Lab 15(comprehensive)_partial keys

Lab 15(comprehensive)_partial keys - ECONOMICS 207 FALL...

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Unformatted text preview: ECONOMICS 207 FALL 2009 LABORATORY EXERCISE 15 14th December 2009 Problem 1: Differentiate each of the following functions with respect to ac a_62x2+4a;+2 b. (6312) 4 [1 C6578 ’5 9n,” . 6 -67<. c. 111(5m2 + 3:1:z + 8) 1/4 mic) 46 3m + 2 + 6. (8x [(284 91+ 9.): + 3 43: + 3:3) 22:2 (3332 + g.e awry] z . [2sz i_e3z1n [(4332 + 3?] 2 3x. 6 Q’Hix +9 +- C éx+q>} éb+5x+> — Gag—+5 C, (1191+ ’9 1L X- +3x +1) 312 12+2x 2 l (_L ' 3z2+2z+5 Problem 2 : Find the second derivative of each of these functions with respect to ac my = —400 + 20096 + 20132 — 2953 by = 180x1/4z2/5 — 15:1: — 8z Problem 3: Consider the following function: y=3332—12w+9 a. What is the slope of the function at x = 1 b. What is the slope of the function at a: = ~2 d. At What value of a: is the slope of the function 0. What is this value of 3: called? Problem 4: Consider the following function: 1 y=2m3—§$2—$+3 a. What is the slope of the function at w = —1 SLOPE (S TH—E DERIVATIVE <——- Kernembcr‘ l 2. “5; got ~9L—l (w) W .2 M é of w) -' g + I ~ 1 = 5 b. What is the slope of the function at 9: = 0 e (01— C0) —’ ‘1 n W '9 c. At what values of a: is the slope of the function 0. What are these values of as called? ‘ “I ad; which VAIMEI Here 03% “we I; 6 f’awo' 0 o . of x I do“; 2 1- —- 0 Store, = O ::> 61' — 2:“ q; @wordadna 00‘ get ' (2’1") ) mesa vwwku 06 9C. “my” “V3‘9wro ML -9/ dz (62.0 5 2/ * —= ”‘ 5 V5 -5/5 ’ 8x >x — afi- %_ (62° 2 J 2' ay— :3 7“ J «29.5 -9/ ’2. &_ mo .H 5 /5‘ , Partfimufi ~ "‘53" 61:2 _ L J % 162°C): x/ng'g/S an wet“ regroxat E) :6 y 5' 5 ~ 522 “V5 9 . ' “awn. 4—— : (£3 2. - /5 ‘q/r ’ fiwfiaud ufi ayam <5 5 7L J 622 2 Mb 3; l at 55/5 ne‘ with marvel: L_—__r——3way—_1 5 S a g Rh“ mu rmbluns In M CM! 5' Thw- two sum: rmmh uf Z=$7_y7 6z 6?: %_ 6y— 622 $2 92“ 0y2_ 62z _ 6y6m_ 622 z = 1620x1/5y2/5 _ 16m _ 243g away 2 82 6:5 822 63:2 ayaa; : (93/69: I z=x5lny away : Byax : 63/63: : O z=$5—3$2y+y6 z=x2+e2y 10 2e, 7'3 Problem 7: Consider the following function: 1 y = 5 — 11174 3.. Find the critical points of the ginction. l U = “ 7‘ J:0 =2.) “‘7‘— onfj mat-cw [’5 . b. Do we have a local maximum or a local minimum or a point of inflection at this critical point. II J = "3’3 J"@) ' , " J” evMuaruU‘ at 1: 0 86m 0, “(he Second can" Make” mt ‘LS C6 CV5 US av ' rrynqh'on ‘ go a wyfo J'" (a) = —- 6 (O) = O I) I U3 r\ o k) 'l 0 H0 kl d a —. G'x. kc ‘ fl 0‘) “6 740 Since. we we. aim. aim, we. 80 C6 alum . J r. .. T‘M A,ch (H) M when we. 3!! q mn~icro aw. US even V _£<O I we, Problem 8: Consider the following function: —— km a. MAXIMUM «tho 0 3124—326 a. Find the critical points of the function. b. Do we have a local maximum or a local minimum or a. point of inflection at this critical point. 