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You may not need to use \ ", StyleBox["all ", FontSlant->"Italic"], "symbols included in the palette for this exam.\n* If you want to make a \ given interim graph visible, just delete the \"DisplayFunction\[Rule]Identity\ \" part of the command.\n* The maximum score is 40 points.\n* ", StyleBox["The exam has to be handed in ", FontWeight->"Bold"], StyleBox["in person", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[" on Friday, May 6 between 9:45 and 11:45 am in my office.", FontWeight->"Bold"] }], "Text", CellDingbat->None, Background->RGBColor[0.996078, 0.905882, 0.811765]], Cell[TextData[StyleBox["Symbolizing:", FontSize->16]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(<< Utilities`Notation`;\)\)], "Input"], Cell[BoxData[ RowBox[{"Symbolize", "[", TagBox[\(q\_1\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}]], "Input"], Cell[BoxData[ RowBox[{"Symbolize", "[", TagBox[\(q\_2\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}]], "Input"], 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bank. His income for the year is \ somewhat uncertain. If his band stays together, he will earn an additional \ $10000. If the band breaks up, he will earn only $5000. The probability \ that the band ", StyleBox["stays together", FontVariations->{"Underline"->True}], " is " }], "Text"], Cell[BoxData[ \(\(p = 0.8;\)\)], "Input"], Cell["\<\ Assume for now that Ronnie has no influence over the probability of a \ break-up.\ \>", "Text"], Cell[TextData[StyleBox["Q1 (1 point): What is Ronnie's certainty equivalent \ of wealth?", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]]], "Text"], Cell[TextData[StyleBox["Q2 (1 point): What is his expected wealth?", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]]], "Text"], Cell[TextData[StyleBox["Q3 (1 point): Is he risk averse? How can you tell?", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]]], "Text"], Cell[TextData[{ StyleBox["Q4 (2 points): An insurance company offers Ronnie a contract to \ cover the risk of his band breaking up. Assuming the contract is actuarilly \ fair, and that only full insurance is offered, what premium will the company \ charge? Will Ronnie purchase this insurance?", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" (Hint: Compare expected utility levels)", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[TextData[{ "Assume now that Ronnie can affect the probability of his band breaking up. \ Specifically, if he behaves grossly inappropriatly on stage (use your \ imagination), which he happens to enjoy, the probability of a ", StyleBox["break-up", FontVariations->{"Underline"->True}], " increases to 0.4. If he behaves, the original break-up probability holds.\ \n\n", StyleBox["Q5 (2 points): The insurance company assesses the probability \ that Ronnie misbehaves at q=0.5. What is the new premium the company would \ charge for full insurance?", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" (Hint: you may find it useful to draw a probability tree, \ similar to an extensive form game. There are now two layers of \ probabilities...)", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[TextData[{ StyleBox["Q6(2 points): If misbehaving adds the equivalent of $1000 to \ Ronnie's wealth, will he i) misbehave and buy the insurance, ii) misbehave \ and not buy the insurance, iii) behave and buy the insurance, or iv) behave \ and not buy the insurance?", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" (Assume the original contract from before is no longer \ available).", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[TextData[StyleBox["Part II: Game Theory - One-period game of complete \ information (5 points)", FontSize->16]], "Text"], Cell[TextData[StyleBox["Based on Gibbons ex. 1.13", FontSize->14, FontWeight->"Bold"]], "Text"], Cell[BoxData[ \(\(Clear[p, q];\)\)], "Input"], Cell[TextData[{ "Use the following numerical values: ", Cell[BoxData[ \(TraditionalForm\`w\_1 = 20, \ w\_2 = 30\)]], ". This yields the following game box:" }], "Text"], Cell[BoxData[ StyleBox[GridBox[{ {"\[Placeholder]", "F1", "F2"}, {"F1", \(10, 10\), RowBox[{ StyleBox["20"], StyleBox[",", Background->None], StyleBox["30"]}]}, {"F2", RowBox[{ StyleBox["30"], StyleBox[",", Background->None], StyleBox["20"]}], \(15, 15\)} }, GridFrame->True, RowLines->True, ColumnLines->True], FontSize->18]], "Input"], Cell[TextData[{ StyleBox["Q7 (1 point) Are there any pure strategy (PS) Nash Equilibria? \ Which, if any? Explain why they are NEs.