Lecture 18. Comparative Statics

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Lecture 18. Comparative Statics

• Learning objectives. By the end of this lecture
  you should
• Understand some basic ideas about comparative
  statics.
• Introduction. In most economic problems we have
  exogenous variables
• E.g. prices and income for the consumer
• E.g. wages and prices for the competitive firm
• And endogenous variables.
• E.g. demand variables, x, for the consumer
• Supply and input demand for the firm.
• Exogenous variables are often call parameters.
• Often we want to know how the endogenous
  variables change as the exogenous variables alter
• E.g. how does demand for oranges change if the
  price of apples rises?
• The technique for finding the answer is called
  comparative statics
• It is closely related to the second order
  conditions we have been studying.
• Note that often we dont need a specific number
  for the comparative static, just a direction e.g.
  orange demand rises if the apple price rises

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Demand and supply example.

• Suppose we are interested in two variables x
  (equilibrium output) and p (equilibrium price).
  We have two equations that determine the
  variables
• Demand x 0.5m/p
• Supply p 4 2x
• How does a marginal change in income (m) alter
  the equilibrium price?
• A rise in m raises demand, x
• Thus prices must rise to meet this demand
• But then x must fall because prices have risen.
• But then prices will fall
• So what is the net effect on x and p?
• To put it another way. Suppose we ran a
  regression of
• P abM.
• What would we expect for the sign of b?
• To find the answer, we need to implicitly
  differentiate this system of equations. First we
  rewrite the two equations in the right kind of
  format. Here m is our only exogenous variable. X
  and p are endogenous and we have two equations

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2. The implicit function theorem.

• An explicit function x f(b). So x is
  explicitly a function of b. To find dx/db we
  differentiate this function by b.
• An implicit function f(x,b) 0. So x is
  implicitly a function of b. To find dx/db we
  differentiate this function by b
• If x and b are single variables, then dx/db will
  exist provided
• exist and
• This proposition is known as the implicit
  function theorem. This is what we have been
  (implicitly) using.

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3. The implicit function theorem x is a vector.

• Let f(x,b) 0 where x (x1,,xn) and 0 is a
  vector of n zeros. So x is implicitly a function
  of b. We write this out in detail in vector form
  as
• Note that in our examples up until now the fis
  have been the derivatives of f with respect to
  xi. They dont have to be derivatives. E.g. The
  equations for a consumers demand for x1 and x2
• To find dxi/db we differentiate the function by
  b

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3. The implicit function theorem x is a vector.

• Rearrange
• This first matrix is known as the Jacobean (J).
• So, for dxi/db to exist we require,
• fi to be differentiable with respect to the xis
• J to be non-singular,
• Then,
• Or alternatively, use Cramers rule

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4. Example.

• Suppose
• Then
• Exercise find dx1/db

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5. Demand and supply example again.

• Suppose we are interested in two variables x
  (equilibrium output) and p (equilibrium price).
  We have two equations that determine the
  variables
• Demand x 0.5m/p
• Supply p 4 2x
• To find the answer, we need to implicitly
  differentiate this system of equations. First we
  rewrite the two equations in the right kind of
  format. Here m is our only exogenous variable. X
  and p are endogenous and we have two equations

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5. Demand and supply example.

• Implicitly differentiate with respect to m
• That is,
• Or

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5. Demand and supply example.

• Invert the Jacobian
• So,
• yields

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5. Interpreting the demand and supply example.

• Our solution (after some tidying up)
• Everything is positive, so dx/dm gt 0 and dp/dm gt
  0. i.e. equilibrium output and price rise.
• In our regression p a bM, we would expect b
  to be positive.

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6. Comparative statics when there is maximization.

• A competitive firm faces a market price p.
• It has total costs of 2x2 where x output.
• What is the optimal output?
• Answer
• Profits p px 2x2
• Differentiate to get first order condition dp/dx
  p 4x 0
• So x p/4 an explicit solution for x.
• How does output change as the price alters?
• In this question this is easy because we have the
  explicit solution for x
• dx/dp 1/4

p
x
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6. Comparative statics when there is maximization.

• A competitive firm faces a market price p.
• It has total costs of c(x) where x output.
  c(x) gt 0 and c(x) gt 0.
• What is the optimal output in this more general
  case?
• Answer
• Profits p px c(x)
• Differentiate to get first order condition dp/dx
  p c(x) 0
• So x is given implicitly by the solution to this
  equation.
• How does output change as the price alters?
• This is harder, but we know
• Note that weve written p as a function of p and
  x, but x depends on p.
• If we totally differentiate the first order
  condition

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6.

• Rearrange
• First note that if the first order condition
  represents a maximum,
• So will have the same sign as
• Here
• So as long as the second order condition is
  satisfied, output rises as prices rise i.e. the
  supply curve is upward sloping.
• The sign of a variable or function is whether
  it is positive () or negative (-). Its
  shorthand for whether the value of the function
  is greater than zero or less than zero.

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6. continued..

• Lets just check that the second order condition
  holds
• given our assumption about c.
• Question c gt 0 is equivalent to
• Increasing returns
• Constant returns
• Decreasing returns?

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7. Maximization with respect to one variable.

• If x is just a single variable, then
• If the first order conditions represent a
  maximum, then the denominator must be negative so
  the sign of df/db is the same as the sign of
• This is often written
• Remember that this right hand term is the
  derivative of the first order condition with
  respect to b.

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8. Comparative statics with maximization
general points.

• More generally. Suppose we maximize f(x,b) with
  respect to x, where b is a parameter. Then our
  starting point for the comparative statics is the
  set of first order conditions
• It follows that the Jacobean is the Hessian
  matrix for f. In other words
• Typically knowing this gives us additional
  information about the sign of the comparative
  statics.

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9. Summary.

• In todays lecture you learnt
• How to do comparative statics with a functions of
  one and two variables.
• Next more on comparative statics in the context
  of optimization.