Continuous time formalism

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Continuous time formalism
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Exhaustible Problem

• Max ? (e-rt p(t) h(t)) s.t. dx/dt -h
• Hamiltonian
• Objective function multiplier rhs of diffeq.
• H e-rt p(t) h(t) l h

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Rules
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Exhaustible Resource
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• Since x gt0 it cant be that hinf for any
  measurable length of time.
• If h0 forever then x doesnt go to zero, so by
  transverality l must 0, and by costate it is
  always zero, which cant be greater than or equal
  to pe-rt unless p is zero.
• It must be that at least for long enough to
  exhaust h any and l pe-rt

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• During the time h any, l e-rtp
• By costate -re-rtp e-rt dp/dt 0
• Or dp/dt rp
• Which is Hotellings rule.

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Plausibility for costate

• Let length of time between periods be n. We will
  make n small and see what happens to costate
  equation in discrete time problem.

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Steady State Fish

• State equationhold constant for n seconds
• xtn xt f(xt)n - htn.
• U(x,h) St (1r)-t pt ht n
• here take p as constant
• L(x,h,l) St (1r)-t pt ht n
• St lt (xt f(xt)n - htn
  -xt1n)
• Lx lt(1f(xt)n ) - lt-1n0

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• lt(1f(xt)n ) - lt-n0
• lt - lt-n - ltf(xt)n
• Limn-gt0 (lt - lt-n)/n - ltf(xt)
• dl/dt-lf
• Hamiltonian H pe-rth l(f(x)-h)
• dH/dx - lf dl/dt Costate equation!

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Finish Fish

• H pe-rth l(f(x)-h)
• restrict h to hmin, hmax
• maxh H implies
• pe-rt l 0 and h any
• pe-rt l lt 0 and h hmin
• pe-rt l gt 0 and h hmax

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Exceptional Control

• Suppose pe-rt l 0 and h any
• hence -r pe-rt dl/dt 0
• hence -r pe-rt lf 0
• hence -r l lf
• hence r f(x)
• So the only way to have an interval with h not at
  hmin or hmax is for x to be x.

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• Suppose that x(0) is gt x.
• Set h hmax until x x is an optimal control
• x above x means that f(x) lt f(x) so
• dl/dt gt - lr
• when x x, pe-rt l
• so l falls slower than pe-rt it must have started
  below it, which is what is needed for hhmax

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• Optional At home finish this upshow what
  happens if x(0) lt x, a moratorium. Also show
  that once one gets to x it is indeed optimal to
  stay there. (h f(x) so dx/dt 0.

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• Note that the moratorium is a function of the
  linearity of the problem in h.
• Alternative-- monopoly max ? (p(h)h e-rt)
• Try with p h-a and compare competition and
  monopoly
• Competition for renewable resource. Berck Jeem
  1981

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Competition Sketch

• max int( p(t)h(t)e-rt s.t. dx/dt f(x) h
• H p(t)h(t)e-rt l(f(x) h)
• Costate