Lecture nine

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Lecture nine

• Economic applications of matrix algebra

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Plan of the session

• Using matrix algebra to find the equilibrium
  (endogenous) values of micro and macro-economic
  models (as a function of exogenous variables)
• Using matrix algebra in comparative statics
  analysis i.e. showing how the change in
  exogenous variables changes the equilibrium
• Using matrix algebra in optimization (typically
  micro) analysis
• Reference Chapters 5.3. and 7.3. of Ian Jacques

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Equilibrium in the simple one commodity market
model

• The one commodity market model
• Example The linear supply and demand equations
  for a one-commodity market model are given by
• P aQsb
• P-cQdd
• With the use of matrix algebra, find the
  equilibrium values of P and Q.

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Equilibrium in the simple one commodity market
model cntnd.

• Solution

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The idea behind a comparative statics analysis
Q
Q
b increases
a increases
P
P
P
P
P
Q
c increases
P
Q
d increases
P
P
P
P
P
P
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The national income accounting model

• Fundamental national income accounting identity
• YCIGNX
• Since in equilibrium ADY ADYCIGNX
• But

AD
YAD
Y
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The national income accounting model cntnd.

• Extension one Taxes and transfers

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The national income accounting model cntnd.

• Extension two The goods market (IS schedule)

i
i1
IS
i2
Y
AD
Y
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The national income accounting model cntnd.

• Extension three The money market (LM schedule)

i
LM
i
i2
L2
i1
L1
Y
Y1
Y2
Y
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The national income accounting model cntnd.

• Putting the goods and money market together..

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Example of finding an equilibrium in the national
income accounting model

• Let
• Find the equilibrium values of the endogenous
  variables.

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Example of finding an equilibrium in the national
income accounting model cntnd.

• Solution

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Further examples (to be worked out in class)

• Example one Find the equilibrium values in the
  following two-commodity market model

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Further examples (to be worked out in class)

• Example two Find the value of the interest rate
  in the following national income accounting
  model

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Further examples (to be worked out in class)

• Example three Find the equilibrium value of Y in
  the following national income accounting model

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Comparative statics analysis

• When we studied the production function Qf(K,L),
  we found out that MPK?Q/ ?K and MPL ?Q/ ?L. But
  we also found that
• ?K / ?L-MPK/MPL has a meaning of its own.
  We can express these relationships as F(Q,K,L)
  and ?K / ?L-Fk/FL

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Comparative statics analysis continued

• In general if we have a simultaneous system of
  equations
• where

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Comparative statics analysis continued

• We can totally differentiate them to find

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Comparative statics analysis continued

• Substituting (2) into (1) and dividing by dx1

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Comparative statics analysis continued

• Which we can rearrange in matrix form as

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Example one

• Example 1 The market model

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Example one continued

• Solution

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Example two (to be worked out in class)

• Example 2 The national income model Find the
  impact of exogenous change in government
  expenditures on output with the use of matrix
  algebra

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Using matrix algebra in problems of unconstrained
optimization

• When we talked about optimization of functions
  with several variables we saw that a sufficient
  condition for maximization (minimization) was
  that d2zfxxdx22fxydxdyfyydy2lt (gt) 0
• We used mechanically the condition of
  fxxfyygtfxy2 together with fxxlt0 and fyylt0 for a
  maximum and fxxgt0 and fyygt0 for a minimum.
• Matrix algebra and in particular, the so
  called Hessian matrix can help us understand
  where these come from and give us a more compact
  way of determining whether a function reaches a
  minimum or a maximum.

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Using matrix algebra in problems of unconstrained
optimization

• We can express d2zfxxdx22fxydxdyfyydy2 as
  qau22huvbv2
• By adding and subtracting h2v2/a we obtain
• qau22huv h2v2/a bv2- h2v2/a
  a(u22huv/ah2v2/a2)(b-h2/a)v2
• a(u(h/a)v)2(ab-h2)/av2
• Given a ab-h2gt0, the only condition for a
  positive q would be that agt0 and the only
  condition for a negative q would be that alt0.
• But this is really the condition that we
  discussed earlier!

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Using matrix algebra in problems of unconstrained
optimization

• In a matrix form we can define the Hessian
  determinant (or simply Hessian as a matrix of the
  second partial derivatives of a system of
  equations
• We have a maximum if fxxlt0 and Hgt0 and a minimum
  if fxxgt0 and Hgt0

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Using matrix algebra in problems of unconstrained
optimization

• For the case of more than two equations, using
  exactly the same exercise we can show that for
• The condition for a maximum is
• The condition for a minimum is

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Example

• If the demand functions in a monopolistic market
  are
• P155-Q1-Q2 and P270-Q1-2Q2 and the cost
  function is CQ12Q1Q2Q22, find whether the
  profit function reaches a minimum or a maximum.

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Solution

• Problem 2 We saw previously that our first order
  conditions for the two profit functions are
• ?155-3Q2-4Q1
• ?270-3Q1-6Q2
• This gives second order conditions
• ?11-4, ?12 ?21 -3, ?22-6
• The Hessian is therefore

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Using matrix algebra in problems of constrained
optimization

• Let qau22huvbv2 be subject to ?u?v0
• Since from the constraint we get v(- ? /
  ?)u, we can rewrite
• q(a ? 2-2h ? ? b ? 2)u2/ ? 2
• q would be positive (negative) if the
  expression in brackets is positive (negative).
  But this is equivalent to

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Using matrix algebra in problems of constrained
optimization

• In other words, if our bordered Hessian is
  positive we have a maximum. If the bordered
  Hessian is negative, we have a minimum.
• Example Ux1x2, subject to x1x2/(1r)B