(*^ ::[paletteColors = 128; automaticGrouping; magnification = 125; currentKernel; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8, 24, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6, 18, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6, 14, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20, 18, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15, 14, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12, 12, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = input, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1, 12, "Courier"; ; fontset = output, output, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = message, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = print, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = info, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = Left Header, nohscroll, cellOutline, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, L1, 12; fontset = Left Footer, cellOutline, blackBox, 12; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Courier"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; next21StandardFontEncoding; ] :[font = title; inactive; preserveAspect; startGroup; ] Profit Function :[font = subsubtitle; inactive; preserveAspect; ] Hal R. Varian June, 1992 :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] Profit function :[font = text; inactive; preserveAspect; ] Recall the profit function for the Cobb-Douglas technology that we derived in the last chapter. We will repeat the derivation here without comment, since we already saw it done in the previous chapter. :[font = input; preserveAspect; ] f[x1_,x2_] := (x1^(1/4))*(x2^(1/4)) Profit[x1_,x2_,p_,w1_,w2_] := p*f[x1,x2] - w1*x1 - w2*x2 solution=Solve[{D[Profit[x1,x2,p,w1,w2],x1]==0, D[Profit[x1,x2,p,w1,w2],x2]==0}, {x1,x2}][[1]] pi[p_,w1_,w2_] := Evaluate[Profit[x1,x2,p,w1,w2]/.solution] pi[p,w1,w2] :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] Convexity of the profit function :[font = text; inactive; preserveAspect; ] Let's graph this function to make sure that it is really convex. First we'll fix w1 and w2 and look at how the function behaves with respect to p. ;[s] 7:0,0;82,1;84,2;89,3;91,4;144,5;146,6;149,-1; 7:1,11,8,Times,0,12,0,0,0;1,10,8,Courier,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,1,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; ] Plot[pi[p,1,1],{p,1,4}] :[font = text; inactive; preserveAspect; ] Now let's look at a 3D-plot and a contour plot, fixing p at 1. ;[s] 3:0,0;55,1;56,2;63,-1; 3:1,11,8,Times,0,12,0,0,0;1,10,8,Courier,1,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; ] Plot3D[pi[1,w1,w2],{w1,1,50},{w2,1,50}] :[font = input; preserveAspect; endGroup; ] ContourPlot[pi[1,w1,w2],{w1,1,50},{w2,1,50},ContourShading->False] :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] Hotelling's identity :[font = text; inactive; preserveAspect; ] Finally, we want to verify that we satisfy Hotelling's identity, so we'll compute the derivatives of the profit function with respect to w1 and compare it to the factor demand functions. ;[s] 3:0,0;137,1;139,2;189,-1; 3:1,11,8,Times,0,12,0,0,0;1,10,8,Courier,1,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; ] deriv[p_,w1_,w2_]:=-Evaluate[Simplify[D[pi[p,w1,w2],w1]]] deriv[p,w1,w2] factorDemand1[p_,w1_,w2_] := Evaluate[Simplify[x1/.solution]] factorDemand1[p,w1,w2] :[font = text; inactive; preserveAspect; ] It is not obvious that these expressions are equal to each other. But if we apply the right simplification, we see that Hotelling's identity works. :[font = input; preserveAspect; endGroup; endGroup; endGroup; ] PowerExpand[deriv[p,w1,w2]] ^*)