C SR, then frm wll be makng exactly at the lowest pont of AT C SR,

3 3 1.2 Long Run Supply wth Entry Barrer Smlar to the short run problem, we can examne the ndustry supply curve n the long run by aggregate each frm s long run supply functon. Recall that n the long run, each frm wll have a convex cost functon c LR. Suppose that each frm has a strctly convex cost functon. Then each frm s long run supply, y LR (p) s gven by the soluton to the frst order condton p = MC LR (y). The long run ndustry supply wth an entry barrer s then Y LR (p) = N =1 y LR (p). Agan, the graph of the long run supply wll smply be the horzontal sum of ndvdual frm s supply and dfferent frms may earn dfferent profts. However, n the long run, snce margnal cost curve s always above the average cost curve, no frm wll operate wth negatve proft. 2 Industry Supply wth Free Entry Instead of assumng an entry barrer, another scenaro one can magne for an ndustry s that t s of free entry that s, each frm can choose whether or not to stay n the ndustry or leave, n addton to ts producton decsons. Also, every frm that s outsde of the ndustry can choose whether or not to enter. To smplfy the problem, let us consder the case where all the frms and potental frms are homogeneous. That s, each frm has access to the same producton functon (and thus the same ( short run and long run) cost functons, denoted by c SR (y) and c LR (y)) once t s operatng n the ndustry. 2.1 Short Run Supply wth Free Entry 1 1 In Varan s textbook, free entry and long run are ndstngushable. I fnd ths odd. In ths lecture note, I wll separate these to concepts, although t mght seem a lttle bt not ntutve to talk about enter and ext n the short run. After all, the dscusson n secton 24.4 n Varan s textbook s effectvely about the short run ndustry supply wth free entry presented here.

4 4 Let y SR be the short run supply functon of each frm n the ndustry. Snce the frms are homogeneous, ther supply functons are the same. Therefore, f there are n frms n the ndustry, ndustry supply s smply ny LR (p) when the market prce s p. However, ths s not the actual supply functon. Snce the frms can decde to enter or to leave the ndustry freely, whenever there s postve proft for the frms n the ndustry. That s, whenever p > AT C SR (y SR (p)), all the frms wll want to enter and therefore total supply s nfnty. On the other hand, whenever the frms n the ndustry are makng negatve proft. That s, p < AT C ysr (p), all the frms wll leave the ndustry and thus ndustry supply s zero. Fnally, when the frms are makng zero proft. That s, when p = AT C SR (y SR (p)), then all the frms are ndfferent between enter and ext and therefore the ndustry can supply any amount of output. Therefore, let p be the lowest pont of AT C SR, the ndustry supply wth free entry s exactly Y SR (p) = 0, f p < p [0, ), f p = p., f p > p [Fgure 3.] [Fgure 4.] The dscussons can be summarzed by Fgure 3 and Fgure Long Run Industry Supply wth CRS Technology and Free Entry Smlarly, we can examne the ndustry supply wth free entry n the long run. Notce that the above analyses can stll go through f we replace all the SR by LR. However, recall that n the long run, cost functon s convex. If the frms do not have constant return to scale and the producton s concave, average total cost s strctly ncreasng n y. As a result, the only possble prce p for the frms to make zero proft s p = 0. In ths case, ndustry supply s always nfnty. As such, t wll be more nterestng to look at the ndustry supply for constant return to scale technology f we allow free entry. Recall that f a frm has constant return to scale

5 5 technology, the long run cost functon s lnear. That s c LR (y) = cy for some c > 0. Therefore, AT C LR (y) = MC LR (y) = c for all y. Under ths scenaro, f the market prce p s greater than c, not only each ndvdual frm wll produce nfnty amount of output, but nfnty many frms wll also want to enter the ndustry, makng the ndustry supply nfnty. On the other hand, f p < c, all the frms wll not be wllng to produce any output and no frms wll want to enter, makng the ndustry supply zero. When p = c, each frm n the ndustry wll be ndfferent n producng any amount and all the frms wll be ndfferent between enter and ext. Therefore, the ndustry can supply any amount f p = c. To sum up, the long run ndustry supply n ths case s smlar to th short run. 0, f p < c Y LR (p) = [0, ), f p = c, f p > c [Fgure 5.] It s n fact ntutve to explan why there s a qualtatve dfference n ndustry supply dependng on whether there s entry barrer or not. When there s free entry, the zero proft condton essentally says that every frm, both operatng frm and potental frms, s ndfferent between jonng/stayng n the ndustry or leave t for elsewhere. Ths actually means that every frm has a proft that s the same as ts opportunty cost. The supply analyss tells us ths s the only possble way for the ndustry to operate. As we can see from Fgure 4 and Fgure 5, when there s no entry barrer, the ndustry supply s perfectly elastc. Indeed, free entry generates a great amount of competton among frms, yeldng the aggregate supply extremely elastc. On the other hand, when there s entry barrer, there wll be not as much competton and the frms n ths ndustry can stll earn some economc rent the fact that they are allowed to stay n the ndustry s tself a prvlege and therefore they can earn postve proft n the long run, and the ndustry supply wll be less elastc.