Economics D101: Lecture 12

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Economics D10-1 Lecture 12

• Cost minimization and the cost function

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The minimum cost function is the workhorse of the
theory of the firm

• The cost function expresses the minimized costs
  of the firm as a function of output level(s) and
  input prices.
• The cost function is exactly analogous to the
  expenditure function of consumer theory.
• The cost function can also be viewed as a
  restricted profit function, with the level of
  output taken as parametric.
• The properties of the cost function -e.g., linear
  homogeneity and concavity in factor prices- can
  be developed using the algebraic approach,
  duality, or the Neoclassical approach.

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Cost minimization the algebraic approach

• The cost minimization problem produce a
  specified output level with the least expenditure
  on inputs.
• C(q,w) minz?V(q)w?z
• Or, in the single output case,C(q,w)
  minz0w?z f(z) q
• C is linearly homogeneous in factor pricesPf
  C(q,tw) minz?V(q)tw?z tminz?V(q)w?z tC(q,w)

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Solutions to this problem are conditional factor
demands

• z(q,w) argminz0w?z f(z) q
• Law of Downward-Sloping Conditional Factor Demand
• Let z0?z(q,w0) and z1?z(q,w1). Then,
• ?z?w (z1-z0)(w1-w0) (w1z1 - w1z0) (w0z0 -
  w0z1) 0
• Concavity of costs in factor prices
• Let wt tw0 (1-t)w1 and zt?z(q,wt)
• C(q,wt)wtzttw0zt(1-t)w1zttw0z0(1-t)w1z1tC(q,
  w0)(1-t)C(q,w1)
• Concavity of f (convexity of Y) implies C convex
  in q
• Let z0?z(q0,w), z1?z(q1,w), qt tq0 (1-t)q1,
  zt?z(qt,w), and z tz0(1-t)z1. Note that, by
  concavity, f(z) f(zt). Therefore,
• C(qt,w) w?zt w?z tw?z0 (1-t)w?z1
  tC(q0,w) (1-t)C(q1,w)

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Cost minimization the dual approach

• The dual approach to cost minimization derives
  differential comparative statics results using a
  Mirrlees construction.
• The derivative property (Shephards Lemma)
• Let z0 z(q,w0) and define the function
  g(w)C(q,w)-wz0
• Clearly, g(w) 0 and g(w0) 0, so that g is
  maximized at w0. Therefore, if g is
  differentiable at w0
• Dg(w0) DwC(q,w0) - z0 0 and DwC(q,w0) z0
  z(q,w0)
• Law of Downward-sloping conditional factor demand
• If g is twice differentiable at w0, its Hessian
  matrix D2g(w0)D2wC(q,w0)Dwz(q,w0) is negative
  semi-definite there.
• Thus, all diagonal elements ?zi/?wi are non
  positive.

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Cost minimization the Neoclassical approach

• Assume that f?C2. The LaGrangian expression for
  the cost minimization problem is
• L wz - ?(f(z)-q)
• with FONCs of
• Li wi - ?fi 0 zi 0 ziLi 0
• L ? f(z) - q 0 ? 0 ?L? 0
• Assume z(q,w) is differentiable and strictly
  positive. By ET
• ?L/?q ?C/?q ?(q,w)
• Total differentiation of the FONCs yields

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The Neoclassical approach two input example
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The restricted (short-run) cost function

• Let f ?m???? .
• variable cost function
• Cv(q,k,w) minzwz f(z,k)q
• zv(q,k,w)argminzwz f(z,k)q
• short-run cost function
• Cs(q,k,w,r) Cv(q,k,w)rk.
• long-run cost function
• C(q,w,r)mink0Cs(q,k,w,r)
• k(q,w,r)argmink0Cs(q,k,w,r)
• z(q,w,r)zv(q,k(q,w,r),w)

• Breaking down an optimization problem into 2 or
  more stages
• Makes it possible to focus on variables of
  interest
• Short-run - long-run distinction traditional in
  economics, but time really plays no role
• Short-run and variable cost formulations can be
  used to focus on any decision variable

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SRMC LRMC(Jacob Viners Envelope Theorem)

• Long-run cost equals short-run at the plant size
  efficient for that output level.
• By the Envelope Theorem, SRMCLRMC there.
• Intermediate Micro diagrams also show SRMC curves
  to be steeper than LRMC where they intersect
• This fact --that value functions are less steep
  in the long-run-- is an example of the Le
  Chatlier Principle

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The Le Chatlier Principle IValue Functions

• Partition the decision variables into fixed k
  and variable x
• Define the value function for the sub
  optimization problem
• Set fixed variables to a level optimal for some
  arbitrary (interior) parameter value a0
• Construct a Mirrlees function equal to the
  difference between short run and long-run value
  functions
• Le Chatlier Principle follows from SONC

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Le Chatlier Principle IIDecision Variables

• Vector of parameters and the objective functions
  takes the special linear form.
• Again, use Mirrlees construction.
• FONC establish equality of SR and LR value
  functions
• Envelope Theorem and linearity establish equality
  between SR and LR decision variables
• SONCs establish responsiveness result