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T* /F3 1 Tf 0.5019 0 TD (c)Tj >> /F6 1 Tf 0.1697 Tc (w)Tj (u)Tj /F10 1 Tf /F7 1 Tf /F3 1 Tf 0.0003 Tc >> endobj (1)Tj 7.9701 0 0 7.9701 265.08 672.84 Tm /F7 1 Tf (=)Tj 0.3513 0 TD -0.003 Tc /F3 1 Tf (\)+)Tj 7.9701 0 0 7.9701 364.68 485.04 Tm /F3 1 Tf 7.9701 0 0 7.9701 322.92 363.36 Tm /F3 1 Tf 0.3513 0 TD /F3 1 Tf 1.5357 0 TD (\()Tj /F12 1 Tf 0.5019 0 TD 20.145 0 TD (€)Tj 0.3764 0 TD (,)Tj -1.5759 -2.3889 TD 7.9701 0 0 7.9701 318.48 658.4401 Tm /F14 1 Tf /F3 1 Tf 0 Tc (t)Tj 0.3814 0 TD (2)Tj -0.0035 Tc (X)Tj -0.0012 Tc /F6 1 Tf (5)Tj (dt)Tj /F7 1 Tf 0.8281 0 TD (. /F6 1 Tf 0 Tc -4.286 -1.9272 TD -24.8126 -4.4165 TD (c)Tj ()Tj (as)Tj /F6 17 0 R /F7 1 Tf 4.0852 0 TD 0 Tc /F9 1 Tf (3)Tj 11.9552 0 0 11.9552 249.48 349.8 Tm 11.9552 0 0 11.9552 333.48 262.32 Tm ()Tj /F6 1 Tf /F3 1 Tf 1.4655 -1.2045 TD (\()Tj [(\)\). ()Tj -0.0002 Tc /F6 1 Tf )]TJ )Tj /F8 1 Tf (c)Tj 11.9552 0 0 11.9552 86.4 699 Tm ()Tj 0.5571 0 TD 0.9837 0 TD /F6 1 Tf /F3 6 0 R 0.3814 0 TD /F3 1 Tf (N)Tj /F6 1 Tf (=)Tj (V)Tj (t)Tj /F7 1 Tf (=)Tj 7.9701 0 0 7.9701 312.96 677.88 Tm (5)Tj /F3 1 Tf [(the)3.9(n)2.3(,)-330(f)-1.6(or)-321.5(an)32.4(y)-326.1(s)5.1(e)3.9(q)5.2(ue)3.9(nc)3.9(e)-357.5(o)-2(f)-322.8(c)3.9(on)32.4(trols)5.1(,)]TJ 1.0338 0 TD 0 Tc 0 Tc 11.9552 0 0 11.9552 284.16 651.36 Tm [(2\))-209.1(+)-219.8(0)]TJ /F10 1 Tf 7.9701 0 0 7.9701 495.36 133.32 Tm 0.8281 0 TD 0 Tc 0.7327 0 TD (t)Tj 11.9552 0 0 11.9552 397.44 291.24 Tm 0.3727 Tc 0.3814 0 TD ()Tj (})Tj (,x)Tj 1.2246 0 TD 7.9701 0 0 7.9701 234 460.08 Tm ()Tj [(that)-321.6(is)]TJ ()Tj -0.0017 Tc /F6 1 Tf /F6 1 Tf 0.272 Tc -0.1205 0 TD /F10 1 Tf 0.0015 Tc 0.3608 Tc ()Tj The aim of this book is to teach topics in economic dynamics such as simulation, sta-bility theory, and dynamic programming. 0.7428 0 TD 0.3814 0 TD 0.3814 0 TD 3.0413 0 TD (,)Tj 0.3814 0 TD ()Tj /F6 1 Tf (t)Tj (u)Tj ()Tj 0 Tc (\)\))Tj 4.2558 0 TD /F6 1 Tf /F6 1 Tf 0.3814 0 TD 1.0037 0 TD 0 Tc /F1 1 Tf 0.7127 0 TD 0.3814 0 TD endobj (\()Tj (\(1\))Tj 0.3235 Tc -33.3446 -3.0815 TD ()Tj 0 Tc 0 Tc 11.9552 0 0 11.9552 86.4 372.6 Tm (w)Tj /F6 1 Tf (X)Tj /F10 1 Tf /F14 1 Tf 0.0023 Tc /F6 1 Tf /F3 1 Tf (Š)Tj /F10 1 Tf 0.7829 0 TD /F3 1 Tf [(s)-438.7(h)0.1(a)26(v)33.1(e)-450(p)-30.1(erf)-3.8(ect)-444.2(inf)-3.8(o)-4.2(rma)-4.2(t)-2.5(io)-4.2(n)-441.6(o)26(v)33.1(e)1.7(r)-454.2(t)-2.5(he)]TJ /F8 1 Tf /F10 1 Tf 0 Tc /F7 1 Tf 0.6324 0 TD /F7 1 Tf 0 Tc 2.7402 0 TD /F7 1 Tf 0 -1.2045 TD (u)Tj 1.6963 0 TD (t)Tj /F6 1 Tf 0.7829 0 TD (\)=)Tj (=0)Tj 0 Tc 0 Tc (,x)Tj (=1)Tj 0.271 0 TD (+1)Tj (\()Tj << /F7 1 Tf �������q��czN*8@`C���f3�W�Z������k����n. /F3 1 Tf 0.0006 Tc 7.9701 0 0 7.9701 278.76 625.8 Tm ()Tj /F7 1 Tf (0)Tj The chapter covers both the deterministic and stochastic dynamic programming. /F1 1 Tf /F12 29 0 R 0.9937 0 TD (\()Tj /F3 1 Tf 11.9552 0 0 11.9552 267.48 599.52 Tm /F6 1 Tf (0)Tj 0.9837 0 TD 0.7829 0 TD (t)Tj 0.6023 0 TD /F10 1 Tf /F10 1 Tf 11.9552 0 0 11.9552 376.44 311.64 Tm 7.9701 0 0 7.9701 386.28 127.2 Tm 0.3814 0 TD 7.518 0 TD (1)Tj 0.2215 Tc [(a)-325.8(r)-2.9(ecursiv)32.7(e)-350(m)0.6(ec)31.4(ha)-4.6(nism)-350.7(in)-331.6(a)-4.6(l)-1.3(l)-322.5(o)-4.6(r)-334.2(i)-1.3(n)-321.5(a)-335.8(subset)-344.2(o)-4.6(f)-325.4(the)-330(s)2.5(ta)-4.6(te)-330(spa)-4.6(ce. 0.2221 Tc /F3 1 Tf (1)Tj /F12 1 Tf -0.0032 Tc 0 Tc 7.9701 0 0 7.9701 371.76 349.32 Tm /F5 16 0 R 0.9937 0 TD /F3 1 Tf /F3 1 Tf /F6 1 Tf (,x)Tj 7.8895 -2.5194 TD -0.0005 Tc -0.0001 Tc /F6 1 Tf [(It)-242.3(is)-246.9(still)-250.8(v)34.2(e)2.8(ry)-246.9(rare)-248.2(to)-233.9(“nd)-249.8(explicit)-262.4(solutions)-246.9(f)-2.7(or)]TJ -0.0029 Tc /F6 1 Tf (d)Tj (0)Tj /F6 1 Tf 0.5621 0 TD 1.5859 0 TD 1.8268 0 TD /F10 27 0 R endobj /F6 1 Tf (,x)Tj 3.764 0 TD /F3 1 Tf /F3 1 Tf 0.1697 Tc /F3 1 Tf 7.9701 0 0 7.9701 310.92 435.24 Tm (. /F6 1 Tf [(:)-371.3(m)2(eans)-387.6(“nding)-384.6(the)-378.8(v)54.2(alue)-388.8(function)]TJ ()Tj 11.9552 0 0 11.9552 309.12 174.6 Tm /F12 1 Tf ([)Tj 0 Tc 0.0023 Tc /F6 1 Tf )Tj 0.0006 Tc 0.2716 Tc /F6 1 Tf /F10 1 Tf 0 Tc (t)Tj 0.8632 1.4153 TD (t)Tj /F3 1 Tf 1.4454 0 TD 0 Tc -0.0008 Tc 0.2743 Tc ({)Tj (u)Tj /F3 1 Tf (h)Tj (\)=)Tj 10.1076 0 TD /F6 1 Tf (D)Tj 0.3513 0 TD 0.1697 Tc 0.2743 Tc 7.9701 0 0 7.9701 453.12 474.12 Tm 11.9552 0 0 11.9552 397.56 172.56 Tm -0.0008 Tc 0 Tc /F8 25 0 R 11.9552 0 0 11.9552 298.68 312.6 Tm )Tj 0 Tc /F8 1 Tf 0.6022 0 TD -0.0018 Tc (x)Tj (dt)Tj 7.9701 0 0 7.9701 401.16 719.4 Tm -0.0073 Tc -0.0018 Tc 0.2768 Tc (t)Tj /F7 1 Tf << (+1)Tj 0.3513 0 TD /F6 1 Tf /F3 1 Tf 0.0031 Tc Q 11.9552 0 0 11.9552 278.04 455.16 Tm 7.9701 0 0 7.9701 397.92 348 Tm /F6 1 Tf (w)Tj 1.9874 0 TD 0 Tc (\()Tj /F6 1 Tf ()Tj /F6 1 Tf (N)Tj (t)Tj /F6 1 Tf (Ru)Tj /F3 1 Tf 0.2215 Tc (and)Tj /F14 1 Tf /ExtGState << 0.8281 0 TD 0 Tc /F6 1 Tf (f)Tj 11.9552 0 0 11.9552 351.84 229.68 Tm 7.9701 0 0 7.9701 273.24 635.28 Tm 0 Tc 7.9701 0 0 7.9701 157.44 136.92 Tm (,)Tj -19.4724 -2.3889 TD 7.9701 0 0 7.9701 320.64 528.72 Tm 1.2145 0 TD 0 Tc /F6 1 Tf 1.9171 0 TD (\()Tj ET (,)Tj /F7 1 Tf (,u)Tj (T)Tj endstream 7.9701 0 0 7.9701 287.16 719.4 Tm (g)Tj /F6 1 Tf /F3 1 Tf (r)Tj -0.0001 Tc (\))Tj >> BT (\()Tj /F6 1 Tf 1.4554 -2.0376 TD /F8 1 Tf 0.9837 1.1041 TD 0.0001 Tc (d)Tj 0.0006 Tc /F10 1 Tf [(. (\012)Tj [(function)-330.1(\(not)-322.6(a)-324.3(n)31.3(um)32.3(b)-29(e)2.8(r)-342.7(a)-3.1(s)-327.2(i)0.2(n)-320(t)-1.4(he)-328.5(case)-328.5(of)-323.9(functions\). >> /F6 1 Tf 11.9552 0 0 11.9552 116.52 423.24 Tm [(Mark)34.1(et)-433.1(timing,)-441.6(w)33.8(e)-428.9(a)-3.2(s)3.9(s)3.9(u)1.1(me)-428.9(that)-413.1(the)-418.9(g)-3.2(o)-33.3(o)-33.3(d)-410.5(mark)34.1(et)-423.1(op)-29.1(ens)-427.7(i)0.1(n)-410.5(t)-1.5(he)-418.9(b)-29(e)2.7(ginning)]TJ /F9 1 Tf (A)Tj 7.9701 0 0 7.9701 313.44 334.92 Tm /F6 1 Tf [(is,)-321.4(i)-0.2(n)-320.4(t)-1.8(his)-327.6(s)3.6(imple)-328.9(c)2.4(ase,)-331.5(the)-318.8(q)3.7(uan)30.9(t)-1.8(it)28.3(y)-337.6(o)-3.5(f)-314.3(t)-1.8(he)-318.8(asset)-333.1(b)-29.4(ough)30.9(t)-323(a)-3.5(t)-313(t)-1.8(he)-318.8(b)-29.4(e)2.4(ginning)-324.7(of)]TJ 7.9701 0 0 7.9701 221.76 153.36 Tm (y)Tj (u)Tj 5.9776 0 0 5.9776 288.48 551.76 Tm (V)Tj 0.0017 Tc /F6 1 Tf ()Tj 11.9552 0 0 11.9552 264.24 612.84 Tm (R)Tj 0 Tc (1+)Tj (\))Tj 0.8221 Tc /F10 1 Tf 5.9776 0 0 5.9776 201 262.92 Tm /F3 1 Tf (P)Tj (:)Tj 0 Tc /F7 1 Tf -0.0007 Tc /F7 1 Tf -0.0022 Tc 0.4306 Tc 0 Tc Applied dynamic programming by Bellman and Dreyfus (1962) and Dynamic programming and the calculus of variations by Dreyfus (1965) provide a good introduction to the main idea of dynamic programming, -0.0023 Tc [(and,)-330.8(therefore)-328.2(t)-1.1(he)-328.2(optimal)-330.8(p)1.5(ath)-319.7(f)-2.4(or)-322.3(consumption)-339.8(i)0.5(s)]TJ /F6 1 Tf /Im1 Do -0.0035 Tc 0.1697 Tc 0.4216 0 TD Let's try to understand this by taking an example of Fibonacci numbers. 