University of California, Davis Date: June 22, 2009 Department of Agricultural and Resource Economics. PRELIMINARY EXAMINATION FOR THE Ph.D.

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1 Unversty of Calforna, Davs Date: June 22, 29 Department of Agrcultural and Resource Economcs Department of Economcs Tme: 5 hours Mcroeconomcs Readng Tme: 2 mnutes PRELIMIARY EXAMIATIO FOR THE Ph.D. DEGREE Part I Please answer four of the followng fve parts a) There are goods. Good s lesure, measured n hours; ts consumpton s denoted x, and ts dollar prce per hour) p >. Goods 2 to are consumpton goods, wth dollar prces p 2,, p, all postve. The consumer s endowed wth ω > unts of lesure and zero unts of any other good ω 2 = = ω = ). The consumer s also endowed wth m dollars of prce-ndependent wealth. We defne the varable L ω - x and call t the supply of labor. The consumer s preferences are represented by a dfferentable utlty functon u: X R, ncreasng n all ts arguments n the nteror of X, where X R+ denotes her consumpton set. Denote the Jevonsan supply-of-labor functon by Lp ˆ, p2,..., p ) we take the parameters m and ω as gven, so that they do not appear as arguments). ˆL a). Wrte the Slutsky equaton for the decomposton of the total effect of a p change n p on the supply of labor, expressng the substtuton and the wealth effects n terms of the partal dervatves of the Hcksan and Walrasan demand-for-lesure functons. a) 2. Assume that lesure s a normal good. What can you say about the sgn of the ˆL substtuton and wealth effects of a)? What can you say about the sgn of the total effect? p Interpret n words. b) We now consder a new model. We stll have prces p, p 2,, p ) >>, an endowment vector ω,, ), and the consumer s prce-ndependent wealth m. But we ntroduce two dfferences. Frst, lesure no longer enters the utlty functon, so that preferences are defned on an open subset Z R, wth typcal element x2,, x ), and represented by a dfferentable, strctly + quasconcave and strctly ncreasng utlty functon β : Z R. Second, the consumpton of any commodty {2,, } takes tme, so that the avalable amount of lesure ω must be allocated between the supply of labor n the market, denoted L, whch s pad at rate p per hour, and unpad tme devoted to consumpton actvtes. More

2 specfcally, the data of the economy nclude, for = 2,,, a nonnegatve coeffcent t expressng the per-unt amount of tme requred by the consumpton of good of, so that the consumpton of x unts of good requres spendng t x unts of tme, n addton to spendng p x dollars. b). Wrte the consumer optmzaton problem that yelds her demand for goods x p, p,..., p ), = 2,..., and her supply of labor L p, p2,..., p ) agan, we take the parameters 2 m and ω as gven). Assume that the soluton exsts. b) 2. Argue that the tme endowment ω s totally spent. b) 3. Combne the budget constrant and the tme constrant nto a sngle equalty constrant nvolvng x 2,, x ), and nterpret. b) 4. Denote a -) dmensonal prce vector by π2,..., π ), and a wealth magntude by w, and defne the Walrasan demand functon x% 2 π2, L, π, w), K, x% π2, L, π, w)), as usual, by the soluton to the problem max β x2,..., x ) subect to π 2 x = = w. Smlarly, defne the Hcksan demand functon h 2 π 2,..., π, u),..., h π 2,..., π, u)) by the soluton to the problem mn π 2 x = subect to β x2,..., x ) = u. Usng b)3, express the functons x p, p,..., p ), = 2,..., and L p, p2,..., p ) n terms of the Walrasan demand functons 2 x%, = 2, L,. %,, % ). x2 L x b) 5. Express L p n terms of the dervatves of the Walrasan demand functons x% b) 6. By usng the Slutsky decomposton of the total effects, k = 2, K, ), wrte πk L h as the sum of a substtuton term nvolvng the dervatves of Hcksan demand, and a p πk x% wealth term nvolvng the dervatves of Walrasan demand. w b) 7. Assume that goods 2,, are normal. What can you say about the sgn of the substtuton and wealth effects obtaned n b).6? What can you say about the sgn of the total L effect? Compare wth a) above. p 2

