Department of Agricultural Economics. PhD Qualifier Examination. August 2011

Transcription

1 Departmet of Agrcultural Ecoomcs PhD Qualfer Examato August 0 Istructos: The exam cossts of sx questos You must aswer all questos If you eed a assumpto to complete a questo, state the assumpto clearly ad proceed Be as clear as possble your aswer You have four hours to complete the exam If a aswer requres complcated mathematcal calculatos, studets wll be gve full credt f they smply wrte dow the fucto that could have bee typed to a calculator Importat procedural structos: Be sure to put your assged letter ad o other detfyg formato o each page of your aswer sheets Also, put the questo umber ad aswer page umber (eg 4) at the top of each page Wrte o oly oe sde of your paper ad the leave at least margs o all sdes Tur your fal copy wth all pages order Good Luck!

2 I aswerg some parts of the ext questo, t may be helpful to recall that: a b d = c d ad bc c b a (0 pots) A appled ecoometrca wshes to estmate the followg relatoshp: () y =x β + z γ + ε where x ad z are scalars (assume the data have bee demeaed to remove the costat) a Defe the matrx X = [x z], where x ad z are vectors (X s a matrx) Express the smplest possble way each elemet of the matrces plm ( ) X ' X ad plm ( ) X ' ε as goes to fty b Suppose that the ecoometrca estmates equato () usg OLS Derve the probablty lmt of β (o credt wll be gve for leavg the probablty lmt matrx otato) Call ths β c Suppose E[xε]=0 ad E[zε] 0 Provde codtos uder whch β s a lower boud o the true value of β d Suppose that stead of estmatg (), the ecoometrca estmates the followg usg OLS: () y =x β + ν where ν = z γ + ε Derve the probablty lmt of β (o credt wll be gve for leavg the probablty lmt matrx otato) Call ths β e Provde codtos uder whch β s a upper boud o the true value of β whe β s a lower boud o the true value of β f Provde a real-world example (e provde examples of y, x, ad z) where estmatg () ad () usg OLS would yeld upper ad lower bouds o the true value of β

3 (5 pots) Cosder the followg parametrc form for a caddate cost fucto for a frm: a c d bw + w w3 g C( y, w) = m d y, hw4 y a c where y s the frm s output, whch s produced usg puts x, x, x 3, ad x 4, wth prces w, w, w 3, ad w 4 a Fd restrctos o the scalars a, b, c, d, g ad h such that C( ) s a vald cost fucto for some covex mootoc techology b Fd a form of producto fucto, F(x, x, x 3,x 4 ) that geerates C(y,w) as ts cost fucto 3 (5 pots) Cosder the followg system of equatos: Y X β u =, + Y = X β + u where Y j s a vector, X j s a k j matrx, u j s a vector for j =, It s assumed that E u j = 0 Var u j = σ I, 0 < σ j < for j=,, where I s a, ( ) -dmesoal dmesoal detty matrx, ad Cov( u u ) σ are ukow parameters, σ,i =, σ, a Costruct a smultaeous equato estmator of the two equatos Preset the estmator explctly a matrx form b Preset a feasble effcet smultaeous equato estmator c Suppose that σ, = 0 Dscuss the relato betwee the effcet smultaeous σ ad equato estmator gve part (b) ad the estmators obtaed by smple OLS o the two equatos separately d Suppose that σ, 0 Dscuss the relato betwee the effcet smultaeous equato estmator gve part b ad the estmators obtaed by smple OLS o the two equatos separately

4 4 (5 pots) Cosder a publc goods ecoomy wth oe prvate good, x, oe publc good, y, ad agets Each aget's cosumpto set s the oegatve quadrat Aget s edowed wth w uts of prvate good, ad hs preferece s deoted by There are o tal edowmets for the publc good, but the publc good ca be produced by usg prvate cotrbutos to the publc good accordg to a producto techology y = v/q wth q> 0, v = g ad x + g = w = a Defe the Ldahl equlbrum AND Pareto effcecy for ths ecoomy b Prove every Ldahl equlbrum allocato s Pareto effcet If ay addtoal assumpto s eeded, state t explctly ad show where t s used the proof c Now suppose prefereces of aget ca be represeted by utlty fucto u (x, y) that are dfferetable ad have postve margal utltes Derve the frst order codto(s) that are ecessary for teror Pareto Optmalty (Iteror here meas that x ad y are strctly postve umbers) Note: Lagrage multplers should ot appear the fal form of the codto(s) d Whe utlty fuctos of agets are gve by ( ) u x y = x y wth α > 0 α β, ad β > 0, fd the Ldahl equlbrum Is t Pareto optmal? 5 (5 pots) A expected utlty maxmzer over lotteres of moey has utlty dex u ad tal wealth, w Lottery offers a payoff of x>0 dollars wth probablty α ad a payoff of y>x dollars wth probablty -α a If the aget ows the lottery, what s the mmum prce p at whch she s wllg to sell t (fd the equato that characterzes p ) b If the aget does ot ow the lottery, what s the maxmum prce p that she s wllg to pay for t (fd the equato that characterzes p ) c If the aget s rsk averse ad exhbts o-creasg relatve rsk averso, what ca we say about the relato betwee p ad p? d If the aget s rsk averse ad exhbts o-creasg absolute rsk averso, what ca we say about the relato betwee p ad p? 3

5 6 (0 pots) A frm s cosderg hrg a worker, who ca be oe of two types: hgh or low (e θ = { H, L} ) The worker kows hs ow type ad the frm oly kows that the probablty of the worker beg of hgh type s 3 The hgh ablty worker geerates a reveue of π ( H ) = for the frm ad the low ablty worker- π ( L) = 0 If hred, the frm wll pay the worker a fxed wage w = The hgh ablty worker has the possblty of a alteratve occupato earg hm a payoff of whle the value of the low ablty worker's outsde opto s 0 a Let h = {, 0} deote the frm's hrg decso where stads for the frm's choce to make a job offer to the employee Specfy Equlbrum (BNE) of ths game * h a Bayesa Nash Suppose ow that before the frm makes ts hrg decso, the worker ca choose to vest educato Let e = {,0} deote the educato strategy of the worker where e = stads for the worker's choce to acqure educato The cost for the hgh type of acqurg educato s c( H ) = ad the cost for the low type of acqurg 6 educato s c( L ) = Whle the educato does ot affect the workers' 3 productvty sde or outsde the frm, t s observable to the frm Let µ ( θ e) deote the frm's belef that the worker s of type θ after observg the educato choce b Draw a game tree depctg the extesve form of ths game c Specfy a fully separatg Perfect Bayesa Equlbrum (PBE) or expla why such equlbrum does ot exst d Specfy a poolg PBE or expla why such equlbrum does ot exst e Specfy a (o-degeerate) partally separatg PBE ( α( e θ ), hˆ ( e), µ ( θ e)), where α( e θ ) stads for the probablty of type θ choosg e ad h ˆ( e ) stads for the probablty of the frm makg a job offer to the worker after observg e, or expla why such equlbrum does ot exsts 4