0.3764 0 TD (F)Tj 0.5019 0 TD /F3 1 Tf (,x)Tj (u)Tj (1)Tj /F3 1 Tf 0.5019 0 TD /F6 1 Tf )]TJ (Programming)Tj -31.2262 -1.2045 TD (\)=)Tj /GS1 gs 0 Tc (,...)Tj /F7 21 0 R (=0)Tj /F6 1 Tf -0.0038 Tc ()Tj (,\015)Tj (t)Tj /F3 1 Tf 0.3814 0 TD /F6 1 Tf (8)Tj /F3 1 Tf 11.9552 0 0 11.9552 423.6 116.76 Tm 0.8281 0 TD (0)Tj (4)Tj /F6 1 Tf 7.9701 0 0 7.9701 239.16 684.6 Tm 0 Tc (t)Tj 0.8281 0 TD /F5 1 Tf (\))Tj 0.7729 0 TD -0.0024 Tc 7.9701 0 0 7.9701 119.52 505.44 Tm 4.9785 0 TD /F6 1 Tf 0.7428 0 TD [(w)4.6(h)2(ic)33.7(h)-369.4(g)-2.3(iv)35(e)3.6(s)-356.5(an)-349.4(optim)2.9(a)-2.3(l)-360.4(r)-0.6(ule)-347.7(f)-1.9(or)-351.9(c)33.7(h)2(anging)-363.6(the)-347.7(o)-2.3(ptim)2.9(al)-360.4(c)3.6(o)-2.3(n)32.1(trol,)-360.4(giv)35(e)3.6(n)-359.4(t)-0.6(he)-357.8(s)4.8(t)-0.6(ate)-347.7(o)-2.3(f)]TJ (and)Tj /F3 1 Tf -0.0035 Tc /F6 1 Tf /F3 1 Tf /F6 1 Tf -33.3446 -3.0815 TD /F7 1 Tf /F3 1 Tf -0.0034 Tc /F6 1 Tf 7.9701 0 0 7.9701 297.36 523.8 Tm /F5 1 Tf ()Tj /F10 1 Tf ()Tj ()Tj -0.0044 Tc 0.3814 1.1142 TD /F6 17 0 R 0 Tc -0.0005 Tc /F6 1 Tf 0.532 -0.0502 TD [(D)-7(e)1.6(“ne)]TJ 11.9552 0 0 11.9552 275.04 692.04 Tm 0 Tc /F6 1 Tf (0)Tj (R)Tj /F12 1 Tf (0)Tj (c)Tj /F12 1 Tf 11.9552 0 0 11.9552 359.28 703.8 Tm 11.9552 0 0 11.9552 305.28 331.2 Tm /F7 1 Tf 0.6524 0 TD 11.9552 0 0 11.9552 230.4 142.32 Tm (})Tj 15.7187 0.261 TD 13.7213 -2.2083 TD /F7 1 Tf )]TJ /F6 17 0 R /F8 1 Tf /F10 1 Tf 0.542 0 TD -0.0016 Tc /F6 1 Tf ()Tj 0 Tc /F8 1 Tf 0 Tc 5.7916 -2.8406 TD (})Tj (\()Tj 11.9552 0 0 11.9552 403.8 638.16 Tm /F8 1 Tf /F6 1 Tf 7.9701 0 0 7.9701 245.04 629.88 Tm (Š)Tj 11.9552 0 0 11.9552 185.04 469.08 Tm (5)Tj (t)Tj /F8 1 Tf )]TJ 0 Tc >> 0.271 0 TD 0.9435 0 TD [(u,)-164.9(x)]TJ 11.9552 0 0 11.9552 449.16 442.92 Tm 1.5859 0 TD /F3 1 Tf /F3 1 Tf /F8 1 Tf (t)Tj 0.5621 0 TD 11.9552 0 0 11.9552 420.48 131.88 Tm 0 Tc (B)Tj 1.9573 0 TD /F7 1 Tf 6.6246 0 TD 11.9552 0 0 11.9552 423.48 494.88 Tm 7.9701 0 0 7.9701 346.2 399.84 Tm 11.9552 0 0 11.9552 487.08 668.4 Tm /F3 1 Tf 1.4554 0 TD (\()Tj (T)Tj 11.9552 0 0 11.9552 423.72 249.72 Tm /F9 1 Tf /F12 1 Tf /F6 1 Tf /F6 1 Tf ()Tj [(dyna)-4.7(mic)-340.1(p)-0.4(ro)-4.7(g)-4.7(r)-3(a)-4.7(mming)-325.9(metho)-34.8(d)-0.4(s:)]TJ 0 Tc (T)Tj /F9 1 Tf /F6 1 Tf 0.2716 Tc (|)Tj /F8 1 Tf (t)Tj /F7 1 Tf 0.4416 0 TD 7.9701 0 0 7.9701 167.88 499.8 Tm /F3 1 Tf 0.9937 0 TD >> /F10 1 Tf 7.9701 0 0 7.9701 242.04 538.92 Tm ()Tj /F10 1 Tf /F14 1 Tf (\()Tj (2)Tj /F7 1 Tf /Length 2176 0.5019 0 TD 0.282 Tc /F8 1 Tf (7)Tj 0.2768 Tc 0 Tc (\()Tj /F8 1 Tf [(pro)-35.3(cesses)-339.4(a)-5.2(nd)]TJ /F7 1 Tf (3)Tj 1.5759 0 TD -0.0035 Tc -3.1718 -1.2045 TD (x)Tj /F6 1 Tf (U)Tj 0.5019 0 TD /F8 1 Tf /F3 1 Tf [(ark)35.4(e)4(ts)-275.8(for)-271.3(t)-0.2(he)-267(go)-32(o)-32(d)-258.6(and)-268.7(f)-1.5(or)-271.2(a)-262.8(“nanc)4(ial)]TJ 4.6874 0 TD /F3 1 Tf ET /GS1 gs (,)Tj /F6 1 Tf (+)Tj /F6 1 Tf 11.9552 0 0 11.9552 249.48 155.16 Tm /F7 1 Tf 0.7026 0 TD 7.9701 0 0 7.9701 403.68 411.72 Tm 0 Tc [(get)-242(t)-1.1(he)-237.8(optimal)-240.4(w)34.2(ealth)-249.5(a)-2.8(ccum)32.6(u)1.5(lation)]TJ ()Tj 0.0017 Tc 7.9701 0 0 7.9701 273.6 398.16 Tm /F1 1 Tf 0.2215 Tc (V)Tj (w)Tj 11.9552 0 0 11.9552 307.32 623.64 Tm (\015)Tj (RA)Tj /F6 1 Tf -0.0023 Tc 0 Tc (F)Tj 0.3814 0 TD 4.8982 0 TD /F7 1 Tf 26.7997 0 TD /F6 1 Tf 0 Tc (+1)Tj /F6 1 Tf /F8 1 Tf 11.9552 0 0 11.9552 301.32 437.04 Tm /Im1 Do 10.6195 0 TD (t)Tj Budget Constraint. An economic agent chooses a random sequence {u∗ t,x ∗ t} ∞ t=0 that maximizes the sum max u E0 ∞ t=0 βtf(u t,x t) subject to the contingent sequence of budget constraints x t+1 = g(x t,u t,ω t+1),t=0..∞, x0 given where 0 <β<1. (\)+)Tj 11.9552 0 0 11.9552 137.76 116.76 Tm /F8 1 Tf (,)Tj 0.6524 0 TD (2)Tj endstream [(pre)4.2(s)5.4(e)4.2(n)32.7(t)-341.3(v)55.7(alue)-327.1(of)-322.5(the)-327.1(v)55.7(ariation)-318.6(of)-322.5(the)-337.1(s)5.4(tate)4.2(. 1.345 0 TD /F3 1 Tf 0.1697 Tc (t)Tj /F11 1 Tf /F7 1 Tf ()Tj 0 Tc /F8 1 Tf -0.0002 Tc [(Det)-6(e)-0.5(rmining)-382.9(t)-6(he)-371.9(opt)-6(i)-0.6(mal)-392(v)59.9(alue)-381.9(funct)-6(i)-0.6(on)]TJ (E)Tj 0 Tc )Tj 0 Tc /F3 1 Tf -0.0019 Tc (t)Tj 0.002 Tc 11.9552 0 0 11.9552 378.84 463.56 Tm /F3 1 Tf /F3 1 Tf ()Tj 0.4407 Tc (2)Tj /F8 1 Tf BT (B)Tj 0 Tc 7.9701 0 0 7.9701 311.04 690.48 Tm (E)Tj (+)Tj 0.5119 0 TD /F6 1 Tf /F6 1 Tf 0.9435 0 TD 0 Tc -26.1878 -3.0213 TD /F3 6 0 R [(,t)319.6(h)322.2(a)317.9(t)-11.7(i)321.2(s)]TJ /F3 1 Tf 0.9234 0 TD /F6 1 Tf (})Tj 5.37 0 TD << /F6 1 Tf -0.0026 Tc [(When)-339.4(the)-327.8(i)0.9(nformation)-329.4(is)-336.6(giv)34.9(e)3.5(n)-339.4(b)32(y)-326.5(a)-333.6(“ltration)]TJ (,)Tj 11.9552 0 0 11.9552 331.44 503.4 Tm /Im1 Do /F11 1 Tf (\()Tj (k)Tj -0.0027 Tc /F3 1 Tf (f)Tj 11.9552 0 0 11.9552 348 241.68 Tm /ExtGState << 0.3814 0 TD )Tj /F10 1 Tf [(t,)-173.1(x,)-173.1(u)]TJ 0 Tc /F10 1 Tf ()Tj /F9 1 Tf 12.4063 -2.2082 TD )Tj (X)Tj 11.9552 0 0 11.9552 220.92 332.16 Tm 7.9701 0 0 7.9701 235.44 594.96 Tm In each instance the authors present the specific optimization problem as a dynamic programming problem, characterize the optimal policy functions, estimate the parameters, and use models for policy evaluation. 0.1697 Tc (A)Tj stream 1.5959 0 TD 11.9552 0 0 11.9552 226.2 210.24 Tm >> 7.9701 0 0 7.9701 324.84 316.56 Tm ({)Tj 0.552 0 TD (if)Tj [(wher)-4.1(e)]TJ /F3 1 Tf (y)Tj /F3 1 Tf 0.271 0 TD 0.803 -1.7064 TD (+)Tj 0.0017 Tc 0.5219 0 TD (=0)Tj 16.8226 0 TD (\)])Tj /F6 1 Tf ( )Tj 2.9108 0 TD 0 Tc ([)Tj /F6 1 Tf 0.7026 0 TD /F3 1 Tf 11.9552 0 0 11.9552 281.64 399.96 Tm (u)Tj /F7 1 Tf [(,...)10.1(,\015)]TJ ()Tj -0.0035 Tc (\))Tj /ExtGState << /F8 1 Tf /F10 1 Tf /F3 1 Tf (c)Tj (,)Tj 1.5859 0 TD (\()Tj 0.1697 Tc 1.7164 0 TD 11.9552 0 0 11.9552 240.24 378.96 Tm 7.9701 0 0 7.9701 202.8 603.96 Tm 0 Tc )-924.7(If)-485.9(w)32.5(e)-500.5(co)-4.5(nsider)-504.7(a)-486.3(s)2.6(ma)-4.5(ll)]TJ 2.1881 0 TD (\))Tj ()Tj /ProcSet [/PDF /Text ] 0 Tc 0 Tc (\()Tj 0.5019 0 TD 11.9552 0 0 11.9552 86.4 418.08 Tm /F9 26 0 R 11.9552 0 0 11.9552 249.96 196.68 Tm ET 7.9701 0 0 7.9701 489.72 697.2001 Tm 0.3814 0 TD /F10 1 Tf 0.3814 0 TD /F6 1 Tf /F3 1 Tf (h)Tj /F12 1 Tf (t)Tj (\()Tj (Y)Tj 0.0017 Tc (\()Tj (2)Tj 11.9552 0 0 11.9552 255.6 190.08 Tm ()Tj 0.4819 0 TD /F6 1 Tf 11.9552 0 0 11.9552 403.2 131.88 Tm /F10 1 Tf 0.0031 Tc ()Tj 2.8305 0 TD -0.0021 Tc /F9 1 Tf 13.37 -2.2083 TD >> /F6 1 Tf 135 0 obj [(dis)4(c)2.8(rete)-338.5(time)-328.5(and)-330.1(c)2.8(on)31.3(tin)31.3(u)1.2(ous)-337.3(t)-1.4(ime;)]TJ (\)=)Tj 7.9701 0 0 7.9701 320.4 405.6 Tm /F7 1 Tf /F8 1 Tf (\012c)Tj -32.3107 -2.7202 TD /F9 26 0 R 7.9701 0 0 7.9701 306.12 125.52 Tm 11.9552 0 0 11.9552 266.4 452.28 Tm (\()Tj 56 0 obj (t)Tj /F6 1 Tf 5.9776 0 0 5.9776 262.44 299.88 Tm >> 0.6324 0 TD 0 Tc (. (})Tj (0)Tj (A)Tj 0.1697 Tc /F6 1 Tf 0 Tc 11.9552 0 0 11.9552 368.52 545.64 Tm (=0)Tj 0 Tc ()Tj -6.8053 -3.1417 TD (€)Tj 0 Tc 0.5219 0 TD /F8 1 Tf ()Tj endstream 0.2817 Tc (\()Tj 1.5658 0 TD 0.0595 Tc /F6 1 Tf (})Tj 0 Tc [(set)-393.7(o)-3.9(p)-29.8(e)2(rations)-378.2(\()-2.2(complemen)30.5(t)-2.2(s,)-432.2(unions)-388.3(a)-3.9(nd)-381.1(in)30.5(tersections\))-413.8(a)-3.9(nd)-381.1(build)-391.1(a)]TJ -0.0019 Tc /F8 1 Tf 11.9552 0 0 11.9552 238.44 522.48 Tm /F6 1 Tf /F7 1 Tf 45.48 0 0 -0.48 270.348 632.028 cm (V)Tj 11.9552 0 0 11.9552 296.16 391.08 Tm 11.9552 0 0 11.9552 205.8 677.88 Tm /F8 25 0 R 7.9701 0 0 7.9701 110.04 625.8 Tm (\(1)Tj endobj 0.2717 Tc 1.345 0 TD /F8 1 Tf /F7 1 Tf 0.8281 0 TD (t)Tj 0.8281 0 TD 1.2145 0 TD /GS1 gs 0.0006 Tc 0.8321 Tc 1.1543 0 TD /F10 1 Tf 0.3814 0 TD (Š)Tj 0.0017 Tc /F3 1 Tf 0.0018 Tc /F6 1 Tf -0.0034 Tc (8)Tj [(equa)-4.5(tio)-4.5(n)]TJ 11.9552 0 0 11.9552 485.04 758.04 Tm 7.9701 0 0 7.9701 267.36 521.04 Tm 0 Tc 0.3814 0 TD 11.9552 0 0 11.9552 257.28 417.36 Tm /F10 1 Tf 7.9701 0 0 7.9701 247.8 271.2 Tm 30.0117 0 TD (\()Tj /F12 1 Tf (T)Tj [(\)=)-311.8(m)836.6(a)831.4(x)]TJ (t)Tj /F6 1 Tf )Tj 0.8281 0 TD /F10 1 Tf (X)Tj /F3 1 Tf 0.5019 0 TD q 0.3814 0 TD 0 Tc 0.3513 0 TD /F6 1 Tf /F8 1 Tf 0.3814 0 TD /F9 1 Tf 11.9552 0 0 11.9552 224.4 166.8 Tm 0.1697 Tc [(consists)-337.7(in)-330.5(“nding)-334.7(an)]TJ 0.3814 0 TD /F6 1 Tf 0.3814 0 TD (u)Tj ()Tj 5.9776 0 0 5.9776 200.64 562.2 Tm [(Then,)-343.3(f)-4.9(o)-5.3(r)-324.8(t)-3.6(i)-2(me)]TJ In undergrad-uate courses economic arguments are often made using graphs. -0.0009 Tc (>)Tj ({)Tj >> [(st)-5.4(o)-37.2(c)28.9(k)]TJ (. /F3 1 Tf /F8 1 Tf (F)Tj (\015)Tj /F6 1 Tf /F6 1 Tf 0.0017 Tc 0.5922 0 TD 1.5357 0 TD 5.9776 0 0 5.9776 367.92 683.88 Tm /F3 1 Tf 11.9552 0 0 11.9552 378 394.8 Tm /F7 1 Tf 0.9837 0 TD /F8 1 Tf 0 Tw 0.1697 Tc 2.1881 0 TD /F14 1 Tf /F3 1 Tf [(. /F6 1 Tf (t)Tj 0.3814 0 TD (+)Tj (\()Tj (23)Tj [(m)3.9(e)4.6(tho)-31.4(d)3(ology)]TJ /F3 1 Tf -0.0025 Tc 0.5621 0 TD 1.0439 0 TD -0.0011 Tc 0.3814 0 TD 0.3513 0 TD -19.9043 -1.6261 TD /F3 1 Tf /F6 1 Tf 0 Tc /F6 1 Tf /F3 1 Tf 0 Tc 0 -1.2045 TD /F1 1 Tf 0.5219 0 TD 0 Tc 0.5119 0 TD 0.3814 0 TD 0 Tc (x)Tj ()Tj /F10 1 Tf (t)Tj ()Tj (Š)Tj /F6 1 Tf [(the)-227.5(p)1.8(ath)-239.1(o)-2.5(f)-223(t)-0.8(he)-237.5(co-s)4.6(tate)-227.5(and)-229.1(s)4.6(tate)-227.5(v)54.9(a)-2.5(riables)-236.3(a)-2.5(s)-226.3(i)0.8(n)-239.1(t)-0.8(he)-227.5(P)33.6(o)-2.5(n)31.9(try)4.7(iagins)-246.3(c)3.4(as)4.6(e,)-260.2(s)4.6(o)-2.5(metimes)]TJ (t)Tj 0.0085 Tc 7.9701 0 0 7.9701 259.56 617.76 Tm 0.0008 Tc (u)Tj [(In)-331.1(this)-328.2(case)]TJ 0.5521 0 TD 0.3764 0 TD /F7 1 Tf 11.9552 0 0 11.9552 310.8 112.8 Tm -33.5654 -1.2045 TD (})Tj /F6 1 Tf 0.5922 0 TD /F6 1 Tf 0.2723 Tc /F3 1 Tf /F3 1 Tf 7.9701 0 0 7.9701 210.72 488.64 Tm [(,)-320.3(i)0.9(t)-321.9(h)1.9(as)-326.5(the)-327.8(p)1.9(rop)-28.3(e)3.5(rties)]TJ ({)Tj (s)Tj (t)Tj 2.8306 0.6826 TD [(w)361.4(e)-11(h)328.7(a)354.6(v)361.7(e)]TJ /F3 1 Tf 4.2458 0 TD (\()Tj (Š)Tj 2.8305 0 TD (u)Tj (2)Tj 23.4974 0 TD (A)Tj (t)Tj /F6 1 Tf 0.1697 Tc 7.9701 0 0 7.9701 444.84 590.88 Tm /F3 1 Tf (w)Tj (6)Tj ()Tj 13.56 0 0 -0.36 287.388 328.668 cm (t)Tj 7.9701 0 0 7.9701 483.72 697.2001 Tm (\()Tj 0.7628 0 TD 0.1697 Tc /F7 1 Tf q /F3 1 Tf /F6 1 Tf 0 Tc /F10 1 Tf 7.9701 0 0 7.9701 402.84 349.32 Tm /F8 1 Tf (€F)Tj (c)Tj [(Ev)33.9(en)-420.7(for)-403.2(e)2.5(xplicit)-423.3(utilit)28.4(y)-417.8(a)-3.4(nd)-410.7(pro)-33.5(duction)-400.6(functions)-417.9(t)-1.7(he)-409.1(HJB)-411.8(h)0.9(as)-407.8(not)-403.2(a)-3.4(n)-400.6(e)2.5(xplicit)]TJ 0.9937 0 TD 0.5119 0 TD /F3 1 Tf 0.8337 Tc (A)Tj 0.0023 Tc /F3 1 Tf (=1)Tj /F8 1 Tf /F10 1 Tf /F6 1 Tf 0.5922 0 TD 0.0031 Tc /F6 1 Tf 0.0006 Tc /F7 1 Tf -0.0012 Tc 0.3814 0 TD -0.0005 Tc /F6 1 Tf (1)Tj (\()Tj /F13 1 Tf 0 Tc /F6 1 Tf /F10 1 Tf /F9 1 Tf 0 Tc -0.0008 Tc (})Tj (B)Tj -0.0012 Tc 0.3513 0 TD 0 Tc -0.001 Tc (=)Tj -5.4703 -2.941 TD /F13 30 0 R [(or)-320.8(not\);)]TJ /F6 1 Tf 4.4666 0 TD (t)Tj 0.1697 Tc 0.3814 0 TD -0.0007 Tc /F8 1 Tf /F6 1 Tf 0.3814 0 TD (+)Tj /F6 1 Tf 0.8281 0 TD ()Tj 2.3487 0 TD (e)Tj [(\)=)-10.7(m)274.5(a)269.3(x)]TJ )Tj (0)Tj 1.2045 0 TD /F8 1 Tf ()Tj 0 Tc )Tj 7.9701 0 0 7.9701 260.76 330.24 Tm 7.9701 0 0 7.9701 339.12 593.88 Tm -0.0044 Tc (t)Tj (Š)Tj /F10 27 0 R (\)=)Tj -0.0021 Tc /F6 1 Tf (u)Tj )Tj /F8 1 Tf [(\)\))-272.9(is)-277.4(con)30.9(t)-1.8(in)30.9(uous,)-311.4(d)0.8(ieren)30.9(tiable,)-311.4(i)-0.2(ncreasing)]TJ (\):)Tj /F6 1 Tf -0.0035 Tc 0.0595 Tc /F10 1 Tf 0 Tc /F10 1 Tf 7.9701 0 0 7.9701 140.76 690.48 Tm (\()Tj /F3 1 Tf /F3 1 Tf (\()Tj 1.2045 0 TD )Tj /F6 1 Tf ()Tj /F7 1 Tf 11.9552 0 0 11.9552 335.16 357 Tm 0.0452 Tc /F6 1 Tf 0.2716 Tc 0.5019 0 TD (t)Tj 0 -1.2145 TD (t)Tj (. <> ET /F3 1 Tf (dB)Tj ()Tj (,)Tj 11.9552 0 0 11.9552 290.76 514.6801 Tm 0.8281 0 TD 0.282 Tc (=)Tj 1.0037 0 TD ()Tj /F3 1 Tf /F11 1 Tf 0.3814 0 TD /F3 1 Tf 0.4416 0 TD 0 Tc /F3 1 Tf /F3 1 Tf 0.0001 Tc 2.3487 0 TD 0.0011 Tc [(:)-452.6(I)0(n)-331.2(t)-2.5(he)-329.6(previo)-4.2(us)-338.4(exa)-4.2(m)1(ple,)-342.2(the)-329.6(s)2.9(to)-34.3(c)31.8(h)0.1(a)-4.2(s)2.9(tic)-339.6(p)0.1(ro)-34.3(cess)]TJ /F6 1 Tf 7.9701 0 0 7.9701 234.96 655.5601 Tm (X)Tj 0.3814 0 TD /F3 1 Tf /F15 51 0 R ()Tj 0.5119 0 TD 0.282 Tc 0.1697 Tc /F10 1 Tf /F6 1 Tf -29.3392 -1.2045 TD 2.8305 0 TD /F15 1 Tf 10.1478 0 TD 0.3235 Tc (R)Tj 1.2747 0 TD /F8 1 Tf 0 Tc /F3 1 Tf ({)Tj /F8 1 Tf (V)Tj -0.0349 Tc (+)Tj BT /F6 1 Tf 0.5019 0 TD /F3 1 Tf (x)Tj /F3 1 Tf (t>)Tj )Tj 11.9552 0 0 11.9552 137.76 579.72 Tm -0.0008 Tc /F8 1 Tf 0.2321 Tc 11.9552 0 0 11.9552 448.56 494.88 Tm (V)Tj 0.006 Tc /F3 1 Tf }��eީ�̐4*�*�c��K�5����@9��p�-jCl�����9��Rb7��{�k�vJ���e�&�P��w_-QY�VL�����3q���>T�M`;��P+���� (+1)Tj -0.001 Tc 0 Tc /F10 1 Tf (u)Tj /F3 1 Tf 11.9552 0 0 11.9552 257.28 399.96 Tm 11.9552 0 0 11.9552 187.08 699 Tm [(is)-376.8(the)-368(n)1.8(ominal)-370.6(rate)-368(of)-373.5(return. 