UNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS. M.Sc. in Economics MICROECONOMIC THEORY I. Problem Set II

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1 Mcroeconomc Theory I UNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS MSc n Economcs MICROECONOMIC THEORY I Techng: A Lptns (Note: The number of ndctes exercse s dffculty level) ()True or flse? If V( y ) s convex set, then the ssocted producton set Y must be convex ()Prove tht convex nput requrement set s equvlent to qus-concve producton functon 3 () () Wht s the elstcty of substtuton for the generl CES technology y = ( x + x ) when? ρ ρ / (b) Show tht we cn lwys wrte the CES technology n the form f( x, x ) = A( ρ)[ bx + ( b) x ] ρ ρ / 4 () () Defne the output elstcty of fctor to be f( x) ε ( x) = x x f( x) Wht s the output elstcty of ech fctor of the producton functons: y k x x β = ( + ), y = xx β, y = ( x + x )? ρ ρ / ρ (b) If ε ( x) s the elstcty of scle nd ε ( x) s the output elstcty of fctor, show tht bove technologes? n = ε( x) = ε ( x) Wht s the elstcty of scle of ech of the

2 Mcroeconomc Theory I 5 () If f ( x ) s homothetc technology nd x nd x produce the sme level of output, then tx nd tx must lso produce the sme level of output Cn you prove ths rgorously? 6 ()Show tht proft mxmzng bundle wll typclly not exst for technology tht exhbts ncresng returns to scle s long s there s some pont tht yelds postve proft 7 () For ech nput requrement set determne f t s regulr, monotonc, nd/or convex Assume tht the prmeters nd b nd the output levels re strctly postve () V( y) = { x, x : x log y, bx log y} (b) V y x x x bx y x ( ) = {, : +, > 0} V ( y) = { x, x : x + x x + bx y} (c) (d) V ( y) = { x, x : x + bx y} (e) V( y) = { x, x : x ( y), x ( y) b} V ( y) = { x, x : x x x + bx y} (f) (g) V( y) = { x, x : x + mn( x, x ) 3 y} 8 () Clculte explctly the proft functon for the technology y = x, for 0< < nd verfy tht t s homogeneous of degree nd convex n ( p, w ), where p nd w re output nd nput prce respectvely 9 () Let f ( x, x be producton functon wth two fctors nd let w nd w be ther respectve prces Show tht the elstcty of the fctor shre ( wx / wx ) wth respect to ( x / x ) s gven by /σ nd wth respect to ( w / w ) s gven by σ, where σ s the elstcty of substtuton 0 () The producton functon s f ( x) 0x x = nd the prce of output s normlzed to Let w be the prce of the nput x 0

3 Mcroeconomc Theory I () Wht s the frst order condton for proft mxmzton f x > 0? (b) For wht vlues of w wll the optml x be zero? (c) For wht vlues of w wll the optml x be 0? (d) Wht s the fctor demnd functon? (e) Wht s the proft functon nd wht s ts dervtve wth respect to w? () A compettve proft mxmzng frm hs proft functon π ( w, w ) = ϕ ( w) + ϕ ( w ) The prce of output s normlzed to () Wht do we know bout the frst nd second dervtves of the functons ϕ ( w )? (b) If x ( w, w s the fctor demnd functon for fctor, wht s the sgn of x / w? j (c) Let f ( x, x be the producton functon tht generted the proft functon of ths form Wht cn we sy bout the form of ths producton functon? () Consder the technology descrbed by y = 0 for x nd y = ln x for x > Clculte the proft functon for ths technology 3 () () Gven the producton functon f ( x, x = lnx+ lnx, clculte the proft mxmzng demnd nd supply functons, nd the proft functon Assume n nteror soluton nd > 0 (b) Do the sme for the producton functon f ( x, x ) = x x Assume > 0 Wht restrctons must nd stsfy? (c) Do the sme for the producton functon f ( x, x = mn{ x, x} Wht restrctons must stsfy? 3

