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It's the starting point not only for growth theory but also \ for modern business cycle theory. Ask an economist how the economy will \ react to an increase in government purchases or to a change in the tax rate \ on capital and the first model he will reach for in a search for answers is \ the Ramsey model. In this paper I briefly explain the Ramsey model and put \ the model through its paces using ", StyleBox["Mathematica'", FontSlant->"Italic"], "s extensive numerical and graphical abilities. \n\tThe model requires \ that we solve two paired differential equations with a boundary condition. \ The solution methods used, the shooting and time elimination method, are of \ wide applicability and may be of interest to researchers in fields other than \ economics. \n\tSince the Ramsey model is well known I will focus on how ", StyleBox["Mathematica", FontSlant->"Italic"], " can be used to solve the model. Introductions to the Ramsey model can be \ found in the texts of Romer (1996), Barro and Sala-i-Martin (1995), and \ Blanchard and Fischer (1989) and of course the original articles by Ramsey \ (1928), Cass (1965), and Koopmans (1965). My notation will follow that of \ Romer's most closely." }], "Text"], Cell[TextData[ "\tWe assume that there are a large number of identical firms. Each firm \ uses capital, K, and labor, L to produce output, Y, according to the \ production function Y=F[K,A*L]. The parameter A measures the state of \ \"technology.\" Technology allows L workers to produce as if there were \ actually A*L workers. We will assume that A grows exogeneously at the rate \ g. \n\tProfit maximizing firms hire capital and labor in competitive markets \ which they use to produce output. The price of output is normalized to 1 \ which implies that a firm's profit function can be written as below where w \ is the wage rate of effective labor, r is the interest rate and \[Delta] the \ depreciation rate on capital. To be competitive with other assets capital \ must pay its owners a return of r+\[Delta] which is therefore the rental rate \ or interest rate on capital."], "Text"], Cell[BoxData[ \(Off[General::spell]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(profit = F[K, AL] - \ w\ AL\ - \ \((r + \[Delta])\)\ K\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(-AL\)\ w - K\ \((r + \[Delta])\) + F(K, AL)\)], "Output"] }, Open ]], Cell["\<\ \tTo maximize profit the firm hires capital and labor until the first \ derivatives of the profit function with respect to capital and labor \ respectively are zero.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(D[profit, K] == 0\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(-r\), "-", "\[Delta]", "+", RowBox[{ SuperscriptBox["F", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "(", \(K, AL\), ")"}]}], "==", "0"}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[profit, AL] == 0\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["F", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "(", \(K, AL\), ")"}], "-", "w"}], "==", "0"}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "\tWe will also assume that the production function F is linearly \ homogeneous or in economic terms that it exhibits constant returns to scale \ (CRS).The CRS assumption means that if inputs increase by a factor of c, \ output will also increase by a factor of c, ", Cell[BoxData[ \(TraditionalForm\`F[cK, cAL] = \ cF[K, AL]\)]], ". Euler's theorem says that if F[K,AL] is linearly homogeneous then ", Cell[BoxData[ \(TraditionalForm\`Y = \(F\_K\) K + \(F\_AL\) AL\)]], ". Using the above first order conditions, Euler's theorem implies that \ factor payments exhaust the product, ", Cell[BoxData[ \(TraditionalForm\`Y = \((r + \[Delta]\ )\) K + w\ AL\)]], ". Setting ", Cell[BoxData[ \(TraditionalForm\`c = 1/AL\)]], " allows us to write the production function in terms of one variable, \ capital per unit of effective labor ", Cell[BoxData[ \(TraditionalForm\`k = K/AL\)]], ". Thus, output per unit of effective labor can be written ", Cell[BoxData[ \(TraditionalForm\`Y/AL = \(y = f[k]\)\)]], ". Rewriting the first order condition for capital we have\n1) ", Cell[BoxData[ \(TraditionalForm\`r + \[Delta] = \(f'\)[k]\)]], "\nand using (1) and Euler's theorem\n2) ", Cell[BoxData[ \(TraditionalForm\`w = f[k] - \(kf'\)[k]\)]], "." }], "Text"], Cell["\<\ \tHousehold behavior is more complicated. There are a large number, H, of \ identical households. Each member of the household supplies 1 unit of labor \ at every point in time. Households own the capital stock which they rent to \ firms. Each household begins with captial holdings of K[0]/H where K[t] is \ the amount of capital at time t. At each moment in time the household \ chooses how to divide its income - from wages and the renting of capital - \ between consumption and savings in order to maximize lifetime utility. The \ household's lifetime utility is written;\ \>", "Text"], Cell[TextData[{ "\nU=", Cell[BoxData[ \(TraditionalForm \`\[Integral]\+0\%\[Infinity]\ \(e\^\(-\[Rho]t\)\) \(u(C[t])\) L[t]/H\ dt\)]], " " }], "Text", TextAlignment->Center], Cell[TextData[{ "\tWhere C[t] is the consumption of each household member at time t. \ u[C[t]] is the instantaneous utility created by the consumption of C[t] at \ time t. L[t] is the total population of the economy and L[t]/H the number of \ members of the household at time t. ", Cell[BoxData[ \(TraditionalForm\`\[Rho] > 0\)]], " is the household's discount rate (assumed the same for all households). \ The higher is \[Rho] the more the household discounts future consumption \ relative to current consumption. It is common to assume that instantaneous \ utility is of the form ", Cell[BoxData[ \(TraditionalForm\`u(C[t]) = C[t]\^\(1 - \[Theta]\)\/\(1 - \[Theta]\), \ with\ \[Theta] > 0\ and\ \[Rho] - n - \((1 - \[Theta])\) g > 0\)]], ". \[Theta] governs how willing individuals are to substitute consumption \ from one time period into another. A lower \[Theta] implies a greater \ willingness to substitute consumption tomorrow for consumption today. The \ condition ", Cell[BoxData[ \(TraditionalForm\`\[Rho] - n - \((1 - \[Theta])\) g > 0\)]], " is a technical condition needed to ensure that utility is bounded below \ infinity. It's convenient to write the budget constraint in terms of \ consumption per unit of effective labor. After some algebraic manipulations \ and taking into account the fact that L and A are growing we have (see Romer \ (1996) for the derivation)." }], "Text", TextAlignment->Left], Cell[TextData[{ "\n", "U=", Cell[BoxData[ \(TraditionalForm \`B \(\[Integral]\+0\%\[Infinity]\( e\^\(-\[Beta]t\)\) c[t]\^\(1 - \[Theta]\)\/\(1 - \[Theta]\)\ dt\)\)]], " " }], "Text", TextAlignment->Center], Cell[BoxData[ \(TextForm \`where\ B = \(\(A[0]\^\(1 - \[Theta]\)\) L[0]/H\ and\ \[Beta] = \[Rho] - n - \((1 - \[Theta])\) \(g . \)\)\)], "Text"], Cell["\<\ \tB is an unimportant constant and will be normalized to 1 in what follows.\ \>", "Text"], Cell["\<\ \tThe household faces two constraints when maximizing utility. \ \>", "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\(\(1)\)\ \ \(k'\)[t] = \)\)]], Cell[BoxData[ \(TraditionalForm\`r[t]\ k[t]\)]], " + ", Cell[BoxData[ \(TraditionalForm\`e\^\(\((n + g)\) t\)\)]], "(w[t]-c[t])\n\n2) ", Cell[BoxData[ \(TraditionalForm\`Lim\+\(t\[LongRightArrow]\[Infinity]\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(e\^\(-R[t]\)\) \(e\^\(\((n + g)\) t\)\) k[t]\)]], "\[GreaterEqual]0\n\nWhere R[t]=", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\%t\)]], "r[\[Tau]]d\[Tau]" }], "Text"], Cell["\<\ \tThe first constraint is closer to an accounting identity than a constraint \ it says that the growth in the household's capital stock (per unit of \ effective labor) is equal to the interest earnings on the current capital \ stock plus the difference between the household's wages in period t and its \ consumption in period t (the exponential term adjusts for growth in effective \ labor). Notice that 1) does not forbid the household from consuming more \ than its wages by borrowing (creating a negative capital stock). It follows \ that the optimal solution to the household's problem is to borrow an infinite \ amount and live it up! Since the latter strategy is unrealistic we need \ constraint 2 which says that the household's capital stock (adjusted per unit \ of effective labor) cannot be negative in the limit. In other words, the \ household must live within its means, if not on any given day then in the \ limit.