3 Part II There are two goods, good and good 2, and good 2 s the numerare good. There are I consumers. For =,, I, consumer s utlty functon s u : R + R R : u x, x2) = b x) + x2, where b s dfferentable, ncreasng and strctly concave. All consumers are prce takers. We assume that the prce of the numerare good s equal to, and we denote by p the prce of good. Denote by x% ) p consumer s Walrasan demand for good. There s a sngle frm whch produces good by usng the numerare as an nput, wth cost functon Cy), where y s the amount of good produced. We assume n what follows that frstorder equaltes characterze the soluton to every optmzaton problem. p C ' Gven a prce p, we defne the markup as, where the margnal cost C s evaluated p at the amount of output equal to aggregate demand. a) Wrte the frst order condton of the consumer s optmzaton problem that yelds her Walrasan demand for good. b) Suppose that prces are regulated n order to maxmze the sum of consumer surplus and profts. What can you say about the resultng markup? I c) Let consumer own a share θ n the profts of the frm =,, I, θ = ). = As a consumer, she buys the good n the market, where she s a prce taker, but she can vote at the shareholders meetng on the prce that the frm wll charge. Derve the equaton that characterzes the best prce for consumer-shareholder. d) Suppose that, at the shareholders meetng, all shareholders unanmously agree on a prce. What can you say about the resultng markup? What can you say about the share θ of consumer-shareholder? Interpret. Hnt: Add up the FOC. e) We now specalze the model to a very smple case, where b x ax x 2,, I, and Cy) = cy, where a > c. But we assume that only a fracton σ of the populaton of I consumers are shareholders n the frm, each ownng a share θ= n the frm s profts. σ I e). Compute the monopoly proft-maxmzng prce p M. 2 ) = ), = e) 2. Show that all shareholders agree on a prce pσ), and compute t. How does pσ) vary wth σ? What are the lmts of pσ) as σ? As σ? Comment. 3

4 Part 3 When dscussng the market for electrcty t s often asserted that the demand should be ratoned at peak tmes because of the lmted capacty of producton. We show that compettve prces can do the ob provded they are suffcently refned. To see ths, consder an economy wth T + goods: good s a numerare good and goods,..., T represent consumpton of electrcty at tme t =,..., T for example consumpton n the day tme and consumpton at nght). The producton of electrcty requres buldng a plant of capacty K, where K represents the maxmum amount of electrcty that can be produced at any tme. Buldng a plant of capacty K requres ρk unts of numerare good and then the cost of producng one unt of electrcty at any tme s γ unts of numerare. Snce t s not optmal to buld a larger capacty than the maxmum amount of electrcty produced at any tme, the producton set for electrcty s { Y = z, y, y 2,..., y T ) R R T + z ρ max t T y t ) + γ } T y t t= a) Show that the producton set for electrcty s convex and exhbts constant returns. b) Snce t s equvalent to assume that there s one prce-takng frm or a lot of small frms producng electrcty, to smplfy we assume that there s one frm whch maxmzes ts proft takng prces as gven. To smplfy the study of the frm s proft maxmzaton, from now on we take T = 2 and p =, p, p 2 ). ) In the plane y, y 2 ), draw the so-cost curves of the frm. Make sure that you get the shape rght. ) Add an so-revenue lne and fnd the condton on the rato of the prces p /p 2 whch guarantees that proft maxmzaton occurs along the dagonal y = y 2. ) In the case consdered n ) fnd the addtonal condton on the prces whch ensures that there s a non zero soluton to proft maxmzaton. c) There are I agents n the economy. Agent I) has an endowment ω of the numerare good and a utlty functon We assume that u x ) = x ) α x ) α x 2 ) α 2, α + α + α 2 =. < I= α ω I= α 2 ω < γ + ρ γ.e. tme s the perod when electrcty s most needed day tme) but the dfference n needs s not extreme. Show that there s a compettve equlbrum such that the consumpton s the same at perod and at perod 2, so that the capacty s optmally used n equlbrum. Actually ths s the only equlbrum but we do not have tme to prove t.) 4

