Resources and Dynamics

1
Resources and Dynamics

• Discrete Time

2
Lagrange

• A Saddle and an Optimum.
• x in Rn l in R. (positive or zero)
• Problem P Maxx U(x) s.t. g(x) ? 0.
• When g(x) ? 0, we say x satisfies the constraints
  or x is feasible.
• Define L(x,l) U(x) l g(x)

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L2

• Define A Saddle of L is a pair (x,l) that
  satisfies
• SI L(x,l) ? L(x,l)
• SII L(x,l) ? L(x,l)
• Comment 95 of heroin addicts started with
  marijuana. 100 with milk. Heroin implies milk
  but milk does not imply heroin. Can you define
  necessary, sufficient, converse, contrapositive?

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Theorem.

• If (x,l) is a saddle of L then x solves
  problem P.
• Remark I bet Leo talked about If x solves P
  then it is a saddle of L. That was the stuff
  about there being multipliers and the existence
  of prices. This is the converse

5
Sketch of Proof

• a. g(x) ? 0. if not make l l 1 and the
  value of L would go down. contradicts SII.
• b. g(x) 0 implies l 0. if not make l
  smaller and value of L would go down.
• therefore g(x) l 0.
• called complementary slackness.

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End of Proof

• c. SI means U(x) l g(x) U(x) ? U(x)
  l g(x) where the first equality is complementary
  slackness and the ? is SI. Thus For all x for
  which g(x) ? 0, lg(x) ? 0, and U(x) ? U(x).
  This is exactly problem P. The end.

7
Dynamic models.

• (What I do is dynamic. What you do is sh...)
  Warning, making something dynamic for the fun of
  it gains you nothing and really annoys your
  reader.

8
Definitions

• Instead of many goods, same good or thing in many
  time periods.
• Start with xt being the stock of fish in the sea
  or coal in the mine.
• x is called a state variable, it is the amount
  of stuff.
• A control variable or control, ht is the action
  taken, like harvesting or mining.

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State Equations

• state variable next period in terms of the state
  variable this period and control.
• Example xt1 xt - ht
• x 0, is called a state space constraint and
  causes no trouble here, but in continuous time,
  look out.
• h0 is a control constraint, and they never cause
  problems.

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Exhaustible as Optimum

• In the exhaustible resource case let U(x,h) St
  (1r)-t pt ht be the objective function. pt is
  price.
• The initial stock of resource is given x0. Both
  state and control variable are non-negative.

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Exhausted Lagrange

• Problem of exhaustible resources max U s.t.
  state equations
• in Lagrangian form
• L(x,h,l) St (1r)-t pt ht
• St lt (xt - ht -xt1)

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L(x,h,l) St (1r)-t pt ht St lt (xt - ht
-xt1)

• Expand a little of this out
• p0 h0 p1 h1/(1r) l0 (x0 -h0 -x1) l1 (x1 -
  h1 - x2) ...
• don't talk about where we stop the summations.
  Note that xt - ht -xt1 0 works out fine.
  Since the stuff is valuable it will always zero.

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Now to Lh

• L(x,h,l) St (1r)-t pt ht St lt (xt - ht
  -xt1)
• either h0 or (1r)-tpt lt

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Lx 0

• L(x,h,l) St (1r)-t pt ht St lt (xt - ht
  -xt1)
• L p0 h0 p1 h1/(1r)
• l0 (x0 -h0 -x1) l1 (x1 - h1 - x2)
• lt - lt-1 0
• or the constraint is slack, which it never is.

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Hotellings rule

• Putting our two conditions together
• lt - lt-1 0
• (1r)-tpt lt
• (1r)-t-1pt-1 (1r)-tpt or
• pt1 (1r) pt
• whenever harvesting is positive

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Steady State Fish

• State equation
• xt1 xt f(xt) - ht.
• U(x,h) St (1r)-t pt ht
• here take p as constant
• L(x,h,l) St (1r)-t pt ht
• St lt (xt f(xt) - ht -xt1)

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Growth
xt1 xt f(xt) - ht.
Critical Depensation
f
Largest Stock
Max sustainable yield
x
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The r f rule

• L(x,h,l) St (1r)-t p ht
• St lt (xt f(xt) - ht -xt1)
• Lx lt(1f) - lt-1
• Lh (1r)-t p - lt
• Assuming equality (true in steady state)
• (1r)-t p (1f) (1r)-t1 p
• r f(x) The steady state stock.

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Steady State

• One can show that the full solution is
• x x, harvest down to x in one period and then
  set h f(x)
• x then set h f(x)

20
Growth
The truth is that fisheries that are nominally
regulated to produce MaxSustainableYield regularly
get pushed to the brink of extinction. There
is an interesting political problem here.
rf(x)
f
Max sustainable yield
x