\ \>", "Text"], Cell["\<\ \tThe household's problem has been set up as a standard problem in optimal \ control theory which can be solved using the method of Hamiltonians (see the \ references cited above or Kamien and Schwartz (1981) for an introduction to \ optimal control theory). Write the Hamiltonian as:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"H", "=", RowBox[{ RowBox[{\(E\^\(\(-\[Beta]\)\ t\)\), FormBox[\(c[t]\^\(1 - \[Theta]\)\/\(1 - \[Theta]\)\), "TraditionalForm"]}], "+", RowBox[{\(\[Lambda][t]\), RowBox[{"(", RowBox[{\(r[t]\ k[t]\), "+", " ", RowBox[{ FormBox[\(E\^\(\((n + g)\) t\)\), "TraditionalForm"], \((w[t] - c[t])\)}]}], ")"}]}]}]}]], "Input"], Cell[BoxData[ \(TraditionalForm \`\(\[ExponentialE]\^\(\(-t\)\ \[Beta]\)\ \(c(t)\)\^\(1 - \[Theta]\)\)\/\(1 - \[Theta]\) + \((\(k(t)\)\ \(r(t)\) + \[ExponentialE]\^\(\((g + n)\)\ t\)\ \((w(t) - c(t))\))\)\ \(\[Lambda](t)\)\)], "Output"], Cell[TextData[{ "\tOptimal control theory tells us that the solution to our maximization \ problem must satisfy ", Cell[BoxData[ \(TraditionalForm\`H\_c = 0, \ H\_k = \(-\(\[PartialD]\(\[Lambda](t)\)\/\[PartialD]t\)\)\)]], " and the transversality condition ", Cell[BoxData[ \(TraditionalForm\`Lim\_\(t \[Rule] \[Infinity]\)\)]], "\[Lambda](t)k(t)\[Rule]0. (It can be shown that the transversality \ condition implies that the no infinite debt constraint of the household's \ problem will be satisfied in equilibrium, see Barro and Salai-i-Martin, \ 1995)." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(H\_c = D[H, c[t]] == 0\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\[ExponentialE]\^\(\(-t\)\ \[Beta]\)\ \(c(t)\)\^\(-\[Theta]\) - \[ExponentialE]\^\(\((g + n)\)\ t\)\ \(\[Lambda](t)\) == 0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(H\_k = D[H, k[t]] == \(-D[\[Lambda][t], t]\)\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{\(\(r(t)\)\ \(\[Lambda](t)\)\), "==", RowBox[{"-", RowBox[{ SuperscriptBox["\[Lambda]", "\[Prime]", MultilineFunction->None], "(", "t", ")"}]}]}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "\tWe now rearrange the first-order conditions to find a particularly \ convenient representation. First solve ", Cell[BoxData[ \(TraditionalForm\`H\_c\)]], " for \[Lambda](t) then differentiate with respect to t." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(sol1 = Solve[H\_c, \[Lambda][t]]\ // Simplify\)], "Input"], Cell[BoxData[ \(TraditionalForm \`{{\[Lambda](t) \[Rule] \[ExponentialE]\^\(\(-t\)\ \((g + n + \[Beta])\)\)\ \(c(t)\)\^\(-\[Theta]\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(sol2 = D[sol1, t]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["\[Lambda]", "\[Prime]", MultilineFunction->None], "(", "t", ")"}], "\[Rule]", RowBox[{ \(\[ExponentialE]\^\(\(-t\)\ \((g + n + \[Beta])\)\)\ \((\(-g\) - n - \[Beta])\)\ \(c(t)\)\^\(-\[Theta]\)\), "-", RowBox[{ \(\[ExponentialE]\^\(\(-t\)\ \((g + n + \[Beta])\)\)\), " ", "\[Theta]", " ", \(\(c(t)\)\^\(\(-\[Theta]\) - 1\)\), " ", RowBox[{ SuperscriptBox["c", "\[Prime]", MultilineFunction->None], "(", "t", ")"}]}]}]}], "}"}], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell["Now divide sol1 by sol2 and simplify.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(tmp = sol2[\([1, 1, 1]\)]/sol1[\([1, 1, 1]\)] == sol2[\([1, 1, 2]\)]/sol1[\([1, 1, 2]\)]\ // Simplify\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"g", "+", "n", "+", "\[Beta]", "+", FractionBox[ RowBox[{"\[Theta]", " ", RowBox[{ SuperscriptBox["c", "\[Prime]", MultilineFunction->None], "(", "t", ")"}]}], \(c(t)\)], "+", FractionBox[ RowBox[{ SuperscriptBox["\[Lambda]", "\[Prime]", MultilineFunction->None], "(", "t", ")"}], \(\[Lambda](t)\)]}], "==", "0"}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "\tFrom the first order condition for ", Cell[BoxData[ \(TraditionalForm\`\(H\_k\ \)\)]], "we can make the substitution ", Cell[BoxData[ RowBox[{ FormBox[ FractionBox[ RowBox[{ SuperscriptBox["\[Lambda]", "\[Prime]", MultilineFunction->None], "(", "t", ")"}], \(\[Lambda](t)\)], "TraditionalForm"], "->", \(-r[t]\)}]]], "and from the definition of \[Beta] we can make the substitution \[Beta]->\ \[Rho]-n-(1-\[Theta])g", "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"tmp", "=", RowBox[{"tmp", "/.", RowBox[{ FormBox[ FractionBox[ RowBox[{ SuperscriptBox["\[Lambda]", "\[Prime]", MultilineFunction->None], "(", "t", ")"}], \(\[Lambda](t)\)], "TraditionalForm"], "->", \(-r[t]\)}]}]}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"g", "+", "n", "+", "\[Beta]", "-", \(r(t)\), "+", FractionBox[ RowBox[{"\[Theta]", " ", RowBox[{ SuperscriptBox["c", "\[Prime]", MultilineFunction->None], "(", "t", ")"}]}], \(c(t)\)]}], "==", "0"}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[tmp, \(c'\)[t]]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["c", "\[Prime]", MultilineFunction->None], "(", "t", ")"}], "\[Rule]", \(-\(\(\(c(t)\)\ \((g + n + \[Beta] - r(t))\)\)\/\[Theta]\)\)}], "}"}], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(% /. {\[Beta] -> \[Rho] - n - \((1 - \[Theta])\) g}\ // Simplify\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["c", "\[Prime]", MultilineFunction->None], "(", "t", ")"}], "\[Rule]", \(-\(\(\(c(t)\)\ \((g\ \[Theta] + \[Rho] - r(t))\)\)\/\[Theta]\)\)}], "}"}], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell["\tRearranging we have:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"tmp", "=", RowBox[{\(\(c'\)[t]/c[t]\), "==", FractionBox[ FormBox[\((\ r(t) - \[Rho] - g\ \[Theta])\), "TraditionalForm"], FormBox[\(\[Theta]\ \), "TraditionalForm"]]}]}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["c", "\[Prime]", MultilineFunction->None], "(", "t", ")"}], \(c(t)\)], "==", \(\(\(-g\)\ \[Theta] - \[Rho] + r(t)\)\/\[Theta]\)}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[ "Using the fact that in equilibrium, r[t]==f'[k[t]]-\[Delta] we have:"], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(tmp = tmp /. r[t] -> \(f'\)[k[t]] - \[Delta]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["c", "\[Prime]", MultilineFunction->None], "(", "t", ")"}], \(c(t)\)], "==", FractionBox[ RowBox[{\(-\[Delta]\), "-", \(g\ \[Theta]\), "-", "\[Rho]", "+", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "(", \(k(t)\), ")"}]}], "\[Theta]"]}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ \tThe dynamics of k are determined from the accounting identity that changes \ in the capital stock equal production minus consumption minus depreciation \ (with suitable adjustments for changes in the stock of effective labor).\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(k'\)[t] == f[k[t]] - c[t] - \((n + g + \[Delta])\) k[t]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["k", "\[Prime]", MultilineFunction->None], "(", "t", ")"}], "==", \(\(-\(c(t)\)\) + f(k(t)) - \((g + n + \[Delta])\)\ \(k(t)\)\)}], TraditionalForm]], "Output"] }, Open ]], Cell[BoxData[{ FormBox[ RowBox[{ RowBox[{ RowBox[{ "The", " ", "two", " ", "key", " ", "equations", " ", "of", " ", "the", " ", "Ramsey", " ", "model", " ", "are", " ", RowBox[{"thus", ":", "\n", "\t\t", "\n", "\t ", RowBox[{"1.", " ", RowBox[{ SuperscriptBox["c", "\[Prime]", MultilineFunction->None], "(", "t", ")"}]}]}]}], "=", RowBox[{\(c(t)\), RowBox[{"(", FractionBox[ RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "(", \(k(t)\), ")"}], "-", "\[Delta]", "-", "\[Rho]", "-", \(g\ \[Phi]\)}], "\[Phi]"], ")"}], " ", "and"}]}], " "}], TraditionalForm], FormBox[ RowBox[{"\t\t", RowBox[{ RowBox[{"2.", " ", RowBox[{ SuperscriptBox["k", "\[Prime]", MultilineFunction->None], "(", "t", ")"}]}], "==", \(f(k(t)) - c(t) - \((g + n + \[Delta])\)\ \(k(t)\)\)}]}], TraditionalForm]}], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Numerically Solving the Model", "Section"], Cell[TextData[{ "\tWe now turn to numerically solving the model for steady states and \ transition paths. To do so we specify that the produciton function is of the \ Cobb-Douglas form, ", Cell[BoxData[ \(TraditionalForm\`f(k(t)) = \(k(t)\)\^\[Alpha]\)]], ". Now define CPrime and KPrime as follows:" }], "Text"], Cell[BoxData[ \(CPrime[\[Alpha]_, \[Delta]_, g_, \[Rho]_, \[Theta]_] := c[t]\ \((\[Alpha]\ k[t]^\((\[Alpha] - 1)\) - \[Delta] - \[Rho] - g\ \[Theta])\)/\[Theta]\)], "Input"], Cell[BoxData[ \(KPrime[\[Alpha]_, \[Delta]_, g_, n_] := k[t]^\[Alpha]\ - c[t] - \((g + n + \[Delta])\)\ k[t]\)], "Input"], Cell[TextData[ "\tWe initially set \[Alpha]=1/3, \[Delta]=.05, g=.02, n=0.01, \[Rho]=.02, \ and \[Theta]=1.75. We now solve for the steady state, the levels of c and k \ such that c'[t]=k'[t]=0."], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(kbar1 = k[t] /. Flatten\ @\ Solve[CPrime[1/3, 0.05, .02, .02, 1.75] == 0, k[t]]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`5.65632258015712974`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(cfunct = c[t] /. 