4 Mcroeconomc Theory I 4 () A prce-tkng frm produces output q from nputs z nd z ccordng to dfferentble concve producton functon f ( z, z The prce of ts output s p > 0 nd the prces of ts nputs re ( w, w >> 0 However, there re two unusul thngs bout ths frm Frst, rther thn mxmzng proft, the frm mxmzes revenue (the mnger wnts her frm to hve bgger dollr sles thn ny other) Second, the frm s csh constrned In prtculr, t hs only C dollrs on hnd before producton nd, s result, ts totl expendtures on nputs cnnot exceed C Suppose one of your econometrcn frends tells you tht she hs used repeted observtons of the frm s revenues under vrous output prces, nput prces, nd levels of the fnncl constrnt nd hs determned tht the frm s revenue level R cn be expressed s the followng functon of the vrbles ( p, w, w, C ): [ γ ] R( pw,, w, C) = p + lnc ln w ( )lnw ( γ nd re sclrs whose vlues she tells you) Wht s the frm s use of nputs z nd z when prces re ( p, w, w nd t hs C dollrs of csh on hnd? 5 () Prove rgorously tht proft mxmzton mples cost mnmzton 6 () A frm hs two plnts wth cost functons Wht s the cost functon for the frm? y c( y ) = nd c ( y ) = y 7 () A frm hs two plnts One plnt produces output ccordng to the producton functon x x Wht s the cost functon for the frm? The other plnt hs producton functon x x b b 4

5 Mcroeconomc Theory I 8 () () Show tht f the producton functon s homogeneous of degree n, then the cost functon cn be wrtten n cwy (, ) = ybw ( ), where n w R ++ (b) Show tht f the producton functon s homothetc, then ε cy = ε, where ε cy s the cost elstcty wth respect to output nd ε s the elstcty of scle 9 () A frm hs producton functon y = xx If the mnmum cost of producton t w = w = s equl to 4, wht s y equl to? 0 () Show tht w x ( w, y) > 0 y f nd only f mrgnl cost t y s ncresng n () A frm produces output y n compettve mrket usng cost functon cy ( ) whch exhbts ncresng mrgnl costs Of the output, frcton s defectve nd cnnot be sold If the output prce s p () Clculte the dervtve of profts wth respect to nd ts sgn (b) Clculte the dervtve of output wth respect to nd ts sgn (c) Suppose tht there re N dentcl producers, let D( p ) be the demnd functon nd let p( ) be the compettve equlbrum prce Clculte ( dp / d)/ p nd ts sgn () Consder proft mxmzng frm tht produces good whch s sold n compettve mrket It s observed tht when the prce of the output good rses, the frm hres more sklled workers but fewer unsklled workers Now the unsklled workers unonze nd succeed n gettng ther wge ncresed Assume tht ll other prces remn constnt () Wht wll hppen to the frm s demnd for unsklled workers? (b) Wht wll hppen to the frm s supply of output? 5

6 Mcroeconomc Theory I 3 () For ech cost functon determne f t s homogeneous of degree one, monotonc, concve nd contnuous If t s derve the ssocted producton functon: () cwy (, ) = y ( ww) / 3/4 (b) cwy = yw+ ww + w (, ) ( ) w (c) cwy (, ) = ywe ( + w) cwy (, ) yw ( ww w) (d) = + (e) cwy (, ) = ( y+ ) ww y β 4 () A frm s cost functon s cw (, w, y) = ww y Wht cn we sy bout nd β? x x 5 () () Show tht the cost functon of Leontef technology, y = mn{, } β β, s gven by cwy (, ) = y( βw+ βw (b) Show tht the cost functon of lner technology, y = x + x, s gven w w cwy (, ) = ymn, by (c) Show tht the cost functon of Cobb-Dougls technology, y = Ax x, s gven by cwy (, ) = ywwb β, where B depends only on A nd For ll the bove cses drw the cost mnmzton problem Fnd the condtonl fctor demnd functons Drw totl, verge nd mrgnl cost functons 6 () Suppose the strctly qus-concve producton functon f ( x, x nd suppose fctor prces w, w Suppose lso tht proft functon s strctly concve wth respect to output Normlze output prce to 6

7 Mcroeconomc Theory I () Solve proft mxmzton problem nd fnd fctor demnd functons x ( w, w ) nd supply functon y ( w, w (b) Solve cost mnmzton problem nd fnd the condtonl fctor demnd functons xˆ ( wy, ) Show tht for y = y, t s ˆ x = x (c) Alterntvely, tke the cost functon cwy (, ) Solve the mxmzton problem Π= R( y) c( y) nd fnd yˆ( w, w Show tht yw ˆ(, w) = y( w, w) 7