11.9552 0 0 11.9552 262.8 114.48 Tm /F8 1 Tf 1.1944 0 TD 11.9552 0 0 11.9552 352.92 556.92 Tm (Š)Tj /F13 1 Tf /F3 1 Tf ET /F8 1 Tf (\()Tj (c)Tj /F7 1 Tf 0.3814 0 TD BT /F6 1 Tf /F6 1 Tf [(wher)-4.1(e)]TJ 0.7327 0 TD (\()Tj /Im1 31 0 R (Š)Tj /F7 1 Tf 0.4407 Tc (. /F1 4 0 R /F8 1 Tf (})Tj (t)Tj (x)Tj /F10 1 Tf (t)Tj 11.9552 0 0 11.9552 278.4 497.76 Tm (=0)Tj ( )Tj 11.9552 0 0 11.9552 368.16 624.24 Tm (=)Tj (1)Tj /F1 4 0 R 11.9552 0 0 11.9552 361.8 401.88 Tm (f)Tj 0 Tc [(\(3\))-492.8(D)-5.3(ieren)31.8(tly)-256.4(f)-2.2(rom)-248.3(t)-0.9(he)-257.7(P)33.5(o)-2.6(n)31.8(try)4.6(iagins)-256.5(p)]TJ (})Tj 1.5959 0 TD /F10 1 Tf (\()Tj 103 0 obj /Im1 Do /F10 1 Tf (})Tj 0.542 0 TD /F10 1 Tf /F3 1 Tf /F6 1 Tf /F3 1 Tf (1)Tj /F6 1 Tf /F8 1 Tf 7.9701 0 0 7.9701 196.08 264 Tm [(b)-30.2(e)1.6(longing)-335.5(t)-2.6(o)-325.5(a)-325.5(s)2.8(equence)-349.7(o)-4.3(f)-335.2(e)1.6(lemen)30.1(t)-2.6(s)-338.5(o)-4.3(f)-335.2()2.6(,)]TJ (u)Tj /F13 1 Tf [(+\()221.5(1)]TJ /F10 1 Tf /F8 1 Tf 5.9776 0 0 5.9776 311.76 339.84 Tm (c)Tj [(whic)31.5(h)]TJ -30.8452 -2.7202 TD (w)Tj 0.7217 Tc /F3 1 Tf 0.3513 0 TD /F3 1 Tf /F3 1 Tf /F6 1 Tf /F6 1 Tf [(are)-319.8(undetermined)-351.6(co)-34.6(ecien)29.9(ts. 11.9552 0 0 11.9552 314.28 346.2 Tm -0.002 Tc /F3 1 Tf -0.0012 Tc 1.6963 0 TD Q 1.084 0 TD -3.1618 -1.2045 TD 11.9552 0 0 11.9552 265.68 600.72 Tm /F5 1 Tf 0 Tc 0.0085 Tc 5.9776 0 0 5.9776 268.8 401.64 Tm 0 Tc 0.8281 0 TD /F6 1 Tf (V)Tj /F8 1 Tf /F6 1 Tf 1.5859 0 TD /F3 1 Tf /F3 1 Tf /F10 1 Tf /Im1 Do /F3 1 Tf (c)Tj (t)Tj 0.3814 0 TD 11.9552 0 0 11.9552 376.56 703.8 Tm -0.001 Tc /F1 1 Tf (T)Tj /F7 1 Tf (+)Tj /F6 1 Tf -24.2004 -4.1957 TD (\()Tj (3)Tj 0.9837 1.4053 TD 7.9701 0 0 7.9701 309.96 112.68 Tm 11.9552 0 0 11.9552 324.84 686.4 Tm 8.1302 0 TD << 0.9234 0 TD /F7 1 Tf << 0.0096 Tc (V)Tj 0.7428 -0.0502 TD -0.0001 Tc T* (t)Tj 0.1697 Tc [(Ex)5.8(a)9.2(m)8.3(p)7(l)5.6(e)]TJ /F3 1 Tf -19.7134 -1.9473 TD q /F6 1 Tf BT (1)Tj (t)Tj 0.9837 0 TD 0.3814 0 TD /F9 1 Tf (\()Tj /F3 1 Tf /F7 1 Tf (,w)Tj /F6 1 Tf (t)Tj BT 8.8932 -2.399 TD (T)Tj 1.0338 0 TD (F)Tj /F8 1 Tf 11.9552 0 0 11.9552 228.48 155.16 Tm /F3 1 Tf /F8 1 Tf 0 Tc 0.3764 0 TD 0.0017 Tc 0.5621 0 TD 0.1697 Tc /F6 1 Tf (\()Tj 11.9552 0 0 11.9552 258.48 522.24 Tm 11.9552 0 0 11.9552 327.84 663.48 Tm 0.9937 0 TD -30.3032 -4.8682 TD 1.4554 0 TD (0)Tj 0 Tc (k)Tj 7.9701 0 0 7.9701 469.56 343.32 Tm 7.9701 0 0 7.9701 234 485.88 Tm /F6 1 Tf 0.793 0 TD /F3 1 Tf (t)Tj (x)Tj (\()Tj 11.9552 0 0 11.9552 353.4 596.76 Tm ()Tj 0 Tc (\))Tj 0.0006 Tc (3)Tj 0.1697 Tc -0.0022 Tc (k)Tj 11.1115 -1.2145 TD 0.0031 Tc (=\()Tj (,)Tj /F8 1 Tf 9.5556 0 TD 11.9552 0 0 11.9552 261 703.8 Tm 0.4617 0 TD /Length 9997 [(t,)-172()]TJ 1.064 0 TD 1.5959 0 TD /F9 1 Tf 0.0007 Tc 1.0037 0 TD -0.0005 Tc (\()Tj (u)Tj 0.8321 Tc /F3 1 Tf -0.001 Tc /F6 1 Tf 7.9701 0 0 7.9701 255.48 636.36 Tm 0 Tc /F10 1 Tf [(,)-321.9(but)-333.6(the)-329.4(p)0.3(ro)-34.1(ces)3.1(s)]TJ 11.9552 0 0 11.9552 127.2 663.48 Tm /F3 1 Tf ET /F3 1 Tf ()Tj 33 0 obj (0)Tj 18.9907 0 TD /F6 1 Tf /Length 6853 -13.2395 -2.8205 TD /F6 1 Tf 11.9552 0 0 11.9552 179.16 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introduction to dynamic programming applied to economics
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introduction to dynamic programming applied to economics

/Im1 31 0 R (0)Tj 9.9873 -1.7566 TD BT [(Substituting)-254.3(in)-240(the)-238.4(budget)-242.6(constrain)31(t,)-271.1(w)33.6(e)]TJ /F14 1 Tf 11.2318 0 TD /F6 1 Tf /F7 1 Tf -0.0034 Tc (:)Tj 0 Tc /F8 1 Tf (+)Tj (t)Tj (X)Tj /F6 1 Tf 1.1744 0 TD 1.7365 0 TD /F9 1 Tf 0.5912 Tc /F10 1 Tf /F8 1 Tf (and)Tj 0 Tc 0.9937 0 TD ()Tj /F15 1 Tf (,)Tj (f)Tj 0.793 0 TD 0.8231 0 TD (,x)Tj (0)Tj 1.4554 -1.8569 TD (})Tj /F6 1 Tf 0.3814 0 TD /F6 1 Tf (=)Tj /F9 1 Tf 0.3202 Tc /F6 1 Tf (\()Tj 7.9701 0 0 7.9701 343.8 256.8 Tm 11.9552 0 0 11.9552 257.04 520.56 Tm �jf��s���cI� /F6 1 Tf /F10 1 Tf 7.9701 0 0 7.9701 248.64 309.84 Tm /F14 1 Tf [(that)-321.6(is)]TJ (\()Tj (,)Tj 7.9701 0 0 7.9701 237 165 Tm -0.6625 -3.0113 TD 10.4189 0.7528 TD 0 Tc 0.5621 0 TD (|F)Tj 11.9552 0 0 11.9552 328.08 112.8 Tm ()Tj 11.9552 0 0 11.9552 307.8 666.96 Tm >> -4.5771 -3.6838 TD 0.3814 0 TD /F6 1 Tf 7.9701 0 0 7.9701 353.52 454.32 Tm (\()Tj 10.0474 0 TD 0 Tc �6��o>��sqrr���m����LVY��8�9���a^XmN�L�L"汛;�X����B�ȹ\�TVط�"I���P�� 0.7729 0 TD (t)Tj (\()Tj /F6 1 Tf 0 Tc /F6 1 Tf ()Tj /F6 1 Tf /F3 1 Tf ()Tj 7.9701 0 0 7.9701 341.16 350.16 Tm 7.9701 0 0 7.9701 96.12 421.44 Tm 30.48 0 0 -0.48 280.908 129.348 cm 0.3814 0 TD (=\()Tj /F6 1 Tf /F7 1 Tf (b)Tj 0 Tw /F3 1 Tf 7.9701 0 0 7.9701 370.92 188.28 Tm (,)Tj (7)Tj 11.9552 0 0 11.9552 280.68 638.4 Tm 0.8337 Tc /F3 1 Tf 0 Tc /F3 1 Tf 2.8305 0 TD (H)Tj (=1)Tj (w)Tj 0.2817 Tc 1.0037 0 TD -0.0011 Tc 0.5521 0 TD 2.68 0 TD BT /F3 1 Tf (,)Tj 4.6874 0 TD /F3 1 Tf /F6 1 Tf (,t)Tj (\)=)Tj /F3 1 Tf /F6 1 Tf (ŠV)Tj -1.2648 -0.9937 TD ()Tj 0.5922 0 TD (t)Tj 7.9701 0 0 7.9701 260.52 539.04 Tm 0.0015 Tc 0.1697 Tc 0.0017 Tc 11.9552 0 0 11.9552 353.76 692.28 Tm (P)Tj 0 Tc 0.4216 0 TD 0.6424 0 TD 0.2717 Tc 1.0037 0 TD (B)Tj /F7 1 Tf -25.9168 -1.2145 TD [(. /F8 1 Tf 9.8567 0 TD /F8 1 Tf (\()Tj /F6 1 Tf 7.9701 0 0 7.9701 328.44 130.08 Tm (1)Tj (7)Tj 0.0006 Tc (X)Tj 7.9701 0 0 7.9701 365.64 352.32 Tm 0.3814 0 TD 0.3814 0 TD 11.9552 0 0 11.9552 366.12 546.24 Tm (Š)Tj 0.2823 Tc 0 Tc 7.9701 0 0 7.9701 473.04 394.08 Tm (Š)Tj /F6 1 Tf /F7 21 0 R /F6 1 Tf 13.68 0 0 -0.48 297.348 469.068 cm (=0)Tj )Tj (t)Tj /F3 1 Tf /F6 1 Tf (\()Tj 0.2823 Tc /F3 1 Tf >> (c)Tj /F3 1 Tf 0.0003 Tc 0.0018 Tc ({\012)Tj /F7 1 Tf /F8 1 Tf /F6 1 Tf (t)Tj 0.3814 0 TD (V)Tj 11.9552 0 0 11.9552 386.28 692.28 Tm (k)Tj )Tj -0.0005 Tc )]TJ /F6 1 Tf (V)Tj (t)Tj /F10 1 Tf 0.5019 0 TD /F7 1 Tf /F6 1 Tf /F3 1 Tf 0.1697 Tc [(0,)-291.3(whic)32.5(h)-300.4(m)1.7(a)26.7(y)-277.4(pro)-33.6(duce)]TJ (s)Tj 0.271 0 TD /F6 1 Tf -0.0011 Tc (t)Tj 1.345 0 TD 1.0138 0 TD 0.0043 Tc >> 0.0017 Tc -0.001 Tc ()Tj 7.9701 0 0 7.9701 447.48 112.68 Tm /F3 1 Tf 0 Tc 0.8331 0 TD 1.5558 0 TD 0.3814 0 TD 0 Tc 1.0338 0 TD /F3 1 Tf /F6 1 Tf 5.9776 0 0 5.9776 315.6 300.96 Tm 2.0175 0 TD 0 Tc (\()Tj 0.7628 0 TD /F10 1 Tf /F3 1 Tf 0 Tc 0.3513 0 TD 0.8281 0 TD /F6 1 Tf 7.9701 0 0 7.9701 252.48 649.08 Tm /F12 1 Tf 0.1697 Tc /F15 1 Tf /F6 1 Tf 0 Tc -0.0004 Tc 7.9701 0 0 7.9701 164.4 688.92 Tm 0 Tc endobj /F7 1 Tf /F6 1 Tf /F6 1 Tf 0.3704 Tc -0.0004 Tc >> 0.793 0 TD 0.271 0 TD /F12 1 Tf (1)Tj 0.0006 Tc (\()Tj (s)Tj [(+\()221.5(1)]TJ 0.5119 0 TD (. ()Tj /F6 1 Tf 0 Tc /F10 1 Tf -2.9108 -1.9172 TD /F3 1 Tf (. 