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.13222 .29645 m .11942 .29354 L .10962 .29001 L .10074 .28571 L .10983 .28658 L .11984 .28669 L .13285 .28617 L F .13254 .29131 m .13254 .29131 .00515 0 360 arc F .71674 .12361 m .70517 .12991 L .69592 .13473 L .68729 .13929 L .69391 .13194 L .70116 .12433 L .71072 .11525 L F .71373 .11943 m .71373 .11943 .00515 0 360 arc F .70797 .13148 m .69658 .13815 L .68748 .14328 L .67902 .14814 L .6854 .14056 L .6924 .1327 L .70171 .1233 L F .70484 .12739 m .70484 .12739 .00515 0 360 arc F .85082 .20576 m .85394 .21895 L .85678 .22981 L .86004 .24031 L .85355 .23095 L .84747 .22124 L .84111 .2092 L F .84597 .20748 m .84597 .20748 .00515 0 360 arc F .70384 .23927 m .70115 .25245 L .69925 .26315 L .69794 .27343 L .69583 .26283 L .69431 .25181 L .69358 .2383 L F .69871 .23879 m .69871 .23879 .00515 0 360 arc F .7921 .04751 m .77884 .04985 L .76807 .05145 L .7578 .05292 L .76705 .04817 L .7768 .04329 L .78904 .03767 L F .79057 .04259 m .79057 .04259 .00515 0 360 arc F .64071 .21494 m .63269 .22569 L .62641 .23424 L .6208 .24242 L .62344 .23251 L .62676 .22223 L .63181 .20975 L F .63626 .21235 m .63626 .21235 .00515 0 360 arc F .274 .47896 m .27484 .49254 L .27578 .50359 L .27717 .51416 L .27239 .50417 L .26807 .49369 L .26385 .48068 L F .26893 .47982 m .26893 .47982 .00515 0 360 arc F .873 .20288 m .87723 .21601 L .88099 .22689 L .88515 .23744 L .87786 .22829 L .87095 .2188 L .86358 .20706 L F .86829 .20497 m .86829 .20497 .00515 0 360 arc F .28645 .34082 m .27606 .34959 L .26793 .35626 L .26064 .36234 L .26542 .35391 L .27104 .34491 L .27892 .33379 L F .28268 .3373 m .28268 .3373 .00515 0 360 arc F .32854 .39118 m .32298 .40372 L .31879 .4137 L .31532 .42312 L .31552 .41265 L .31644 .40162 L .31873 .38803 L F .32363 .38961 m .32363 .38961 .00515 0 360 arc F .80399 .23285 m .80651 .24671 L .80892 .25821 L .8118 .26935 L .80562 .25916 L .79992 .24862 L .7941 .23572 L F .79905 .23429 m .79905 .23429 .00515 0 360 arc F .0877 .27993 m .07697 .27064 L .06876 .26165 L .06128 .25183 L .07056 .25872 L .08058 .2648 L .09312 .27116 L F .09041 .27554 m .09041 .27554 .00515 0 360 arc F .75052 .25031 m .75141 .26442 L .75249 .2761 L .7541 .28741 L .7491 .27666 L .74464 .26555 L .74036 .25199 L F .74544 .25115 m .74544 .25115 .00515 0 360 arc F .58135 .24185 m .57291 .25297 L .56625 .26191 L .5603 .27046 L .5633 .26014 L .56702 .24943 L .57252 .23654 L F .57694 .2392 m .57694 .2392 .00515 0 360 arc F .66119 .26872 m .65879 .2826 L .65716 .29399 L .65613 .30497 L .65373 .29375 L .65194 .28212 L .65092 .268 L F .65605 .26836 m .65605 .26836 .00515 0 360 arc F .21856 .56742 m .22659 .57942 L .23325 .58985 L .23988 .6003 L .23064 .59209 L .22139 .5839 L .21075 .57414 L F .21465 .57078 m .21465 .57078 .00515 0 360 arc F .62209 .1735 m .6103 .18133 L .60078 .18753 L .59193 .19344 L .59857 .18491 L .60587 .17609 L .61545 .16562 L F .61877 .16956 m .61877 .16956 .00515 0 360 arc F .55189 .27736 m .54504 .28981 L .53972 .29992 L .5351 .30962 L .53655 .29858 L .53871 .28714 L .5424 .27336 L F .54714 .27536 m .54714 .27536 .00515 0 360 arc F .37018 .36154 m .36342 .37399 L .35821 .384 L .35376 .39348 L .35504 .38268 L .35708 .37136 L .36067 .35759 L F .36542 .35956 m .36542 .35956 .00515 0 360 arc F .29742 .32358 m .2861 .33198 L .27715 .3384 L .26904 .34428 L .2748 .3359 L .2814 .32697 L .29037 .31607 L F .2939 .31982 m .2939 .31982 .00515 0 360 arc F .60217 .29614 m .59917 .31033 L .59705 .32204 L .59557 .33334 L .59363 .32166 L .59234 .30957 L .59194 .295 L F .59705 .29557 m .59705 .29557 .00515 0 360 arc F .38194 .38151 m .37716 .39511 L .37359 .40618 L .37073 .41674 L .37025 .40537 L .37049 .39349 L .37193 .37908 L F .37693 .38029 m .37693 .38029 .00515 0 360 arc F .35973 .32449 m .34988 .33491 L .34209 .34317 L .3351 .35094 L .33935 .3411 L .34441 .33076 L .35152 .31826 L F .35563 .32138 m .35563 .32138 .00515 0 360 arc F .53117 .30609 m .52579 .31955 L .5217 .33057 L .51831 .34116 L .5184 .32962 L .5192 .31765 L .52127 .30324 L F .52622 .30467 m .52622 .30467 .00515 0 360 arc F .4455 .32453 m .43829 .33711 L .43267 .34733 L .42781 .35709 L .42954 .34593 L .43201 .33432 L .43609 .32034 L F .44079 .32243 m .44079 .32243 .00515 0 360 arc F .53058 .23999 m .52007 .24998 L .51164 .25801 L .50394 .26568 L .50903 .25579 L .51484 .24554 