11 Problem 9: Consider the following function: y = 5 + 3:5 a. Find the critical points of the function. b. Do we have a local maximum or a local minimum or a point of inflection at this critical point. Problem 10: Consider the following function: y = 4 — 3m3 a. Find the critical points of the function. I ‘3 :.- -fix,z=0 r.) x Q t. b. Do we have a local maximum or a local minimum or a point of inflection at this critical point. 7" : we» at): we): 0 So ~4th derivafl'oe, tut ails. w; 39 G, Maker . ". Max/mm . 3'" -— I8 ¢ 0 M Mh‘zuo M. TR “VI. a); mum wen a“ “L a name 315 M “M We, hm wfew: oéwaix:0_ To a" (1): m :7" C“) ‘ Problem 11: Consider the following function: 5 a: 5 3 a. Find the critical points of the function. val - 7c," —. 52:. + ‘l‘ = 6— ~> cat-o ,) 7c . 2/ -2./ i I ‘1 b. Classify the critical points as local maximum or local minimum. O 12-: Ll I .1. a CVL‘BZ‘IL (In all - Msscfi Unese {30an we use, Um 4x? _. ID x L; (2)3 _ 106-) = $2 — 20 > 0 km a. Law max a1- x= “C99~'°(Z’) = ~§1+2o <0 lame a Local ’ern ai' 7c: —2, "0)5 —:o@).- 4-10 < o Rowe 0. «Local. rmM3 0d? 7‘ = i_ ((-03 «1060- ~9+w >0. W m M mum at x:-[ Problem 13 : A firm’s cost function is C(y) = 1000 + 50031 — 603/2 + 33/3 Where 3/ denotes output. Price per unit of the output is $725. a. What is the revenue function? kCVCth = ‘Monefi gmwemw by 28% oustrux‘ m m = Frau. {Wm M 0% owl:wa X “0‘06 WAX: agent” = #255 b. What is the profit function? Profit in manna 4%: much am 0.66:. am hm Fad, for ‘ KW -' Cost ‘- a ’R‘ : ¥253 —- (loco +5003 —- 60d +35) c. How much output should the firm produce if it wants to maximize its profit. mow we a“: 7. 3 ~looo 1- 225'J +50a ‘ gj germ: FWL cream 0% egg) L. R -' 215‘ +1203 —q\y “W “‘W’Wc 5°15 ‘wm OwC out which I}qu v0.04.“ 0% 3 I Jam, ad“ a, MAX k“ Cost = A? Problem 15 : A firm’s production function is y = 3002: + 402:2 — 3w3 Where y denotes output and 3: denotes input. Price per unit of the output is $10 and price per unit of input is 2000. a. What is the revenue function? W ‘7; mt '“° 3 5 7c ) b. What is the cost function? no (fioox 1~ Wit" rm}... 0% X m. at M Keven» = (D h H H c. What is the profit function? : Pro = -. Cost; RW“ d. How much input should the firm employ if it wants to maximize its profit. I 7‘ : 0 Solve . W 9"- PWOL Cmbe V 6% 9/ 3“ x f . o”:- which Use MVW "to We at 4 ' Mimi-Mi 06 new, MOW VW7 0% z, “A m 2 t 2 I ‘1 ~‘l ' 9‘ 2 5 ‘ “Ll _ ‘5' b b . —H *5 5 :‘7‘ | l 2- 1 l ' 2 3 u w 5 v -v «LI ‘5' 3 ‘1 -q «5’ 5 I I Z 5 H l - ' v5 .1 -5 g: L to l l ’L 1 b q “f 4! -5 5 1:? 20 va-s Problem 17:Solve the following system of equations using Cramer’s rule. m+y+2z : 2 3x+4y—4z : 1 —4a:—5y+3z = —3 Problem 19: Solve the following system of equations: ml/3y2/3 _ 2321/23/1/3 — 80 1/ o =) 5 (1600) x — ’0 V3 " 19—6- _l. X . 