\n", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["(circle them in the game box)", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[TextData[StyleBox["Q8 (3 points) Find the MS equilibrium. Solve for \ equilibrium probabilities and payoffs. ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]]], "Text"], Cell[BoxData[ \(u\_1[p_, q_] = your\ turn\)], "Input"], Cell[BoxData[ \(u\_2[p_, q_] = your\ turn\)], "Input"], Cell[TextData[{ StyleBox["Q9 (1 point) Compare the set of Nash Equilibria. Is there one \ that sticks out as the ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["most ", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox["or ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["least", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" likely equilibrium? Explain.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[TextData[{ StyleBox["Part III: IO-applications of dynamic games of complete \ information ", FontSize->16], StyleBox["- \n", FontSize->14, FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`A\ Bertrand\ Economy\ with\ 3\ firms\ \((6\ \ points)\)\)], FontSize->16] }], "Text"], Cell[TextData[{ "The Bertrand model with 2 firms is described in Gibbons, p. 21. You should \ read this to \"warm up\" for this problem. Consider a Bertrand economy with 3 \ firms, where quantity demanded from a given firm follows\n", Cell[BoxData[ \(TraditionalForm\`q\_i = a - p\_i + \(\(b\)\(*\)\)\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(j \[NotEqual] i\)p\_j\)]], ", with b=4/10, and identical and constant marginal cost c"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`p\_1, \ p\_2, \ p\_3, \)]], " ", StyleBox["and", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " ", Cell[BoxData[ \(TraditionalForm\`Pr\_1\)]], ", ", Cell[BoxData[ \(TraditionalForm\`Pr\_2\)]], ", ", Cell[BoxData[ \(TraditionalForm\`Pr\_3\)]], ", ", StyleBox["respectively", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[".", FontColor->RGBColor[0, 0, 1]], " ", StyleBox["Denote the equilibrium prices and profits as ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`\(p\_1\^*\), \ \(p\_2\^*\), \ \(p\_3\^*\)\)]], " ", StyleBox["and", FontColor->RGBColor[0, 0, 1]], " ", Cell[BoxData[ \(TraditionalForm\`\(Pr\_1\^*\)\)]], ",", Cell[BoxData[ \(TraditionalForm\`\(Pr\_2\^*\)\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`\(Pr\_3\^*\)\)]], ", ", StyleBox["respectively", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], "." }], "Text"], Cell[BoxData[ \(\(b = 4/10;\)\)], "Input"], Cell[BoxData[ \(q\_1[\(p\_1\) : _, \(p\_2\) : _, \(p\_3\) : _] = your\ turn\)], "Input"], Cell[BoxData[ \(q\_2[\(p\_1\) : _, \(p\_2\) : _, \(p\_3\) : _] = your\ turn\)], "Input"], Cell[TextData[{ StyleBox["Q11 (3points) Now solve this problem from a social planner's \ perspective, who aims at maximizing the sum of profits over all firms. How \ do the prices and profits compare to those in the Bertrand equilibrium? Why \ does the Bertrand economy fail to generate the social planner's results?\n", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["(Hint: denote the social planner prices as ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`p\_1\^s, \ p\_2\^s, \ p\_3\^s, \)]], " ", StyleBox["respectively", FontColor->RGBColor[0, 0, 1]], ". ", StyleBox["Recycle definitions of profits from above.", FontColor->RGBColor[0, 0, 1]], " ", StyleBox["If it's not immediately clear which prices are higher, try taking \ the difference, as in : ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`\(p\_1\^*\)\)], FontColor->RGBColor[0, 0, 1]], StyleBox["-", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`p\_1\^s\)], FontColor->RGBColor[0, 