0.0085 Tc 0 Tc /F10 1 Tf (X)Tj /F8 1 Tf [(b)-29(e)-398.7(c)32.9(h)1.2(osen)-420.4(optimally)-397.4(for)-392.9(t)-1.4(he)-408.8(remaining)-404.6(p)-29(erio)-33.2(d,)-421.4(if)-404.2(w)33.9(e)-408.8(tak)34.2(e)-398.7(the)-408.8(o)-3.1(ptimal)-401.3(v)54.3(a)-3.1(lues)-397.5(of)]TJ 0.3513 0 TD ()Tj /F3 1 Tf BT [(suc)30.3(h)-342.7(t)-4(h)-1.4(a)-5.7(t)-4(:)]TJ BT ()Tj (w)Tj (t)Tj (+1)Tj 0 Tc (at)Tj 11.9552 0 0 11.9552 235.68 451.8 Tm 0 Tc 0.9937 0 TD (\()Tj /F1 4 0 R (\()Tj (and)Tj /F3 1 Tf /F3 1 Tf )-440.5(In)-329.1(addition,)-340.1(i)1.2(n)-329.1(a)-2.1(n)32.3(y)-336.2(giv)35.2(e)3.8(n)-339.1(m)3.1(om)3.1(e)3.8(n)32.3(t)-341.7(w)34.9(e)]TJ /F6 1 Tf 11.9552 0 0 11.9552 376.68 201.6 Tm (=)Tj (u)Tj 7.9701 0 0 7.9701 343.44 443.76 Tm q /F10 1 Tf /F8 1 Tf 7.9701 0 0 7.9701 314.16 520.6801 Tm (t)Tj 0 -1.2145 TD /F3 1 Tf 11.9552 0 0 11.9552 397.8 592.6801 Tm (t)Tj -0.001 Tc (Ak)Tj (c)Tj (t)Tj 0.1697 Tc 0 Tc (Š)Tj (T)Tj 0 Tc /F6 1 Tf 11.9552 0 0 11.9552 277.8 128.64 Tm /F6 1 Tf ()Tj (,)Tj /F6 1 Tf -17.124 -2.1581 TD 0.1697 Tc 1.5257 2.0878 TD 0 Tc 11.9552 0 0 11.9552 250.8 437.04 Tm /F7 1 Tf (,u)Tj ()Tj /F3 1 Tf 0.9837 0 TD (=)Tj 0.7217 Tc 11.9552 0 0 11.9552 291.72 420.36 Tm 7.9701 0 0 7.9701 117 676.08 Tm )]TJ (h)Tj (s)Tj /F3 1 Tf /F7 1 Tf /F6 1 Tf /F6 1 Tf (The)Tj (E)Tj [(fore,)-370.6(a)-2.5(nalogously)-366.7(t)-0.8(o)-353.8(a)-363.8(random)-368.7(v)54.9(ariable,)]TJ (\()Tj /F10 1 Tf (t)Tj 0 -0.3011 TD (})Tj (t)Tj 0 Tc (s)Tj (\))Tj /Font << /F3 1 Tf [(are)-319.8(undetermined)-351.6(co)-34.6(ecien)29.9(ts. /F10 1 Tf /F3 1 Tf 0.3814 0 TD << 7.9701 0 0 7.9701 273.48 295.68 Tm -28.7473 -1.9172 TD 4.9785 0 TD 7.9701 0 0 7.9701 367.32 417.36 Tm (\()Tj -28.8575 -1.2045 TD 0 Tc Cagan Model (RatEx, Discrete Time) Three Asset Model. /F6 1 Tf /F3 1 Tf -30.7147 -2.0978 TD 0.9937 0 TD 0 Tc (u)Tj 0.0066 Tc -0.0007 Tc ()Tj /F8 1 Tf 0.3814 0 TD /F6 1 Tf /F3 1 Tf (\(1)Tj /F6 1 Tf /F3 1 Tf 0.2817 Tc /F6 1 Tf 19.362 0 TD (t)Tj (\)+)Tj ([0)Tj 0.5019 0 TD 8.7325 0 TD 0.3814 0 TD 0.3814 0 TD 0.5922 0 TD /F6 1 Tf 5.9776 0 0 5.9776 246.72 300.96 Tm /F7 1 Tf (a)Tj 0.3764 0 TD -33.3446 -3.0815 TD [(Observ)60.1(at)-5.8(ions:)]TJ 0 Tc Q 0 Tc 12.9482 0 TD 1.6361 0 TD 11.9552 0 0 11.9552 156.12 378.96 Tm /F3 1 Tf 0.7327 0 TD /F6 1 Tf 0 Tc 0.2817 Tc 24.2 0 TD /F3 1 Tf (\()Tj 0.1572 Tc /F6 1 Tf (RA)Tj 11.9552 0 0 11.9552 295.56 336.72 Tm 7.9701 0 0 7.9701 205.56 696.48 Tm 0.5019 0 TD /F7 1 Tf 1.064 0 TD (5)Tj [(\))-423.7(i)-0.5(s)-428.3(n)0.5(ot)-433.8(directly)-438.3(dep)-29.7(e)2.1(nden)30.6(t)-453.8(on)-431.2(time\))-433.8(a)-3.8(nd)-431.2(time)-429.5(indep)-29.6(e)2.1(ndence)-459.6(a)-3.8(nd)-431.2(the)]TJ ()Tj /F3 1 Tf 0 Tc 0.4416 0 TD (A)Tj (+0)Tj 0.7227 0 TD /F3 1 Tf -0.0006 Tc 0.3814 0 TD (sum)Tj 0.5019 0 TD /F3 1 Tf 24 0 obj 0.8344 Tc 10.3687 0 TD 0.7428 0 TD -0.0009 Tc /F6 1 Tf /F9 1 Tf /F3 1 Tf 0.3727 Tc 5.9776 0 0 5.9776 383.76 484.8 Tm /F3 1 Tf (t)Tj 1.0138 0 TD 0.0006 Tc -7.1566 -2.0376 TD /F3 1 Tf /Im1 Do 1.0539 0 TD >> 11.9552 0 0 11.9552 270.12 311.64 Tm ()Tj 11.9552 0 0 11.9552 327.48 691.5601 Tm 0 Tc 0 Tc 0.3814 0 TD ()Tj (,x)Tj /F3 1 Tf 0.9435 0 TD /F8 1 Tf 7.9701 0 0 7.9701 212.64 456.6 Tm (t)Tj 0.2723 Tc 11.9552 0 0 11.9552 156.72 369.24 Tm 0.0002 Tc 0.5922 0 TD 7.9701 0 0 7.9701 369.36 130.08 Tm -32.3107 -2.4793 TD -18.9407 -1.877 TD 13.7313 -1.7666 TD /F10 1 Tf -0.0003 Tc ()Tj 0.5922 0 TD /F7 1 Tf (. 0 Tc /F11 1 Tf 7.9701 0 0 7.9701 304.68 297 Tm /F8 1 Tf 7.9701 0 0 7.9701 126.96 564.84 Tm (0)Tj (\015)Tj /F6 1 Tf 0 Tc /F3 1 Tf /F3 1 Tf (,)Tj ()Tj /F6 1 Tf [(A)-5.9(ssumpt)-6.4(i)-1(ons:)]TJ /F3 1 Tf (\))Tj 0.522 -1.4153 TD 7.9701 0 0 7.9701 274.2 378.72 Tm 11.9552 0 0 11.9552 277.56 188.4 Tm 11.9552 0 0 11.9552 188.04 142.32 Tm (5)Tj 7.9701 0 0 7.9701 292.08 440.52 Tm 0.9034 0 TD /F6 1 Tf /F3 1 Tf 0 Tc 0.3764 0 TD 18.9806 0 TD /GS1 8 0 R 0.5922 0 TD >> 0 Tc /F6 1 Tf ({)Tj /F3 1 Tf (\()Tj )]TJ 0 Tc 1.0338 0 TD /F3 1 Tf 0 Tc endobj -16.4111 -1.2145 TD 11.9552 0 0 11.9552 245.64 540.72 Tm /F10 1 Tf 0.0002 Tc [(The)-338.5(o)-3.1(ptimalit)28.7(y)-327.2(c)2.8(ondition)-330.1(is:)]TJ /F7 1 Tf 7.9701 0 0 7.9701 306.48 378.36 Tm /F8 1 Tf ET 0.3513 0 TD /F6 1 Tf /F6 1 Tf /F12 1 Tf [(t,)-172(x)]TJ /F3 1 Tf (T)Tj -5.8318 -1.4554 TD (\))Tj (,)Tj (8)Tj 0 Tc 0.527 0 TD (du)Tj 0 Tw >> << 0.0006 Tc 0.3764 0 TD (\()Tj 0.3513 0 TD (t)Tj 11.9552 0 0 11.9552 293.64 117.84 Tm /F6 1 Tf (. T* /F3 1 Tf 0.5019 0 TD (c)Tj >> /F6 1 Tf 0.1697 Tc (w)Tj (u)Tj /F10 1 Tf /F7 1 Tf /F3 1 Tf 0.0003 Tc >> endobj (1)Tj 7.9701 0 0 7.9701 265.08 672.84 Tm /F7 1 Tf (=)Tj 0.3513 0 TD -0.003 Tc /F3 1 Tf (\)+)Tj 7.9701 0 0 7.9701 364.68 485.04 Tm /F3 1 Tf 7.9701 0 0 7.9701 322.92 363.36 Tm /F3 1 Tf 0.3513 0 TD /F3 1 Tf 1.5357 0 TD (\()Tj /F12 1 Tf 0.5019 0 TD 20.145 0 TD (€)Tj 0.3764 0 TD (,)Tj -1.5759 -2.3889 TD 7.9701 0 0 7.9701 318.48 658.4401 Tm /F14 1 Tf /F3 1 Tf 0 Tc (t)Tj 0.3814 0 TD (2)Tj -0.0035 Tc (X)Tj -0.0012 Tc /F6 1 Tf (5)Tj (dt)Tj /F7 1 Tf 0.8281 0 TD (. /F6 1 Tf 0 Tc -4.286 -1.9272 TD -24.8126 -4.4165 TD (c)Tj ()Tj (as)Tj /F6 17 0 R /F7 1 Tf 4.0852 0 TD 0 Tc /F9 1 Tf (3)Tj 11.9552 0 0 11.9552 249.48 349.8 Tm 11.9552 0 0 11.9552 333.48 262.32 Tm ()Tj /F6 1 Tf /F3 1 Tf 1.4655 -1.2045 TD (\()Tj [(\)\). ()Tj -0.0002 Tc /F6 1 Tf )]TJ )Tj /F8 1 Tf (c)Tj 11.9552 0 0 11.9552 86.4 699 Tm ()Tj 0.5571 0 TD 0.9837 0 TD /F6 1 Tf /F3 6 0 R 0.3814 0 TD /F3 1 Tf (N)Tj /F6 1 Tf (=)Tj (V)Tj (t)Tj /F7 1 Tf (=)Tj 7.9701 0 0 7.9701 312.96 677.88 Tm (5)Tj /F3 1 Tf [(the)3.9(n)2.3(,)-330(f)-1.6(or)-321.5(an)32.4(y)-326.1(s)5.1(e)3.9(q)5.2(ue)3.9(nc)3.9(e)-357.5(o)-2(f)-322.8(c)3.9(on)32.4(trols)5.1(,)]TJ 1.0338 0 TD 0 Tc 0 Tc 11.9552 0 0 11.9552 284.16 651.36 Tm [(2\))-209.1(+)-219.8(0)]TJ /F10 1 Tf 7.9701 0 0 7.9701 495.36 133.32 Tm 0.8281 0 TD 0 Tc 0.7327 0 TD (t)Tj 11.9552 0 0 11.9552 397.44 291.24 Tm 0.3727 Tc 0.3814 0 TD ()Tj (})Tj (,x)Tj 1.2246 0 TD 7.9701 0 0 7.9701 234 460.08 Tm ()Tj [(that)-321.6(is)]TJ ()Tj -0.0017 Tc /F6 1 Tf /F6 1 Tf 0.272 Tc -0.1205 0 TD /F10 1 Tf 0.0015 Tc 0.3608 Tc ()Tj The aim of this book is to teach topics in economic dynamics such as simulation, sta-bility theory, and dynamic programming. 