L .52273 .23333 L F .52666 .23666 m .52666 .23666 .00515 0 360 arc F .47797 .28809 m .46911 .29957 L .46209 .30886 L .45582 .31772 L .45917 .30705 L .46326 .29596 L .46921 .28267 L F .47359 .28538 m .47359 .28538 .00515 0 360 arc F .26706 .30343 m .25405 .30984 L .24367 .31456 L .23414 .31874 L .2418 .31168 L .25032 .30408 L .26145 .29479 L F .26426 .29911 m .26426 .29911 .00515 0 360 arc F .44645 .28618 m .43639 .29686 L .42837 .30547 L .42111 .31365 L .42564 .30339 L .43093 .29271 L .43825 .27994 L F .44235 .28306 m .44235 .28306 .00515 0 360 arc F .73536 .27551 m .73746 .29033 L .7396 .30278 L .74228 .31488 L .73626 .30358 L .73078 .29193 L .72534 .27791 L F .73035 .27671 m .73035 .27671 .00515 0 360 arc F .44372 .35963 m .43927 .37372 L .43596 .38531 L .43335 .39644 L .4326 .3846 L .43255 .3723 L .43364 .35749 L F .43868 .35856 m .43868 .35856 .00515 0 360 arc F .61469 .30815 m .61328 .32297 L .61251 .33532 L .61236 .34728 L .60907 .33532 L .60642 .32297 L .60439 .30815 L F .60954 .30815 m .60954 .30815 .00515 0 360 arc F .41228 .38932 m .40914 .40391 L .40693 .41595 L .4054 .42752 L .40352 .41554 L .40232 .4031 L .40205 .38812 L F .40716 .38872 m .40716 .38872 .00515 0 360 arc F .38836 .41418 m .38646 .4291 L .38529 .44147 L .38477 .45337 L .38186 .44135 L .3796 .42887 L .37806 .41384 L F .38321 .41401 m .38321 .41401 .00515 0 360 arc F .51997 .34219 m .5171 .35696 L .5151 .36924 L .51377 .3811 L .51168 .3689 L .51026 .35628 L .50972 .34117 L F .51485 .34168 m .51485 .34168 .00515 0 360 arc F .34211 .45949 m .34259 .47464 L .34337 .48728 L .34472 .49948 L .33996 .4877 L .33577 .47549 L .33189 .46077 L F .337 .46013 m .337 .46013 .00515 0 360 arc F .25653 .2907 m .24271 .29621 L .23159 .30018 L .22134 .30362 L .22995 .29716 L .23943 .29017 L .25162 .28164 L F .25408 .28617 m .25408 .28617 .00515 0 360 arc F .81953 .25444 m .82455 .26894 L .82915 .28124 L .83423 .29326 L .82603 .28267 L .81829 .27178 L .81016 .2587 L F .81485 .25657 m .81485 .25657 .00515 0 360 arc F .54134 .20023 m .52857 .20821 L .51814 .21459 L .5084 .22068 L .51602 .21189 L .52432 .20281 L .53497 .19213 L F .53816 .19618 m .53816 .19618 .00515 0 360 arc F .88852 .22837 m .89542 .24216 L .90161 .25391 L .9082 .2654 L .8987 .25574 L .8896 .24581 L .87979 .23384 L F .88416 .23111 m .88416 .23111 .00515 0 360 arc F .62177 .13535 m .60769 .14094 L .59607 .14535 L .58505 .14955 L .59446 .14232 L .60447 .13488 L .61694 .12625 L F .61936 .1308 m .61936 .1308 .00515 0 360 arc F .1706 .26733 m .15557 .26781 L .14353 .26727 L .13243 .26607 L .14298 .26388 L .15445 .26103 L .16893 .25717 L F .16976 .26225 m .16976 .26225 .00515 0 360 arc F .58824 .33029 m .5874 .34558 L .58711 .3584 L .58746 .37085 L .58368 .35853 L .58054 .34583 L .57795 .33066 L F .58309 .33048 m .58309 .33048 .00515 0 360 arc F .69574 .07106 m .68056 .07393 L .66788 .07606 L .65568 .07808 L .66687 .07278 L .67855 .06736 L .69272 .06121 L F .69423 .06613 m .69423 .06613 .00515 0 360 arc F .40558 .42609 m .40518 .44172 L .40527 .45483 L .406 .46753 L .40185 .45505 L .39832 .44215 L .3953 .42674 L F .40044 .42642 m .40044 .42642 .00515 0 360 arc F .37078 .26666 m .35746 .27462 L .34661 .28092 L .33655 .28682 L .34455 .27818 L .35334 .26914 L .36459 .25843 L F .36768 .26255 m .36768 .26255 .00515 0 360 arc F .42717 .24608 m .41396 .25431 L .40315 .2609 L .39311 .26713 L .40104 .25819 L .40974 .24889 L .42084 .23795 L F .424 .24202 m .424 .24202 .00515 0 360 arc F .32859 .27189 m .31474 .27901 L .30346 .28454 L .29299 .28964 L .30157 .28167 L .31096 .27327 L .32292 .26329 L F .32575 .26759 m .32575 .26759 .00515 0 360 arc F .61122 .34104 m .61227 .35692 L .61362 .37038 L .61558 .3835 L .61023 .37091 L .60549 .35797 L .60104 .34262 L F .60613 .34183 m .60613 .34183 .00515 0 360 arc F .09272 .22857 m .07852 .22141 L .06729 .21389 L .05705 .20537 L .06837 .21063 L .08067 .21489 L .09596 .21879 L F .09434 .22368 m .09434 .22368 .00515 0 360 arc F .52948 .3736 m .5295 .38957 L .52999 .4031 L .53111 .41625 L .52657 .4034 L .52266 .39018 L .51922 .3745 L F .52435 .37405 m .52435 .37405 .00515 0 360 arc F .31685 .51611 m .32115 .53175 L .32517 .54512 L .32963 .55819 L .32195 .54633 L .31472 .53416 L .3072 .51972 L F .31203 .51791 m .31203 .51791 .00515 0 360 arc F .71532 .31521 m .71936 .33106 L .72327 .34463 L .72773 .35792 L .72003 .34577 L .71288 .33332 L .7056 .31861 L F .71046 .31691 m .71046 .31691 .00515 0 360 arc F .73688 .01602 m .72039 .01636 L .70632 .0164 L .69261 .01638 L .70589 .01299 L .71953 .00955 L .7356 .0058 L F .73624 .01091 m .73624 .01091 .00515 0 360 arc F .77854 .2978 m .78445 .31337 L .78997 .32679 L .796 .33996 L .78687 .32828 L .77826 .31635 L .76925 .30227 L F .7739 .30003 m .7739 .30003 .00515 0 360 arc F .61748 .09919 m .6014 .10284 L .58782 .10567 L .57477 .10837 L .58672 .10242 L .5992 .09633 L .61417 .08943 L F .61583 .09431 m .61583 .09431 .00515 0 360 arc F .895 .25356 m .90381 .26797 L .91181 .28048 L .92024 .29277 L .90904 .28252 L .89828 .27205 L .88672 .25969 L F .89086 .25662 m .89086 .25662 .00515 0 360 arc F .48316 .18422 m .46787 .19044 L .4551 .19542 L .44299 .20016 L .45349 .19239 L .46466 .18437 L .47834 .17512 L F .48075 .17967 m .48075 .17967 .00515 0 360 arc F .63458 .08149 m .61813 .08441 L .60417 .08664 L .59069 .08876 L .60323 .08334 L .61624 .07781 L .63175 .07159 L F .63317 .07654 m .63317 .07654 .00515 0 360 arc F .5604 .13242 m .54442 .13713 L .53095 .14086 L .51806 .14442 L .52965 .13769 L .54182 .13077 L .5565 .12288 L F .55845 .12765 m .55845 .12765 .00515 0 360 arc F .81467 .29644 m .82224 .312 L .82924 .32553 L .83674 .33885 L .82628 .32729 L .81633 .3155 L .80582 .3017 L F .81025 .29907 m .81025 .29907 .00515 0 360 arc F .1855 .23178 m .1687 .23264 L .15487 .23257 L .14196 .23192 L .15431 .22919 L .16756 .22587 L .18379 .22162 L F .18465 .2267 m .18465 .2267 .00515 0 360 arc F .56769 .39429 m .57081 .41125 L .57401 .42588 L .57782 .44021 L .57069 .42677 L .56418 .41303 L .55774 .39696 L F .56272 .39563 m .56272 .39563 .00515 0 360 arc F .59918 .09377 m .58234 .097 L .568 .0995 L .55417 .10188 L .56702 .09621 L .58038 .09042 L .59624 .08389 L F .59771 .08883 m .59771 .08883 .00515 0 360 arc F .07124 .19284 m .05686 .18326 L .04548 .1733 L .0352 .16208 L .04695 .1702 L .0598 .17705 L .07564 .18353 L F .07344 .18818 m .07344 .18818 .00515 0 360 arc F .08448 .56684 m .09611 .55642 L .10499 .55064 L .11246 .54793 L .10761 .55286 L .10136 .56084 L .09235 .57348 L F .08842 .57016 m .08842 .57016 .00515 0 360 arc F .37179 .50874 m .37659 .52559 L .38118 .5402 L .3863 .55453 L .37796 .5414 L .37016 .528 L .36214 .51235 L F .36696 .51055 m .36696 .51055 .00515 0 360 arc F .35913 .21619 m .34287 .22174 L .32925 .2261 L .31636 .23013 L .32782 .22297 L .34001 .21549 L .35485 .20682 L F .35699 .2115 m .35699 .2115 .00515 0 360 arc F .05117 .26389 m .04434 .24733 L .0388 .23177 L .03354 .21539 L .04181 .23011 L .05035 .24401 L .06018 .25891 L F .05568 .2614 m .05568 .2614 .00515 0 360 arc F .14404 .21077 m .12687 .20903 L .11277 .20653 L .09966 .20333 L .11271 .2031 L .12674 .20216 L .14385 .20047 L F .14394 .20562 m .14394 .20562 .00515 0 360 arc F .74805 .33279 m .75482 .34917 L .76119 .36345 L .76812 .37751 L .75813 .36501 L .74871 .35229 L .73888 .33747 L F .74347 .33513 m .74347 .33513 .00515 0 360 arc F .50054 .14679 m .48376 .1515 L .46953 .15527 L .45589 .15884 L .46828 .15207 L .48127 .14511 L .4968 .13719 L F .49867 .14199 m .49867 .14199 .00515 0 360 arc F .4303 .47774 m .43447 .49503 L .43857 .51004 L .44326 .52476 L .4353 .5111 L .42794 .49715 L .42051 .48093 L F .42541 .47933 m .42541 .47933 .00515 0 360 arc F .2184 .22157 m .20103 .22366 L .18656 .2248 L .17295 .22545 L .18579 .22146 L .19949 .21697 L .21609 .21153 L F .21724 .21655 m .21724 .21655 .00515 0 360 arc F .32005 .56108 m .32746 .57748 L .33426 .59189 L .34147 .60614 L .33126 .59356 L .32146 .58082 L .31105 .56609 L F .31555 .56359 m .31555 .56359 .00515 0 360 arc F .35994 .20369 m .34307 .20877 L .32886 .21276 L .31536 .21644 L .32755 .20958 L .34046 .20242 L .35602 .19416 L F .35798 .19893 m .35798 .19893 .00515 0 360 arc F .51812 .43648 m .5223 .45398 L .52645 .4692 L .53123 .48416 L .52318 .47026 L .51576 .45608 L .50831 .43963 L F .51322 .43805 m .51322 .43805 .00515 0 360 arc F .29941 .21361 m .28217 .21782 L .26768 .22097 L .25395 .22376 L .26654 .21774 L .27989 .21135 L .29599 .2039 L F .2977 .20876 m .2977 .20876 .00515 0 360 arc F .93793 .26019 m .94913 .27463 L .95938 .28737 L .97007 .29996 L .95685 .28969 L .94407 .27927 L .93033 .26716 L F .93413 .26368 m .93413 .26368 .00515 0 360 arc F .42115 .17373 m .40394 .1786 L .38934 .18247 L .37538 .18611 L .38809 .17927 L .40145 .1722 L .41741 .16413 L F .41928 .16893 m .41928 .16893 .00515 0 360 arc F .40666 .50555 m .41207 .523 L .41726 .53826 L .42302 .55326 L .41407 .53952 L .40568 .52553 L .39708 .50935 L F .40187 .50745 m .40187 .50745 .00515 0 360 arc F .45784 .1532 m .44041 .15769 L .42554 .16127 L .41129 .16466 L .42437 .15804 L .43807 .15123 L .45434 .14351 L F .45609 .14835 m .45609 .14835 .00515 0 360 arc F .73238 .35321 m .73974 .37004 L .74669 .38481 L .75422 .39938 L .74366 .38643 L .73368 .37327 L .72329 .35805 L F .72784 .35563 m .72784 .35563 .00515 0 360 arc F .23582 .20458 m .21771 .20689 L .20245 .20829 L .18803 .20926 L .20168 .20495 L .21616 .2002 L .2335 .19455 L F .23466 .19956 m .23466 .19956 .00515 0 360 arc F .64519 .39768 m .65168 .41523 L .6579 .43067 L .66474 .44588 L .65478 .43209 L .64543 .41808 L .63582 .40196 L F .64051 .39982 m .64051 .39982 .00515 0 360 arc F .56482 .43785 m .57077 .4558 L .57655 .47159 L .58296 .48716 L .57338 .47291 L .56443 .45844 L .5553 .4418 L F .56006 .43983 m .56006 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07ooOol0Fgoo00<007ooOol01Goo0006Ool00`00Oomoo`06Ool00`00Ool00002Ool00`00Oomoo`06 Ool00`00Oomoo`0mOol00`00Oomoo`22Ool00`00Oomoo`2XOol00`00Oomoo`1JOol00`00Oomoo`05 Ool000Eoo`8000Qoo`<0009oo`<000Ioo`03001oogoo03eoo`03001oogoo08=oo`03001oogoo0:Qo o`03001oogoo05Uoo`03001oogoo00Eoo`007Goo00<007ooOol0?Goo00<007ooOol0ogooRgoo00<0 07ooOol01Goo000MOol00`00Oomoo`0mOol00`00Oomoo`3oOon;Ool00`00Oomoo`05Ool001eoo`03 001oogoo03eoo`03001oogoo0?mooh]oo`03001oogoo00Eoo`007Goo00<007ooOol0?Goo00<007oo Ool0ogooRgoo00<007ooOol01Goo000MOooo003?0007Ool00?mooo=oo`00ogoolgoo003oOoocOol0 0001\ \>"], ImageRangeCache->{{{0, 497}, {306.625, 0}} -> {-2.47514, -0.00835071, 0.0865077, 0.00596375}}], Cell[BoxData[ FormBox[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ \tThe \"fish field\" tells us both the direction and stength of the flow. \ Notice that the flow is stronger the farther the system is from the k'[t]=0 \ or c'[t]=0 lines and in particular that the flow is slowest nearest the \ equilibrium point. The field also tells us something interesting about the \ solution to the consumer's maximization problem. Suppose that the capital \ stock starts at 1. If consumption is too low then according to the dynamics, \ households begin to add to their capital stock. They keep adding to the \ capital stock even as consumption begins to fall. Eventually consumption \ approaches zero and the capital stock approaches a large constant (the k \ where k'[t]=0 intersects the x axis). Intuitively, this path cannot be \ optimal. Households could attain higher utility by consuming some of their \ capital horde! (More technically, it can be shown that capital hoarding \ violates the transversality condition). If consumption starts too high, \ however, the dynamics indicate increasing consumption financed with a falling \ capital stock. Eventually the capital stock is fully consumed and \ consumption crashes to zero. But this too cannot be optimal. Even if it \ were optimal for the households to consume all of their capital stock they \ would never do it in a way which requires a consumption crash - they would \ seek to smooth consumption instead. We can rule out any solution, therefore, \ which does not converge to the steady state. To understand what happens when \ the capital stock starts away from the steady state we must solve for a level \ of consumption such that the dynamics of the model lead exactly to the steady \ state. We demonstrate two methods for solving this problem. First, the \ trial and error or shooting method and then the more elegant time elimination \ method due to Mulligan and Salai-i-Martin (1991). \ \>", "Text"], Cell["\<\ \tSuppose k[0]=1, the shooting method picks an arbitrary initial consumption \ level, c[0]=a and asks whether given k[0]=1 and c[0]=a the solution path \ converges to the steady state. Below we solve the model given two possible \ initial levels of consumption c[0]=0.6 and c[0]=0.7.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(sol1 = NDSolve[{\(c'\)[t] == CPrime[1/3, 0.05, .02, .02, 1.75], \(k'\)[t] == KPrime[1/3, .05, .02, .01], c[0] == 0.55, k[0] == 1}, {c[t], k[t]}, \ {t, 0, 150}]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{\(c(t)\), "\[Rule]", RowBox[{ RowBox[{"(", TagBox[ RowBox[{"InterpolatingFunction", "(", RowBox[{ RowBox[{"(", GridBo