7L ' -——’—' [60—9- “4 ~22 -2 .. =) x /3 _ 7% — ‘t -z"5/Z 3 22 x q - ‘I 1 flu W 7’ me A; || || Problem 20: Consider the following production function of a perfectly competitive firm y=0utput —30L 1 20K 2L2 : 2LK 3K2 where L stands for labor and K represents capital. The following information are also provided: Price per unit of output = 10 Price per unit of labor 2 20 Price per unit of capital = 40 a. How much labor (L) and capital (K) must the firm employ if it wants to maximize its profits. Use the Hessian matrix to verify that the solution you get actually maximizes profits. HUM there Ls No CONSTKAIMT IMUOLUGD. H—ENCE WE DO NOT USE THE LAQKAW 65pm TECH-NI x05 * sauce, we, want 11> Maxine: (its we £96517 mad. £0 ww'we. down flu, ’10 ‘t 01:30” W ; Ru)an _ QSt ,_ [OJ __ <2DL. +LIDK) 1 -~ 10 (50!. +2014 ~2L7‘+ 2.LK —3K ) C201. +1409 HG» stmrufvdnfl we 6‘45 7(— : Now W C5 0" ‘5‘ 2 Vardmbifls L) g 0 we hcwe '6; me. fowl“, otertvmfuu H a. 2 2.80 L + 160K, — am, 4 510ch ~30K ‘ 23 STE—P i 3 FWIOL 0‘41" W crdbécauL UM of L, K... 1“ L) XL -, 280 ‘L’DL- + ‘RDK’ ‘0 ) 2. 17K 160 1“ ML. —éoK — 0 U , m 2 mkwwh varoabLea L—z’K’ at m orb-em; Values u‘em Mano/:0“ £19-; :9 1940 — 20L + 101’» = D + C2) 160 +2“ “50" ' ° 300 —- 50K = 0 => K: e W +6 — 2L. 2L— 20 : L=Io \Jw-owvbbu W the CmELch quMes 0E m L :— 10, K16 // 0L t5 Von. 5mm 7-60 E; New we, N)- D r mm; mwmaed, ax L: .0, Kzg For M Md flu. HESSLRM marbx Wu. TLK ‘90 20 20. ‘éb TKL 7\‘KK We km“, m fouow—cmd ruin, /H41 > 0 / :0 <0 C2725 thOhCJMSLV‘Q Sadou" [’5 KLL> 0 MIN II b. What is the optimal level of output. 3 0 (to) + 20 C6) = 2.52 0. how much does the firm spend on inputs? C» 52431. -. 02,0 I0) +‘(oCQ ‘— LIL/D + 90K Problem 21: Consider the following production function of a perfectly competitive firm y = output 2 50L + 40K — 2L2 + 2LK — K2 where L stands for labor and K represents capital. The following information are also provided: Price per unit of output 2 3 Price per unit of labor : 120 Price per unit of capital : 90 a. How much labor (L) and capital (K) must the firm employ if it wants to maximize its profits. Use the Hessian matrix to verify that the solution you get actually maximizes profits. 25 - 2001+ z(lo)(6) “3697’ b. What is the optimal level of output. c. How much does the firm spend on inputs? Problem 23: Find the values of :01 and 322 that maximize the function 49— $§~ 26% subject to the constraint 3:1 + 3332 — 10 = 0 Here, there, L6 6» LOhs‘EPo‘Lnb. go we wUJ. use ia—é—f—L‘ 1 "A . . ‘ __ _,9(1 "' which fwd/Eon are, we, mutmema ——> Ll‘) x! 4- ('0 m COhSM Th2, rmaw f‘m’” ’ Z 2 : £101“ 7‘1 "8: " )Gt*5“a_'9 So 05 a, major» 0% 5 Vou—UCOJDW : 2LI; x2, ) 27 W 3* TM w~r-t 9‘1 mm 21 M rat 9(9‘ BH= *2. O 1 M2 [BHl > O M lel <0 M Hm, LBH|= 10 >0' go havoc A MAX Row-e a NlN. MMWK MM ...
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This note was uploaded on 08/01/2011 for the course ECON 207 taught by Professor Staff during the Fall '08 term at Iowa State.

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Lab 15(comprehensive)_partial keys - ECONOMICS 207 FALL...

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