0.7428 0 TD 0.3814 0 TD 0.3814 0 TD 3.0413 0 TD (,)Tj 0.3814 0 TD ()Tj /F6 1 Tf (t)Tj (u)Tj ()Tj 0 Tc (\)\))Tj 4.2558 0 TD /F6 1 Tf /F6 1 Tf 0.3814 0 TD 1.0037 0 TD 0 Tc /F1 1 Tf 0.7127 0 TD 0.3814 0 TD endobj (\()Tj (\(1\))Tj 0.3235 Tc -33.3446 -3.0815 TD ()Tj 0 Tc 0 Tc 11.9552 0 0 11.9552 86.4 372.6 Tm (w)Tj /F6 1 Tf (X)Tj /F10 1 Tf /F14 1 Tf 0.0023 Tc /F6 1 Tf /F3 1 Tf (Š)Tj /F10 1 Tf 0.7829 0 TD /F3 1 Tf [(s)-438.7(h)0.1(a)26(v)33.1(e)-450(p)-30.1(erf)-3.8(ect)-444.2(inf)-3.8(o)-4.2(rma)-4.2(t)-2.5(io)-4.2(n)-441.6(o)26(v)33.1(e)1.7(r)-454.2(t)-2.5(he)]TJ /F8 1 Tf /F10 1 Tf 0 Tc /F7 1 Tf 0.6324 0 TD /F7 1 Tf 0 Tc 2.7402 0 TD /F7 1 Tf 0 -1.2045 TD (u)Tj 1.6963 0 TD (t)Tj /F6 1 Tf 0.7829 0 TD (\)=)Tj (=0)Tj 0 Tc 0 Tc (,x)Tj (=1)Tj 0.271 0 TD (+1)Tj (\()Tj << /F7 1 Tf �������q��czN*8@`C���f3�W�Z������k����n. /F3 1 Tf 0.0006 Tc 7.9701 0 0 7.9701 278.76 625.8 Tm ()Tj /F7 1 Tf (0)Tj The chapter covers both the deterministic and stochastic dynamic programming. /F1 1 Tf /F12 29 0 R 0.9937 0 TD (\()Tj /F3 1 Tf 11.9552 0 0 11.9552 267.48 599.52 Tm /F6 1 Tf (0)Tj 0.9837 0 TD 0.7829 0 TD (t)Tj 0.6023 0 TD /F10 1 Tf /F10 1 Tf 11.9552 0 0 11.9552 376.44 311.64 Tm 7.9701 0 0 7.9701 386.28 127.2 Tm 0.3814 0 TD 7.518 0 TD (1)Tj 0.2215 Tc [(a)-325.8(r)-2.9(ecursiv)32.7(e)-350(m)0.6(ec)31.4(ha)-4.6(nism)-350.7(in)-331.6(a)-4.6(l)-1.3(l)-322.5(o)-4.6(r)-334.2(i)-1.3(n)-321.5(a)-335.8(subset)-344.2(o)-4.6(f)-325.4(the)-330(s)2.5(ta)-4.6(te)-330(spa)-4.6(ce. 0.2221 Tc /F3 1 Tf (1)Tj /F12 1 Tf -0.0032 Tc 0 Tc 7.9701 0 0 7.9701 371.76 349.32 Tm /F5 16 0 R 0.9937 0 TD /F3 1 Tf /F3 1 Tf /F6 1 Tf (,x)Tj 7.8895 -2.5194 TD -0.0005 Tc -0.0001 Tc /F6 1 Tf [(It)-242.3(is)-246.9(still)-250.8(v)34.2(e)2.8(ry)-246.9(rare)-248.2(to)-233.9(“nd)-249.8(explicit)-262.4(solutions)-246.9(f)-2.7(or)]TJ -0.0029 Tc /F6 1 Tf (d)Tj (0)Tj /F6 1 Tf 0.5621 0 TD 1.5859 0 TD 1.8268 0 TD /F10 27 0 R endobj /F6 1 Tf (,x)Tj 3.764 0 TD /F3 1 Tf /F3 1 Tf 0.1697 Tc /F3 1 Tf 7.9701 0 0 7.9701 310.92 435.24 Tm (. /F6 1 Tf [(:)-371.3(m)2(eans)-387.6(“nding)-384.6(the)-378.8(v)54.2(alue)-388.8(function)]TJ ()Tj 11.9552 0 0 11.9552 309.12 174.6 Tm /F12 1 Tf ([)Tj 0 Tc 0.0023 Tc /F6 1 Tf )Tj 0.0006 Tc 0.2716 Tc /F6 1 Tf /F10 1 Tf 0 Tc (t)Tj 0.8632 1.4153 TD (t)Tj /F3 1 Tf 1.4454 0 TD 0 Tc -0.0008 Tc 0.2743 Tc ({)Tj (u)Tj /F3 1 Tf (h)Tj (\)=)Tj 10.1076 0 TD /F6 1 Tf (D)Tj 0.3513 0 TD 0.1697 Tc 0.2743 Tc 7.9701 0 0 7.9701 453.12 474.12 Tm 11.9552 0 0 11.9552 397.56 172.56 Tm -0.0008 Tc 0 Tc /F8 25 0 R 11.9552 0 0 11.9552 298.68 312.6 Tm )Tj 0 Tc /F8 1 Tf 0.6022 0 TD -0.0018 Tc (x)Tj (dt)Tj 7.9701 0 0 7.9701 401.16 719.4 Tm -0.0073 Tc -0.0018 Tc 0.2768 Tc (t)Tj /F7 1 Tf << (+1)Tj 0.3513 0 TD /F6 1 Tf /F3 1 Tf 0.0031 Tc Q 11.9552 0 0 11.9552 278.04 455.16 Tm 7.9701 0 0 7.9701 397.92 348 Tm /F6 1 Tf (w)Tj 1.9874 0 TD 0 Tc (\()Tj /F6 1 Tf ()Tj /F6 1 Tf (N)Tj (t)Tj /F6 1 Tf (Ru)Tj /F3 1 Tf 0.2215 Tc (and)Tj /F14 1 Tf /ExtGState << 0.8281 0 TD 0 Tc /F6 1 Tf (f)Tj 11.9552 0 0 11.9552 351.84 229.68 Tm 7.9701 0 0 7.9701 273.24 635.28 Tm 0 Tc 7.9701 0 0 7.9701 157.44 136.92 Tm (,)Tj -19.4724 -2.3889 TD 7.9701 0 0 7.9701 320.64 528.72 Tm 1.2145 0 TD 0 Tc /F6 1 Tf 1.9171 0 TD (\()Tj ET (,)Tj /F7 1 Tf (,u)Tj (T)Tj endstream 7.9701 0 0 7.9701 287.16 719.4 Tm (g)Tj /F6 1 Tf /F3 1 Tf (r)Tj -0.0001 Tc (\))Tj >> BT (\()Tj /F6 1 Tf 1.4554 -2.0376 TD /F8 1 Tf 0.9837 1.1041 TD 0.0001 Tc (d)Tj 0.0006 Tc /F10 1 Tf [(. (\012)Tj [(function)-330.1(\(not)-322.6(a)-324.3(n)31.3(um)32.3(b)-29(e)2.8(r)-342.7(a)-3.1(s)-327.2(i)0.2(n)-320(t)-1.4(he)-328.5(case)-328.5(of)-323.9(functions\). >> /F6 1 Tf 11.9552 0 0 11.9552 116.52 423.24 Tm [(Mark)34.1(et)-433.1(timing,)-441.6(w)33.8(e)-428.9(a)-3.2(s)3.9(s)3.9(u)1.1(me)-428.9(that)-413.1(the)-418.9(g)-3.2(o)-33.3(o)-33.3(d)-410.5(mark)34.1(et)-423.1(op)-29.1(ens)-427.7(i)0.1(n)-410.5(t)-1.5(he)-418.9(b)-29(e)2.7(ginning)]TJ /F9 1 Tf (A)Tj 7.9701 0 0 7.9701 313.44 334.92 Tm /F6 1 Tf [(is,)-321.4(i)-0.2(n)-320.4(t)-1.8(his)-327.6(s)3.6(imple)-328.9(c)2.4(ase,)-331.5(the)-318.8(q)3.7(uan)30.9(t)-1.8(it)28.3(y)-337.6(o)-3.5(f)-314.3(t)-1.8(he)-318.8(asset)-333.1(b)-29.4(ough)30.9(t)-323(a)-3.5(t)-313(t)-1.8(he)-318.8(b)-29.4(e)2.4(ginning)-324.7(of)]TJ 7.9701 0 0 7.9701 221.76 153.36 Tm (y)Tj (u)Tj 5.9776 0 0 5.9776 288.48 551.76 Tm (V)Tj 0.0017 Tc /F6 1 Tf ()Tj 11.9552 0 0 11.9552 264.24 612.84 Tm (R)Tj 0 Tc (1+)Tj (\))Tj 0.8221 Tc /F10 1 Tf 5.9776 0 0 5.9776 201 262.92 Tm /F3 1 Tf (P)Tj (:)Tj 0 Tc /F7 1 Tf -0.0007 Tc /F7 1 Tf -0.0022 Tc 0.4306 Tc 0 Tc Applied dynamic programming by Bellman and Dreyfus (1962) and Dynamic programming and the calculus of variations by Dreyfus (1965) provide a good introduction to the main idea of dynamic programming, -0.0023 Tc [(and,)-330.8(therefore)-328.2(t)-1.1(he)-328.2(optimal)-330.8(p)1.5(ath)-319.7(f)-2.4(or)-322.3(consumption)-339.8(i)0.5(s)]TJ /F6 1 Tf /Im1 Do -0.0035 Tc 0.1697 Tc 0.4216 0 TD Let's try to understand this by taking an example of Fibonacci numbers. 0.3764 0 TD (F)Tj 0.5019 0 TD /F3 1 Tf (,x)Tj (u)Tj (1)Tj /F3 1 Tf 0.5019 0 TD /F6 1 Tf )]TJ (Programming)Tj -31.2262 -1.2045 TD (\)=)Tj /GS1 gs 0 Tc (,...)Tj /F7 21 0 R (=0)Tj /F6 1 Tf -0.0038 Tc ()Tj (,\015)Tj (t)Tj /F3 1 Tf 0.3814 0 TD /F6 1 Tf (8)Tj /F3 1 Tf 11.9552 0 0 11.9552 423.6 116.76 Tm 0.8281 0 TD (0)Tj (4)Tj /F6 1 Tf 7.9701 0 0 7.9701 239.16 684.6 Tm 0 Tc (t)Tj 0.8281 0 TD /F5 1 Tf (\))Tj 0.7729 0 TD -0.0024 Tc 7.9701 0 0 7.9701 119.52 505.44 Tm 4.9785 0 TD /F6 1 Tf 0.7428 0 TD [(w)4.6(h)2(ic)33.7(h)-369.4(g)-2.3(iv)35(e)3.6(s)-356.5(an)-349.4(optim)2.9(a)-2.3(l)-360.4(r)-0.6(ule)-347.7(f)-1.9(or)-351.9(c)33.7(h)2(anging)-363.6(the)-347.7(o)-2.3(ptim)2.9(al)-360.4(c)3.6(o)-2.3(n)32.1(trol,)-360.4(giv)35(e)3.6(n)-359.4(t)-0.6(he)-357.8(s)4.8(t)-0.6(ate)-347.7(o)-2.3(f)]TJ (and)Tj /F3 1 Tf -0.0035 Tc /F6 1 Tf /F3 1 Tf /F6 1 Tf -33.3446 -3.0815 TD /F7 1 Tf /F3 1 Tf -0.0034 Tc /F6 1 Tf 7.9701 0 0 7.9701 297.36 523.8 Tm /F5 1 Tf ()Tj /F10 1 Tf ()Tj ()Tj -0.0044 Tc 0.3814 1.1142 TD /F6 17 0 R 0 Tc -0.0005 Tc /F6 1 Tf 0.532 -0.0502 TD [(D)-7(e)1.6(“ne)]TJ 11.9552 0 0 11.9552 275.04 692.04 Tm 0 Tc /F6 1 Tf (0)Tj (R)Tj /F12 1 Tf (0)Tj (c)Tj /F12 1 Tf 11.9552 0 0 11.9552 359.28 703.8 Tm 11.9552 0 0 11.9552 305.28 331.2 Tm /F7 1 Tf 0.6524 0 TD 11.9552 0 0 11.9552 230.4 142.32 Tm (})Tj 15.7187 0.261 TD 13.7213 -2.2083 TD /F7 1 Tf )]TJ /F6 17 0 R /F8 1 Tf /F10 1 Tf 0.542 0 TD -0.0016 Tc /F6 1 Tf ()Tj 0 Tc /F8 1 Tf 0 Tc 5.7916 -2.8406 TD (})Tj (\()Tj 11.9552 0 0 11.9552 403.8 638.16 Tm /F8 1 Tf /F6 1 Tf 7.9701 0 0 7.9701 245.04 629.88 Tm (Š)Tj 11.9552 0 0 11.9552 185.04 469.08 Tm (5)Tj (t)Tj /F8 1 Tf )]TJ 0 Tc >> 0.271 0 TD 0.9435 0 TD [(u,)-164.9(x)]TJ 11.9552 0 0 11.9552 449.16 442.92 Tm 1.5859 0 TD /F3 1 Tf /F3 1 Tf /F8 1 Tf (t)Tj 0.5621 0 TD 11.9552 0 0 11.9552 420.48 131.88 Tm 0 Tc (B)Tj 1.9573 0 TD /F7 1 Tf 6.6246 0 TD 11.9552 0 0 11.9552 423.48 494.88 Tm 7.9701 0 0 7.9701 346.2 399.84 Tm 11.9552 0 0 11.9552 487.08 668.4 Tm /F3 1 Tf 1.4554 0 TD (\()Tj (T)Tj 11.9552 0 0 11.9552 423.72 249.72 Tm /F9 1 Tf /F12 1 Tf /F6 1 Tf /F6 1 Tf ()Tj [(dyna)-4.7(mic)-340.1(p)-0.4(ro)-4.7(g)-4.7(r)-3(a)-4.7(mming)-325.9(metho)-34.8(d)-0.4(s:)]TJ 0 Tc (T)Tj /F9 1 Tf /F6 1 Tf 0.2716 Tc (|)Tj /F8 1 Tf (t)Tj /F7 1 Tf 0.4416 0 TD 7.9701 0 0 7.9701 167.88 499.8 Tm /F3 1 Tf 0.9937 0 TD >> /F10 1 Tf 7.9701 0 0 7.9701 242.04 538.92 Tm ()Tj /F10 1 Tf /F14 1 Tf (\()Tj (2)Tj /F7 1 Tf /Length 2176 0.5019 0 TD 0.282 Tc /F8 1 Tf (7)Tj 0.2768 Tc 0 Tc (\()Tj /F8 1 Tf [(pro)-35.3(cesses)-339.4(a)-5.2(nd)]TJ /F7 1 Tf (3)Tj 1.5759 0 TD -0.0035 Tc -3.1718 -1.2045 TD (x)Tj /F6 1 Tf (U)Tj 0.5019 0 TD /F8 1 Tf /F3 1 Tf [(ark)35.4(e)4(ts)-275.8(for)-271.3(t)-0.2(he)-267(go)-32(o)-32(d)-258.6(and)-268.7(f)-1.5(or)-271.2(a)-262.8(“nanc)4(ial)]TJ 4.6874 0 TD /F3 1 Tf ET /GS1 gs (,)Tj /F6 1 Tf (+)Tj /F6 1 Tf 11.9552 0 0 11.9552 249.48 155.16 Tm /F7 1 Tf 0.7026 0 TD 7.9701 0 0 7.9701 403.68 411.72 Tm 0 Tc [(get)-242(t)-1.1(he)-237.8(optimal)-240.4(w)34.2(ealth)-249.5(a)-2.8(ccum)32.6(u)1.5(lation)]TJ ()Tj 0.0017 Tc 7.9701 0 0 7.9701 273.6 398.16 Tm /F1 1 Tf 0.2215 Tc (V)Tj (w)Tj 11.9552 0 0 11.9552 307.32 623.64 Tm (\015)Tj (RA)Tj /F6 1 Tf -0.0023 Tc 0 Tc (F)Tj 0.3814 0 TD 4.8982 0 TD /F7 1 Tf 26.7997 0 TD /F6 1 Tf 0 Tc (+1)Tj /F6 1 Tf /F8 1 Tf 11.9552 0 0 11.9552 301.32 437.04 Tm /Im1 Do 10.6195 0 TD (t)Tj Budget Constraint. An economic agent chooses a random sequence {u∗ t,x ∗ t} ∞ t=0 that maximizes the sum max u E0 ∞ t=0 βtf(u t,x t) subject to the contingent sequence of budget constraints x t+1 = g(x t,u t,ω t+1),t=0..∞, x0 given where 0 <β<1. (\)+)Tj 11.9552 0 0 11.9552 137.76 116.76 Tm /F8 1 Tf (,)Tj 0.6524 0 TD (2)Tj endstream [(pre)4.2(s)5.4(e)4.2(n)32.7(t)-341.3(v)55.7(alue)-327.1(of)-322.5(the)-327.1(v)55.7(ariation)-318.6(of)-322.5(the)-337.1(s)5.4(tate)4.2(. 1.345 0 TD /F3 1 Tf 0.1697 Tc (t)Tj /F11 1 Tf /F7 1 Tf ()Tj 0 Tc /F8 1 Tf -0.0002 Tc [(Det)-6(e)-0.5(rmining)-382.9(t)-6(he)-371.9(opt)-6(i)-0.6(mal)-392(v)59.9(alue)-381.9(funct)-6(i)-0.6(on)]TJ (E)Tj 0 Tc )Tj 0 Tc /F3 1 Tf -0.0019 Tc (t)Tj 0.002 Tc 11.9552 0 0 11.9552 378.84 463.56 Tm /F3 1 Tf /F3 1 Tf ()Tj 0.4407 Tc (2)Tj /F8 1 Tf BT (B)Tj 0 Tc 7.9701 0 0 7.9701 311.04 690.48 Tm (E)Tj (+)Tj 0.5119 0 TD /F6 1 Tf /F6 1 Tf 0.9435 0 TD 0 Tc -26.1878 -3.0213 TD /F3 6 0 R [(,t)319.6(h)322.2(a)317.9(t)-11.7(i)321.2(s)]TJ /F3 1 Tf 0.9234 0 TD /F6 1 Tf (})Tj 5.37 0 TD << /F6 1 Tf -0.0026 Tc [(When)-339.4(the)-327.8(i)0.9(nformation)-329.4(is)-336.6(giv)34.9(e)3.5(n)-339.4(b)32(y)-326.5(a)-333.6(“ltration)]TJ (,)Tj 11.9552 0 0 11.9552 331.44 503.4 Tm /Im1 Do /F11 1 Tf (\()Tj (k)Tj -0.0027 Tc /F3 1 Tf (f)Tj 11.9552 0 0 11.9552 348 241.68 Tm /ExtGState << 0.3814 0 TD )Tj /F10 1 Tf [(t,)-173.1(x,)-173.1(u)]TJ 0 Tc /F10 1 Tf ()Tj /F9 1 Tf 12.4063 -2.2082 TD )Tj (X)Tj 11.9552 0 0 11.9552 220.92 332.16 Tm 7.9701 0 0 7.9701 235.44 594.96 Tm In each instance the authors present the specific optimization problem as a dynamic programming problem, characterize the optimal policy functions, estimate the parameters, and use models for policy evaluation. 0.1697 Tc (A)Tj stream 1.5959 0 TD 11.9552 0 0 11.9552 226.2 210.24 Tm >> 7.9701 0 0 7.9701 324.84 316.56 Tm ({)Tj 0.552 0 TD (if)Tj [(wher)-4.1(e)]TJ /F3 1 Tf (y)Tj /F3 1 Tf 0.271 0 TD 0.803 -1.7064 TD (+)Tj 0.0017 Tc 0.5219 0 TD (=0)Tj 16.8226 0 TD (\)])Tj /F6 1 Tf ( )Tj 2.9108 0 TD 0 Tc ([)Tj /F6 1 Tf 0.7026 0 TD /F3 1 Tf 11.9552 0 0 11.9552 281.64 399.96 Tm (u)Tj /F7 1 Tf [(,...)10.1(,\015)]TJ ()Tj -0.0035 Tc (\))Tj /ExtGState << /F8 1 Tf /F10 1 Tf /F3 1 Tf (c)Tj (,)Tj 1.5859 0 TD (\()Tj 0.1697 Tc 1.7164 0 TD 11.9552 0 0 11.9552 240.24 378.96 Tm 7.9701 0 0 7.9701 202.8 603.96 Tm 0 Tc )-924.7(If)-485.9(w)32.5(e)-500.5(co)-4.5(nsider)-504.7(a)-486.3(s)2.6(ma)-4.5(ll)]TJ 2.1881 0 TD (\))Tj ()Tj /ProcSet [/PDF /Text ] 0 Tc 0 Tc (\()Tj 0.5019 0 TD 11.9552 0 0 11.9552 86.4 418.08 Tm /F9 26 0 R 11.9552 0 0 11.9552 249.96 196.68 Tm ET 7.9701 0 0 7.9701 489.72 697.2001 Tm 0.3814 0 TD /F10 1 Tf 0.3814 0 TD /F6 1 Tf /F3 1 Tf (h)Tj /F12 1 Tf (t)Tj (\()Tj (Y)Tj 0.0017 Tc (\()Tj (2)Tj 11.9552 0 0 11.9552 255.6 190.08 Tm ()Tj 0.4819 0 TD /F6 1 Tf 11.9552 0 0 11.9552 403.2 131.88 Tm /F10 1 Tf 0.0031 Tc ()Tj 2.8305 0 TD -0.0021 Tc /F9 1 Tf 13.37 -2.2083 TD >> /F6 1 Tf 135 0 obj [(dis)4(c)2.8(rete)-338.5(time)-328.5(and)-330.1(c)2.8(on)31.3(tin)31.3(u)1.2(ous)-337.3(t)-1.4(ime;)]TJ (\)=)Tj 7.9701 0 0 7.9701 320.4 405.6 Tm /F7 1 Tf /F8 1 Tf (\012c)Tj -32.3107 -2.7202 TD /F9 26 0 R 7.9701 0 0 7.9701 306.12 125.52 Tm 11.9552 0 0 11.9552 266.4 452.28 Tm (\()Tj 56 0 obj (t)Tj /F6 1 Tf 5.9776 0 0 5.9776 262.44 299.88 Tm >> 0.6324 0 TD 0 Tc (. (})Tj (0)Tj (A)Tj 0.1697 Tc /F6 1 Tf 0 Tc 11.9552 0 0 11.9552 368.52 545.64 Tm (=0)Tj 0 Tc ()Tj -6.8053 -3.1417 TD (€)Tj 0 Tc 0.5219 0 TD /F8 1 Tf ()Tj endstream 0.2817 Tc (\()Tj 1.5658 0 TD 0.0595 Tc /F6 1 Tf (})Tj 0 Tc [(set)-393.7(o)-3.9(p)-29.8(e)2(rations)-378.2(\()-2.2(complemen)30.5(t)-2.2(s,)-432.2(unions)-388.3(a)-3.9(nd)-381.1(in)30.5(tersections\))-413.8(a)-3.9(nd)-381.1(build)-391.1(a)]TJ -0.0019 Tc /F8 1 Tf 11.9552 0 0 11.9552 238.44 522.48 Tm /F6 1 Tf /F7 1 Tf 45.48 0 0 -0.48 270.348 632.028 cm (V)Tj 11.9552 0 0 11.9552 296.16 391.08 Tm 11.9552 0 0 11.9552 205.8 677.88 Tm /F8 25 0 R 7.9701 0 0 7.9701 110.04 625.8 Tm (\(1)Tj endobj 0.2717 Tc 1.345 0 TD /F8 1 Tf /F7 1 Tf 0.8281 0 TD (t)Tj 0.8281 0 TD 1.2145 0 TD /GS1 gs 0.0006 Tc 0.8321 Tc 1.1543 0 TD /F10 1 Tf 0.3814 0 TD (Š)Tj 0.0017 Tc /F3 1 Tf 0.0018 Tc /F6 1 Tf -0.0034 Tc (8)Tj [(equa)-4.5(tio)-4.5(n)]TJ 11.9552 0 0 11.9552 485.04 758.04 Tm 7.9701 0 0 7.9701 267.36 521.04 Tm 0 Tc 0.3814 0 TD 11.9552 0 0 11.9552 257.28 417.36 Tm /F10 1 Tf 7.9701 0 0 7.9701 247.8 271.2 Tm 30.0117 0 TD (\()Tj /F12 1 Tf (T)Tj [(\)=)-311.8(m)836.6(a)831.4(x)]TJ (t)Tj /F6 1 Tf )Tj 0.8281 0 TD /F10 1 Tf (X)Tj /F3 1 Tf 0.5019 0 TD q 0.3814 0 TD 0 Tc 0.3513 0 TD /F6 1 Tf /F8 1 Tf 0.3814 0 TD /F9 1 Tf 11.9552 0 0 11.9552 224.4 166.8 Tm 0.1697 Tc [(consists)-337.7(in)-330.5(“nding)-334.7(an)]TJ 0.3814 0 TD /F6 1 Tf 0.3814 0 TD (u)Tj ()Tj 5.9776 0 0 5.9776 200.64 562.2 Tm [(Then,)-343.3(f)-4.9(o)-5.3(r)-324.8(t)-3.6(i)-2(me)]TJ In undergrad-uate courses economic arguments are often made using graphs. -0.0009 Tc (>)Tj ({)Tj >> [(st)-5.4(o)-37.2(c)28.9(k)]TJ (. /F3 1 Tf /F8 1 Tf (F)Tj (\015)Tj /F6 1 Tf /F6 1 Tf 0.0017 Tc 0.5922 0 TD 1.5357 0 TD 5.9776 0 0 5.9776 367.92 683.88 Tm /F3 1 Tf 11.9552 0 0 11.9552 378 394.8 Tm /F7 1 Tf 0.9837 0 TD /F8 1 Tf 0 Tw 0.1697 Tc 2.1881 0 TD /F14 1 Tf /F3 1 Tf [(. /F6 1 Tf (t)Tj 0.3814 0 TD (+)Tj (\()Tj (23)Tj [(m)3.9(e)4.6(tho)-31.4(d)3(ology)]TJ /F3 1 Tf -0.0025 Tc 0.5621 0 TD 1.0439 0 TD -0.0011 Tc 0.3814 0 TD 0.3513 0 TD -19.9043 -1.6261 TD /F3 1 Tf /F6 1 Tf 0 Tc /F6 1 Tf /F3 1 Tf 0 Tc 0 -1.2045 TD /F1 1 Tf 0.5219 0 TD 0 Tc 0.5119 0 TD 0.3814 0 TD 0 Tc (x)Tj ()Tj /F10 1 Tf (t)Tj ()Tj (Š)Tj /F6 1 Tf [(the)-227.5(p)1.8(ath)-239.1(o)-2.5(f)-223(t)-0.8(he)-237.5(co-s)4.6(tate)-227.5(and)-229.1(s)4.6(tate)-227.5(v)54.9(a)-2.5(riables)-236.3(a)-2.5(s)-226.3(i)0.8(n)-239.1(t)-0.8(he)-227.5(P)33.6(o)-2.5(n)31.9(try)4.7(iagins)-246.3(c)3.4(as)4.6(e,)-260.2(s)4.6(o)-2.5(metimes)]TJ (t)Tj 0.0085 Tc 7.9701 0 0 7.9701 259.56 617.76 Tm 0.0008 Tc (u)Tj [(In)-331.1(this)-328.2(case)]TJ 0.5521 0 TD 0.3764 0 TD /F7 1 Tf 11.9552 0 0 11.9552 310.8 112.8 Tm -33.5654 -1.2045 TD (})Tj /F6 1 Tf 0.5922 0 TD /F6 1 Tf 0.2723 Tc /F3 1 Tf /F3 1 Tf 7.9701 0 0 7.9701 210.72 488.64 Tm [(,)-320.3(i)0.9(t)-321.9(h)1.9(as)-326.5(the)-327.8(p)1.9(rop)-28.3(e)3.5(rties)]TJ ({)Tj (s)Tj (t)Tj 2.8306 0.6826 TD [(w)361.4(e)-11(h)328.7(a)354.6(v)361.7(e)]TJ /F3 1 Tf 4.2458 0 TD (\()Tj (Š)Tj 2.8305 0 TD (u)Tj (2)Tj 23.4974 0 TD (A)Tj (t)Tj /F6 1 Tf 0.1697 Tc 7.9701 0 0 7.9701 444.84 590.88 Tm /F3 1 Tf (w)Tj (6)Tj ()Tj 13.56 0 0 -0.36 287.388 328.668 cm (t)Tj 7.9701 0 0 7.9701 483.72 697.2001 Tm (\()Tj 0.7628 0 TD 0.1697 Tc /F7 1 Tf q /F3 1 Tf /F6 1 Tf 0 Tc /F10 1 Tf 7.9701 0 0 7.9701 402.84 349.32 Tm /F8 1 Tf (€F)Tj (c)Tj [(Ev)33.9(en)-420.7(for)-403.2(e)2.5(xplicit)-423.3(utilit)28.4(y)-417.8(a)-3.4(nd)-410.7(pro)-33.5(duction)-400.6(functions)-417.9(t)-1.7(he)-409.1(HJB)-411.8(h)0.9(as)-407.8(not)-403.2(a)-3.4(n)-400.6(e)2.5(xplicit)]TJ 0.9937 0 TD 0.5119 0 TD /F3 1 Tf 0.8337 Tc (A)Tj 0.0023 Tc /F3 1 Tf (=1)Tj /F8 1 Tf /F10 1 Tf /F6 1 Tf 0.5922 0 TD 0.0031 Tc /F6 1 Tf 0.0006 Tc /F7 1 Tf -0.0012 Tc 0.3814 0 TD -0.0005 Tc /F6 1 Tf (1)Tj (\()Tj /F13 1 Tf 0 Tc /F6 1 Tf /F10 1 Tf /F9 1 Tf 0 Tc -0.0008 Tc (})Tj (B)Tj -0.0012 Tc 0.3513 0 TD 0 Tc -0.001 Tc (=)Tj -5.4703 -2.941 TD /F13 30 0 R [(or)-320.8(not\);)]TJ /F6 1 Tf 4.4666 0 TD (t)Tj 0.1697 Tc 0.3814 0 TD -0.0007 Tc /F8 1 Tf /F6 1 Tf 0.3814 0 TD (+)Tj /F6 1 Tf 0.8281 0 TD ()Tj 2.3487 0 TD (e)Tj [(\)=)-10.7(m)274.5(a)269.3(x)]TJ )Tj (0)Tj 1.2045 0 TD /F8 1 Tf ()Tj 0 Tc )Tj 7.9701 0 0 7.9701 260.76 330.24 Tm 7.9701 0 0 7.9701 339.12 593.88 Tm -0.0044 Tc (t)Tj (Š)Tj /F10 27 0 R (\)=)Tj -0.0021 Tc /F6 1 Tf (u)Tj )Tj /F8 1 Tf [(\)\))-272.9(is)-277.4(con)30.9(t)-1.8(in)30.9(uous,)-311.4(d)0.8(ieren)30.9(tiable,)-311.4(i)-0.2(ncreasing)]TJ (\):)Tj /F6 1 Tf -0.0035 Tc 0.0595 Tc /F10 1 Tf 0 Tc /F10 1 Tf 7.9701 0 0 7.9701 140.76 690.48 Tm (\()Tj /F3 1 Tf /F3 1 Tf (\()Tj 1.2045 0 TD )Tj /F6 1 Tf ()Tj /F7 1 Tf 11.9552 0 0 11.9552 335.16 357 Tm 0.0452 Tc /F6 1 Tf 0.2716 Tc 0.5019 0 TD (t)Tj 0 -1.2145 TD (t)Tj (. <> ET /F3 1 Tf (dB)Tj ()Tj (,)Tj 11.9552 0 0 11.9552 290.76 514.6801 Tm 0.8281 0 TD 0.282 Tc (=)Tj 1.0037 0 TD ()Tj /F3 1 Tf /F11 1 Tf 0.3814 0 TD /F3 1 Tf 0.4416 0 TD 0 Tc /F3 1 Tf /F3 1 Tf 0.0001 Tc 2.3487 0 TD 0.0011 Tc [(:)-452.6(I)0(n)-331.2(t)-2.5(he)-329.6(previo)-4.2(us)-338.4(exa)-4.2(m)1(ple,)-342.2(the)-329.6(s)2.9(to)-34.3(c)31.8(h)0.1(a)-4.2(s)2.9(tic)-339.6(p)0.1(ro)-34.3(cess)]TJ /F6 1 Tf 7.9701 0 0 7.9701 234.96 655.5601 Tm (X)Tj 0.3814 0 TD /F3 1 Tf /F15 51 0 R ()Tj 0.5119 0 TD 0.282 Tc 0.1697 Tc /F10 1 Tf /F6 1 Tf -29.3392 -1.2045 TD 2.8305 0 TD /F15 1 Tf 10.1478 0 TD 0.3235 Tc (R)Tj 1.2747 0 TD /F8 1 Tf 0 Tc /F3 1 Tf ({)Tj /F8 1 Tf (V)Tj -0.0349 Tc (+)Tj BT /F6 1 Tf 0.5019 0 TD /F3 1 Tf (x)Tj /F3 1 Tf (t>)Tj )Tj 11.9552 0 0 11.9552 137.76 579.72 Tm -0.0008 Tc /F8 1 Tf 0.2321 Tc 11.9552 0 0 11.9552 448.56 494.88 Tm (V)Tj 0.006 Tc /F3 1 Tf }��eީ�̐4*�*�c��K�5����@9��p�-jCl�����9��Rb7��{�k�vJ���e�&�P��w_-QY�VL�����3q���>T�M`;��P+���� (+1)Tj -0.001 Tc 0 Tc /F10 1 Tf (u)Tj /F3 1 Tf 11.9552 0 0 11.9552 257.28 399.96 Tm 11.9552 0 0 11.9552 187.08 699 Tm [(is)-376.8(the)-368(n)1.8(ominal)-370.6(rate)-368(of)-373.5(return. 11.9552 0 0 11.9552 262.8 114.48 Tm /F8 1 Tf 1.1944 0 TD 11.9552 0 0 11.9552 352.92 556.92 Tm (Š)Tj /F13 1 Tf /F3 1 Tf ET /F8 1 Tf (\()Tj (c)Tj /F7 1 Tf 0.3814 0 TD BT /F6 1 Tf /F6 1 Tf [(wher)-4.1(e)]TJ 0.7327 0 TD (\()Tj /Im1 31 0 R (Š)Tj /F7 1 Tf 0.4407 Tc (. /F1 4 0 R /F8 1 Tf (})Tj (t)Tj (x)Tj /F10 1 Tf (t)Tj 11.9552 0 0 11.9552 278.4 497.76 Tm (=0)Tj ( )Tj 11.9552 0 0 11.9552 368.16 624.24 Tm (=)Tj (1)Tj /F1 4 0 R 11.9552 0 0 11.9552 361.8 401.88 Tm (f)Tj 0 Tc [(\(3\))-492.8(D)-5.3(ieren)31.8(tly)-256.4(f)-2.2(rom)-248.3(t)-0.9(he)-257.7(P)33.5(o)-2.6(n)31.8(try)4.6(iagins)-256.5(p)]TJ (})Tj 1.5959 0 TD /F10 1 Tf (\()Tj 103 0 obj /Im1 Do /F10 1 Tf (})Tj 0.542 0 TD /F10 1 Tf /F3 1 Tf /F6 1 Tf /F3 1 Tf (1)Tj /F6 1 Tf /F8 1 Tf 7.9701 0 0 7.9701 196.08 264 Tm [(b)-30.2(e)1.6(longing)-335.5(t)-2.6(o)-325.5(a)-325.5(s)2.8(equence)-349.7(o)-4.3(f)-335.2(e)1.6(lemen)30.1(t)-2.6(s)-338.5(o)-4.3(f)-335.2()2.6(,)]TJ (u)Tj /F13 1 Tf [(+\()221.5(1)]TJ /F10 1 Tf /F8 1 Tf 5.9776 0 0 5.9776 311.76 339.84 Tm (c)Tj [(whic)31.5(h)]TJ -30.8452 -2.7202 TD (w)Tj 0.7217 Tc /F3 1 Tf 0.3513 0 TD /F3 1 Tf /F3 1 Tf /F6 1 Tf /F6 1 Tf [(are)-319.8(undetermined)-351.6(co)-34.6(ecien)29.9(ts. 11.9552 0 0 11.9552 314.28 346.2 Tm -0.002 Tc /F3 1 Tf -0.0012 Tc 1.6963 0 TD Q 1.084 0 TD -3.1618 -1.2045 TD 11.9552 0 0 11.9552 265.68 600.72 Tm /F5 1 Tf 0 Tc 0.0085 Tc 5.9776 0 0 5.9776 268.8 401.64 Tm 0 Tc 0.8281 0 TD /F6 1 Tf (V)Tj /F8 1 Tf /F6 1 Tf 1.5859 0 TD /F3 1 Tf /F3 1 Tf /F10 1 Tf /Im1 Do /F3 1 Tf (c)Tj (t)Tj 0.3814 0 TD 11.9552 0 0 11.9552 376.56 703.8 Tm -0.001 Tc /F1 1 Tf (T)Tj /F7 1 Tf (+)Tj /F6 1 Tf -24.2004 -4.1957 TD (\()Tj (3)Tj 0.9837 1.4053 TD 7.9701 0 0 7.9701 309.96 112.68 Tm 11.9552 0 0 11.9552 324.84 686.4 Tm 8.1302 0 TD << 0.9234 0 TD /F7 1 Tf << 0.0096 Tc (V)Tj 0.7428 -0.0502 TD -0.0001 Tc T* (t)Tj 0.1697 Tc [(Ex)5.8(a)9.2(m)8.3(p)7(l)5.6(e)]TJ /F3 1 Tf -19.7134 -1.9473 TD q /F6 1 Tf BT (1)Tj (t)Tj 0.9837 0 TD 0.3814 0 TD /F9 1 Tf (\()Tj /F3 1 Tf /F7 1 Tf (,w)Tj /F6 1 Tf (t)Tj BT 8.8932 -2.399 TD (T)Tj 1.0338 0 TD (F)Tj /F8 1 Tf 11.9552 0 0 11.9552 228.48 155.16 Tm /F3 1 Tf /F8 1 Tf 0 Tc 0.3764 0 TD 0.0017 Tc 0.5621 0 TD 0.1697 Tc /F6 1 Tf (\()Tj 11.9552 0 0 11.9552 258.48 522.24 Tm 11.9552 0 0 11.9552 327.84 663.48 Tm 0.9937 0 TD -30.3032 -4.8682 TD 1.4554 0 TD (0)Tj 0 Tc (k)Tj 7.9701 0 0 7.9701 469.56 343.32 Tm 7.9701 0 0 7.9701 234 485.88 Tm /F6 1 Tf 0.793 0 TD /F3 1 Tf (t)Tj (x)Tj (\()Tj 11.9552 0 0 11.9552 353.4 596.76 Tm ()Tj 0 Tc (\))Tj 0.0006 Tc (3)Tj 0.1697 Tc -0.0022 Tc (k)Tj 11.1115 -1.2145 TD 0.0031 Tc (=\()Tj (,)Tj /F8 1 Tf 9.5556 0 TD 11.9552 0 0 11.9552 261 703.8 Tm 0.4617 0 TD /Length 9997 [(t,)-172()]TJ 1.064 0 TD 1.5959 0 TD /F9 1 Tf 0.0007 Tc 1.0037 0 TD -0.0005 Tc (\()Tj (u)Tj 0.8321 Tc /F3 1 Tf -0.001 Tc /F6 1 Tf 7.9701 0 0 7.9701 255.48 636.36 Tm 0 Tc /F10 1 Tf [(,)-321.9(but)-333.6(the)-329.4(p)0.3(ro)-34.1(ces)3.1(s)]TJ 11.9552 0 0 11.9552 127.2 663.48 Tm /F3 1 Tf ET /F3 1 Tf ()Tj 33 0 obj (0)Tj 18.9907 0 TD /F6 1 Tf /Length 6853 -13.2395 -2.8205 TD /F6 1 Tf 11.9552 0 0 11.9552 179.16 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