Mathematics for Economists
Liquidity modelling Assume that there is some ordering costc. In this case J(S)=cS+hE (SD)+ +pE (DS)+ . Calculating th ...
Liquidity modelling If the distribution ofDis discrete,D= 0 , 1 ,.. ..then one must change the argument. E(DS)+ = ∞ ∑ k=S P(D> ...
Liquidity modelling ∆J(S) = hP(DS)pP(D>S)= = h( 1 P(D>S))pP(D>S)= = h(h+p)P(D>S). Obviously ifS=0 then∆J(S)=h(h+p)P ...
Liquidity modelling S = minfSjh(h+p)P(D>S) (^0) g= = min SjP(D>S) h h+p = = min Sj 1 P(DS) h h+p = = min Sj ...
Liquidity modelling What does happen in a dynamic environment? The state equation is yt+ 1 =(yt+vDt)+, y 0 =x, wherevis the cont ...
Liquidity modelling Theorem If p>c,then there is an optimal strategy vsuch that v(x)= Sx if xS 0 if x>S. The optim ...
Liquidity modelling Example Let the distribution ofDexponential withλ= 2 .Let the penaltyp= 3 and the holding costh=5 and letc= ...
InÖnite dynamic programming Problem The problem ∞ ∑ t= 1 βt^1 r(st,at)!max st+ 1 =f(st,at),t= 1 , 2 ,... at 2 Φ(st),t= 1 , 2 ,.. ...
InÖnite dynamic programming (^1) r(s,a)is bounded and continuous. (^2) f(s,a)is continuous. (^3) Φis continuous and compact valu ...
InÖnite dynamic programming Theorem Under the above conditions there is a stationary optimal strategyπ.The value function solves ...
InÖnite dynamic programming Sometimes the assumption thatris bounded too strong. Theorem If V is an optimal solution then it sat ...
InÖnite cake eating problem The inÖnite cake eating problem is ∞ ∑ t= 0 βtu(ct)!max wt+ 1 =wtct,t= 0 , 1 ,... ct 2 [ 0 ,wt],t= 0 ...
InÖnite cake eating problem By the Bellman equation V(w) = max c 2 [ 0 ,w] (u(c)+βV(wc))= = max s 2 [ 0 ,w] (u(ws)+βV(s)). The c ...
InÖnite cake eating problem Again we will use the Envelope Theorem. We use the second formulation V^0 (w) = u^0 (ws)=u^0 (c(w)) ...
InÖnite cake eating problem Example Solve the problem withu(c)=c 11 σσ We guess thatV(x)=αx 1 ^1 σσ and the optimal policy isπ(x ...
InÖnite cake eating problem Substituting back αw 1 σ 1 σ = (λw) 1 σ 1 σ + βα 1 σ(( 1 λ)w)^1 σ α = λ^1 σ+βα( 1 λ)^1 σ ...
InÖnite cake eating problem If α = 1 β^1 /σ σ λ = 1 β^1 /σ then 1 β^1 /σ σ = 1 β^1 /σ ( 1 σ) +β 1 β^1 /σ σ β(^1 σ)/σ ...
InÖnite cake eating problem Obviously nlim!∞βnV(cn)=^0 so this is the optimal solution. ...
InÖnite cake eating problem Example Solve the problem ifu(c)=lnc. In this case we guess thatV(w)=A+Blnw.Hence A+Blnw= max s 2 [ ...
InÖnite cake eating problem A+Blnw = ln w βB 1 +Bβ w +β A+Bln βB 1 +Bβ w A+Blnw = ln w 1 +Bβ +β A+Bln βB 1 +Bβw ...
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