Compilation - CHIANG act - 11.6 Number 1 Given the change...

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11.6 Number1 Giventhechangeintheproductionfunctionintheformerexample, wesolveforoptimalvaluesofq1 &q2 Settingupourrevenue, cost, andprofitfunctions: In[98]:= Clear @ c D In[99]:= r = p10 * q1 + p20 * q2 c = 2 * q1 + 2 * q2^2 = r - c Out[99]= p10q1 + p20q2 Out[100]= 2q1 + 2q2 2 Out[101]= - 2q1 + p10q1 + p20q2 - 2q2 2 Solvingforthenewoptimlalevelsofq1 &q2, weget: Forq1: In[103]:= D @ , q1 D Out[103]= - 2 + p10 Forq2: D @ , q2 D p20 - 4q2 Solvingforthenewprofit: In[104]:= * = p10 * H - 2 + p10 L + p20 * H p20 - 4q2 L - 2 * H - 2 + p10 L + 2 * H p20 - 4q2 L ^2 Out[104]= - 2 H - 2 + p10 L + H - 2 + p10 L p10 + p20 H p20 - 4q2 L + 2 H p20 - 4q2 L 2 Thenewprofitisnegative, therefore, itseconomicmeaningisthatthefirmwillincurlossesiftheyadoptthenewcostfunction. Number2 Clear @ q1, q2, r, c, , p1, p2 D Giventhe2directdemandfunctions, wesolvefortheinversedemandequations.Bysolvingweget: Solve @ q1 40 - 2 * p1 - p2, p1 D :: p1 fi 1 2 H 40 - p2 - q1 L>> Solve @ q2 35 - p1 - p2, p2 D 88 p2 fi 35 - p1 - q2 << Wesetupthefirm'srevenuefunction, givenby:
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r = 1 2 H 40 - p2 - q1 L * q1 + H 35 - p1 - q2 L * q2 1 2 H 40 - p2 - q1 L q1 + H 35 - p1 - q2 L q2 Wealsodefinethecostfunction, givenby: c = q1^2 + 2 * q2^2 + 10 10 + q1 2 + 2q2 2 Solvingforprofitmaximization, wedefineprofitasTR - TC, definesasr - cinsolution: = r - c - 10 + 1 2 H 40 - p2 - q1 L q1 - q1 2 + H 35 - p1 - q2 L q2 - 2q2 2 Solvingforoptimalvaleuswhenprofit = 0 H FOC L wederivetheprofitfunctionwrttoeachvariable.Weget: Solve @8 D @ , q1 D 0, D @ , q2 D 0 < , 8 q1, q2 <D :: q1 fi 40 - p2 6 , q2 fi 35 - p1 6 >> Wenowhavetheoptimalvaluesforgoods1 &2 Tocheckwhetherthevaluessatisfythesufficientconditionsforamaximum, wesolveforSOC.
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