0, 1]], StyleBox[", and simplify.)", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[BoxData[ \(Clear[Pr]\)], "Input"], Cell[BoxData[ \(Pr[\(p\_1\) : _, \(p\_2\) : _, \(p\_3\) : _] = your\ turn\)], "Input"], Cell[TextData[StyleBox["Part IV: Dynamic games of incomplete information - \ Productive Screening (20 points)", FontSize->16]], "Text"], Cell[TextData[{ "Consider the following screening model: Workers can be either of high type \ ", Cell[BoxData[ \(TraditionalForm\`\(\((\[Theta]\_H)\)\(\ \)\)\)]], "or low type ", Cell[BoxData[ \(TraditionalForm\`\((\[Theta]\_L)\)\)]], ", where ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\_i\)]], ">0, (i=H,L). The constant-returns to scale productivity per unit of labor \ input is given by ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\_i\)]], " + ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\_i\)]], "* ", Cell[BoxData[ \(TraditionalForm\`t\_i\)]], ", where ", Cell[BoxData[ \(TraditionalForm\`t\_i\)]], " is the task level assigned by the competitive firm to type i (i=H,L). \ The cost of performing ", Cell[BoxData[ \(TraditionalForm\`t\_i\)]], " to type ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\_i\)]], " is given as (using generic t and \[Theta]):" }], "Text"], Cell[BoxData[ \(\(c[t_, \[Theta]_] = Exp[t\/\[Theta]] - 1;\)\)], "Input"], Cell[TextData[{ StyleBox["Q12 (2 points) Show that this cost function satisfies all basic \ assumptions made in class, i.e. c(0,\[Theta])=0, ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`c\_t\)], FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["(.)>0, ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ FormBox[\(c\_\[Theta]\), "TraditionalForm"], "(", ".", ")"}], "<", "0"}], ",", \(c\_tt\)}], TraditionalForm]], FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["(.)>0, ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`c\_t\[Theta]\)], FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["(.)<0.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(c[0, \[Theta]]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[c[t, \[Theta]], your\ turn]\)], "Input"], Cell[BoxData[ \(\(\(etc\)\(.\)\)\)], "Input"] }, Open ]], Cell[TextData[{ "Firms pay wage ", Cell[BoxData[ \(TraditionalForm\`w\_i\)]], " to type i. Firm profit per unit of output by using type i is given by " }], "Text"], Cell[BoxData[ \(\(Pr[w_, t_, \[Theta]_] = \[Theta] + \[Theta]*t - w;\)\)], "Input"], Cell["Productive Output per unit of labor is thus ", "Text"], Cell[BoxData[ \(\(P[t_, \[Theta]_] = \[Theta] + \[Theta]*t;\)\)], "Input"], Cell["And workers' utility is given by", "Text"], Cell[BoxData[ \(\(u[w_, t_, \[Theta]_] = w - c[t, \[Theta]];\)\)], "Input"], Cell[TextData[StyleBox["Q13 (1 point) What is thus the wage per unit of labor \ paid by the competitive firms in terms of t and \[Theta]?", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]]], "Text"], Cell[BoxData[ \(\(w[t_, \[Theta]_] = your\ turn;\)\)], "Input"], Cell[TextData[StyleBox["Case 1:", FontWeight->"Bold"]], "Text", FontSize->16], Cell["\<\ Assume the productivity levels and the probability of being a \"high\" type (\ \[Lambda]) are given as follows:\ \>", "Text"], Cell[BoxData[ \(Clear[\[Theta]\_L, \[Theta]\_H, \[Lambda]]\)], "Input"], Cell[BoxData[ \(\({\[Theta]\_L, \[Theta]\_H} = {2, 4};\)\)], "Input"], Cell[BoxData[ \(\(\[Lambda] = 1/3;\)\)], "Input"], Cell[TextData[{ StyleBox["Q14 (2 points) Assume first the ability levels are known to all (\ \"full information case\"). Find the equilibrium task levels and wages for \ both types. Verify that profits are indeed zero in equilibrium.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox["(Hint: You can keep a generic \[Theta] in your expressions. First \ solve for the optimal \"t\", then express this level as an explicit function \ of \[Theta]. Then express the optimal wage as an explicit function of both t \ and \[Theta], etc as shown below. Ignore any \"inverse function\" warnings)", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[BoxData[ \(\(Solve[ D[u[\[Theta] + \[Theta]*t, t, \[Theta]], t] \[Equal] your\ turn, your\ turn]\)[\([1]\)]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(t\^*\)[\[Theta]_] = t /. %\)], "Input"], Cell[BoxData[ \(\[Theta]\ Log[\[Theta]\^2]\)], "Output"], Cell[BoxData[ \(\(\(w\^*\)[\[Theta]_] = w[\(t\^*\)[\[Theta]], \[Theta]];\)\)], "Input"] }, Open ]], Cell[BoxData[ \(\(\(u\^*\)[\[Theta]_] = u[\(w\^*\)[\[Theta]], \(t\^*\)[\[Theta]], \[Theta]];\)\)], "Input"], Cell[BoxData[ \(N[{\(t\^*\)[\[Theta]\_L], \(t\^*\)[\[Theta]\_H], \(w\^*\)[\[Theta]\_L], \ \(w\^*\)[\[Theta]\_H]}]\)], "Input"], Cell[BoxData[ \(check\ for\ zero\ \(profits : \ your\ turn\)\)], "Input"], Cell[TextData[{ StyleBox["Q15 (1 point) These equilibrium contracts rest on a first order \ condition flowing from workers' utility-maximization problem as discussed in \ class. State this condition in words, and show algebraically (with generic \ \[Theta]) OR numerically (using actual ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`\[Theta]\_L\)], FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" and ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`\[Theta]\_H\)], FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" ) that it holds for each type.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell["Graphs:", "Subsubtitle"], Cell[TextData[{ "The following commands generate a graph of this equilibrium. They produce \ indifference curves for both types ", Cell[BoxData[ \(TraditionalForm\`\((IC\_L\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`IC\_H\)]], "), break-even lines ", Cell[BoxData[ \(TraditionalForm\`\((BE\_L\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`BE\_H\)]], ", ", Cell[BoxData[ \(TraditionalForm\`BE\_P\)]], " for \"pooled\"), and the equilibrium points (\"L\" and \"H\")." }], "Text"], Cell[TextData[{ StyleBox["Note:\n", FontVariations->{"Underline"->True}], "I encountered problems combining the following individual graphs in the \ \"show\" command below. For unexplained reasons, ", StyleBox["Mathematica", FontSlant->"Italic"], " produced an error message instead of the full graph. It worked though \ after a \"brute force\" approach of trying it a few times, i.e. re-executing \ all individual graph commands, followed by the \"show\" command.\nThe same \ holds for the other two graphing parts below." }], "Text", CellDingbat->None, Background->RGBColor[0.996078, 0.905882, 0.811765]], Cell[BoxData[ \(<< Graphics`Graphics`\)], "Input"], Cell[BoxData[ \(<< Graphics`ImplicitPlot`\)], "Input"], Cell[BoxData[ \(\(IC\_L = ContourPlot[u[w, t, \[Theta]\_L], {t, 0, 20}, {w, 0, 80}, ContourShading \[Rule] False, Contours \[Rule] {\(u\^*\)[\[Theta]\_L]}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[BoxData[ \(\(IC\_H = ContourPlot[u[w, t, \[Theta]\_H], {t, 0, 20}, {w, 0, 80}, ContourShading \[Rule] False, Contours \[Rule] {\(u\^*\)[\[Theta]\_H]}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[BoxData[ \(\(BE\_L = Plot[P[t, \[Theta]\_L], {t, 0, 20}, PlotStyle \[Rule] {RGBColor[0, 0, 1]}, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[BoxData[ \(\(BE\_H = Plot[P[t, \[Theta]\_H], {t, 0, 20}, PlotStyle \[Rule] {RGBColor[0, 1, 0]}, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[BoxData[ \(\(BE\_P = Plot[\[Lambda]*P[t, \[Theta]\_H] + \((1 - \[Lambda])\)* P[t, \[Theta]\_L], {t, 0, 20}, PlotStyle \[Rule] {RGBColor[1, 0, 0]}, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[BoxData[ \(\(L = LabeledListPlot[{{\(t\^*\)[\[Theta]\_L], \(w\^*\)[\[Theta]\_L], \ "\"}}, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[BoxData[ \(\(H = LabeledListPlot[{{\(t\^*\)[\[Theta]\_H], \(w\^*\)[\[Theta]\_H], \ "\"}}, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Show[Graphics[IC\_L], Graphics[IC\_H], Graphics[BE\_L], Graphics[BE\_H], Graphics[BE\_P], Graphics[L], Graphics[H], DisplayFunction \[Rule] $DisplayFunction, PlotRange \[Rule] {{0, 20}, {0, 80}}];\)\)], "Input"], Cell["If all went well, your graph should look like this:", "Text"], 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00<007ooOol01goo00<007ooOol01goo00<007ooOol01goo00<007ooOol01goo00<007ooOol01goo 