Plan for next 3 lecture slots

1
Plan for next 3 lecture slots

• Today
• Experiment
• Lecture 19
• Monday
• Any remaining lecture material
• Go over last years test (please have a look at
  it before Monday)
• Next Thursday
• Test 2
• 5 questions covering all the material since the
  last test
• Remember to revise your definitions

2
The Experiment

• Taking part is voluntary.
• To take part,
• Read the instructions then
• answers ALL the questions below,
• then remove your lottery ticket and give the
  completed questionnaire to the experimenter.
  Remember to hang on to your lottery ticket.
• The numbers collected will go into a draw and 5
  winners will be picked at random from the people
  here today.
• All the winners who have completed all the
  questions will win prizes.
• You will face a series of SIX different tasks.
  Some tasks are CHOICE questions. Some tasks are
  SWAP questions.
• If you win, then one of these tasks will be
  picked at random and your answer will determine
  your prize.

3
Lecture 19. More comparative statics the
implicit function theorem.

• Learning objectives. By the end of this lecture
  you should
• Know more about how to conduct a comparative
  static exercise with 2 or 3 variables.
• Realise that when we are dealing with the
  solution to a maximization problem, the second
  order conditions play a crucial role in being
  able to sign the effect of a change in parameters
• Introduction Reminder.
• In the last lecture we learnt how to do
  comparative statics with equation systems of the
  form
• Today we problems when the equations fi represent
  first order conditions from a maximum.
• When thats the case, the Jacobean, J, is
  actually the Hessian.

4
3. Example

• Let
• How do the optimal value of x1, x2 and x3 change
  with b?
• The first order conditions are
• (We could solve to get a direct expression for b,
  but thats not the point of the exercise)

5
3. Example

• So
• Notice this implies that b gt 9 if f is to be
  strictly concave.
• And

6
6. Example

• Use Cramers rule
• So,

7
2. f(x,b) where x and b are vectors

• In other words f f(x1,,xn, b1,bn)
• Example In a consumer choice problem with two
  goods, x (x1 x2), while b (m p1 p2)
• Example 2 a firm chooses capital, K and labour L
  to maximize profits, taking wages, w, capital
  price, r and the price of output, p as given.
• In this example profits py wL rk where y
  f(l,k).
• f is the production function f1 means the partial
  derivative of f with respect to l and so on. Wed
  expect f1 gt 0 and f2 gt0 and diminishing returns
  to a factor, i.e. f11 lt 0 and f22 lt 0. Lets
  suppose f l0.5k0.5
• The first order conditions are
• The matrix of second derivatives is
• So for the first order conditions to represent
  the maximum we must have,

8
2. Example where x and b are vectors

• If we change the price, possibly all the input
  variables will also change. So we get something
  like
• We can reformat this system of equations into
  matrices
• Invert H
• Since f12 f21 0, then we get

9
3. Exercise

• Use the information we have about H and f to sign
  dl/dp and dk/dp.
• Hence delete the wrong words
• As prices rise, the demand for labour
  rises/falls.
• As prices rise, the demand for capital
  rises/falls.

10
4. General case f(x,b) where x is a vectors

• In other words f f(x1,,xn, b)
• Recall that the first order conditions are
• If we change the parameters by db, possibly all
  the x variables will also change. We get
  something like

Indirect
direct
11
4. General case f(x,b) where x is a vector

• Put into matrix format
• Solve by inverting H
• Or find an individual dxi/db using Cramers rule
  e.g.

12
5. Quiz

• Suppose we are at a maximum and n 4. What is
  the sign of ?
• Write down the Cramers rule expression for
  finding dxn/db1

13
9. Summing up comparative statics.

• Its a useful method for finding something about
  the relationship between x and parameters b.
• Especially handy when (as is often the case) we
  cannot find an explicit solution for x as a
  function of b.
• Provided the conditions of the implicit function
  are satisfied,
• Distinguishing between J and H.
• The approaches look very similar.
• H is the matrix of second derivatives of f(x,b)
  with respect to x.
• Essentially the fact that we are at a maximum
  tells us something about the sign of which makes
  it easier to sign dxi/db.
• In other words, the Hessian case is a special
  case of the more general problem of comparative
  statics.
• The End of QM II part 1.