5 Part 4 We wll show that the use of lotteres by nsttutons whch provde prvately fnanced publc goods e.g. chartes) results n a hgher level of publc good fnancng than relyng on voluntary contrbutons. Consder a smple model wth two goods, one prvate, one publc, one unt of prvate good producng one unt of publc good. There are I agents wth dentcal endowment ω of prvate good and dentcal, quas-lnear preferences ux, y) = x + γ logy), γ > where x s the consumpton of prvate good, y the quantty of publc good, and γ < ω. Consder only symmetrc outcomes where all agents consume the same amount of the prvate good. a) Fnd the Pareto optmal level of publc good y. b) Calculate the amount ȳ of publc good whch wll be provded by a charty f t reles on voluntary contrbutons by the I agents. Explan why ȳ s less than y. c) Suppose that nstead of relyng on contrbutons, the charty organzes a lottery wth a prze R. It can prnt as many lottery tckets, sold for one unt of prvate good each, as the agents want to buy. If agent buys z tckets and the total number sold s Z, then the probablty that agent wll wn the prze n a random draw of the lottery s πz, Z) = z Z Partcpatng n the lottery makes agent s consumpton x random. Agents are assumed to maxmze expected utlty wth the von-eumann-morgerstern utlty functon u and, snce u s quas-lnear, agents are rsk neutral. An equlbrum wth lottery s an outcome where the charty uses the recepts from the lottery tckets to cover both the cost R of the prze and the cost of producng the publc good y agents know ths), and each agent chooses the optmal number of lottery tckets to buy, gven the number of tckets bought by the other agents and the prze R. ) Wrte the frst-order condton for the optmal number z of lottery tckets for the typcal agent, and usng symmetry express t as a functon of Z = Iz. ) Expressng ths relaton as a second degree polynomal n the varable Z R, show that the total amount of money Z L collected by the lottery satsfes γ < Z L R < Iγ. You do not need to compute Z L ). ) Interpret the result n ). Explan ths result n terms of margnal costs and margnal benefts. 5

6 Part 5 The purpose of ths exercse s to study Betrand-ash equlbra when some consumers do not pay attenton to small prce dfferences and to relate the Bertrand paradox to consumers senstvty to prce dfferences. There are two frms that produce a homogeneous product. Let p be the prce of frm =, 2). Assume for smplcty that the frms have zero costs. There are consumers wth unt demands, each wth the same reservaton prce for the product, denoted by r. That s, each consumer buys one unt at most from one of the two frms f and only f at least one of the prces s less than or equal to r. When p = p 2 r, 5% of the consumers go to frm and 5% to frm 2. What happens when the two prces are dfferent? Some consumers are very senstve to prce dfferences, whle others are not. For example, f p = p 2 +. then some consumers mght prefer frm 2 because they save penny, but probably most consumers would be ndfferent between the two frms. Let f : [, r] R + be a densty functon thus fx), for all x [, r], and r fx)dx = ) that measures the prce-dfference senstvty of consumers. For example, suppose that p < p 2 < r. Then p2 p fx)dx gves the fracton of consumers who prefer the cheaper frm frm ), whle the others are ndfferent between the two frms. Assume that 5% of the ndfferent consumers go to one frm and 5% go to the other frm. In questons b)-e) we focus on the possblty of a ash equlbrum where both frms charge the hghest possble prce, whle n questons f) and g) we look at a ash equlbrum where both frms charge a prce equal to margnal cost. a) Wrte down the demand functon of each frm cover all the possbltes, that s, all pars p, p 2 ) wth p [, ) for every =, 2 ). b) Suppose that r = 2, 2 fx)dx =.4, f2) =.4. Is p = p 2 = 2 a ash equlbrum? c) Suppose that f s contnuous and that the functon gx) = x[ + Fr x)] where F s the x ) c.d.f., that s, Fx) = ft)dt s convex n the nterval, r). Show that p = p 2 = r s a ash equlbrum. d) Suppose that f s contnuous and that the functon g defned n the prevous queston) s concave n the nterval, r). Gve a necessary and suffcent condton for p = p 2 = r to be a ash equlbrum. e) Assume that f s constant unform dstrbuton). Is p = p 2 = r a ash equlbrum? f) Assume that x ft)dt < for x suffcently close to. Is p = p 2 = a ash equlbrum? g) What assumptons on f would yeld the Bertrand paradox? 6