00<007ooOol01goo00<007ooOol01goo00<007ooOol01goo00<007ooOol01goo00<007ooOol01goo 00<007ooOol01goo00<007ooOol01goo00<007ooOol01goo00D007ooOomoo`0000Eoo`03001oogoo 00Moo`03001oogoo00Moo`03001oogoo00Moo`03001oogoo00Moo`03001oogoo00Moo`03001oogoo 00Moo`03001oogoo00Moo`03001oogoo00Moo`03001oogoo00Moo`03001oogoo00Moo`03001oogoo 00Moo`03001oogoo00Moo`03001oogoo00Eoo`03001oo`0000Uoo`03001oogoo00Moo`03001oogoo 00Moo`03001oogoo00Moo`03001oogoo00Moo`03001oogoo00Moo`03001oogoo00Ioo`030n0007oo 00Qoo`03001oogoo00Ioo`03001oogoo00Qoo`0000=oo`00Ool00goo00D007ooOomoo`0000Aoo`03 001oogoo00Aoool007d000030n000000014000Yoo`000Woo100017oo00<007ooOol00Woo00<007oo Ool0ogooWgoo00000goo001oo`03Ool01@00Oomoogoo000017oo00<007ooOol0ogooWgoo00000goo 001oo`03Ool01@00Oomoogoo000017oo00<007ooOol0ogooWgoo00000goo001oo`03Ool01@00Oomo ogoo000017oo00<007ooOol0ogooWgoo0002Ool40005Ool4003oOonROol00?mook5oo`00ogoo/Goo 003oOonaOol00?mook5oo`00ogoo/Goo0000\ \>"], ImageRangeCache->{{{79.5625, 308.938}, {340.125, 110.75}} -> {-8.64902, \ 37.3382, 0.0753584, 0.301433}}] }, Open ]], Cell[TextData[StyleBox["Q16 (2 points) Elaborate on this graph (tangency \ conditions, relative magnitude of low vs. high wages & tasks, zero-profit \ conditions).", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]]], "Text"], Cell[TextData[StyleBox["Q17 (3 points) Can this outcome be sustained as \ separating equilibrium (SE) under asymmetric information? Why or why not? \ Show in the graph the entire set of high type contracts that can be sustained \ in a SE. Of this set, which contract is Pareto optimal from a welfare \ perspective?", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]]], "Text"], Cell[TextData[StyleBox["Case 2:", FontWeight->"Bold"]], "Text", FontSize->16], Cell["\<\ Contimue to assume we operate under full information. The productivity levels \ and cost function are now given as follows:\ \>", "Text"], Cell[BoxData[ \(\({\[Theta]\_L, \[Theta]\_H} = {5, 6};\)\)], "Input"], Cell[BoxData[ \(\(c[t_, \[Theta]_] = Exp[t\/\@\[Theta]] - 1;\)\)], "Input"], Cell[TextData[StyleBox["Q18 (2 points) Re-define the utility function (copy \ from above) and solve again for the optimal wage and task levels for both \ types. Proceed as above.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]]], "Text"], Cell[BoxData[ \(\(u[w_, t_, \[Theta]_] = w - c[t, \[Theta]];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Solve[ D[u[\[Theta] + \[Theta]*t, t, \[Theta]], t] \[Equal] your\ turn, your\ turn]\)[\([1]\)]\)], "Input"], Cell[BoxData[ \(\(\(etc\)\(.\)\)\)], "Input"] }, Open ]], Cell["Graphs:", "Subsubtitle"], Cell["Again, let's illustrate this equilibrium graphically:", "Text"], Cell[BoxData[ \(\(IC\_L = ContourPlot[u[w, t, \[Theta]\_L], {t, 0, 20}, {w, 0, 80}, ContourShading \[Rule] False, Contours \[Rule] {\(u\^*\)[\[Theta]\_L]}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[BoxData[ \(\(IC\_H = ContourPlot[u[w, t, \[Theta]\_H], {t, 0, 20}, {w, 0, 80}, ContourShading \[Rule] False, Contours \[Rule] {\(u\^*\)[\[Theta]\_H]}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[BoxData[ \(\(BE\_L = Plot[P[t, \[Theta]\_L], {t, 0, 20}, PlotStyle \[Rule] {RGBColor[0, 0, 1]}, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[BoxData[ \(\(BE\_H = Plot[P[t, \[Theta]\_H], {t, 0, 20}, PlotStyle \[Rule] {RGBColor[0, 1, 0]}, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[BoxData[ \(\(BE\_P = Plot[\[Lambda]*P[t, \[Theta]\_H] + \((1 - \[Lambda])\)* P[t, \[Theta]\_L], {t, 0, 20}, PlotStyle \[Rule] {RGBColor[1, 0, 0]}, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[BoxData[ \(\(L = LabeledListPlot[{{\(t\^*\)[\[Theta]\_L], \(w\^*\)[\[Theta]\_L], \ "\"}}, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[BoxData[ \(\(H = LabeledListPlot[{{\(t\^*\)[\[Theta]\_H], \(w\^*\)[\[Theta]\_H], \ "\"}}, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Show[Graphics[IC\_L], Graphics[IC\_H], Graphics[BE\_L], Graphics[BE\_H], Graphics[BE\_P], Graphics[L], Graphics[H], DisplayFunction \[Rule] $DisplayFunction, PlotRange \[Rule] {{0, 10}, {0, 80}}];\)\)], "Input"], 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