Transcript Extensive Form - London School of Economics

Frank Cowell: Microeconomics
Revision Lecture
EC202
http://darp.lse.ac.uk/ec202
30th April 2009
Frank Cowell
Overview...
Revision lecture
Frank Cowell: Microeconomics
Styles of
question
How to see what
you need to do
Doing short
questions
Planning
answers
Doing long
questions
Objectives of the lecture
Frank Cowell: Microeconomics
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A look back at Term 1
Exam preparation
Reference materials used (1)
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Exam papers (and outline answers)
2005 1(a)
2006 1(a)
2006 1(d)
2007 1(d)
2008 5
Reference materials used (2)
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CfD presentations 3.3, 4.12, 7.8
All related to past exam questions
The exam paper
Frank Cowell: Microeconomics
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Scope of exam material
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Structure and format of paper
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what’s covered in the lectures…
… is definitive for the exam
follows that of last four years
check out the rubric from, last year’s paper
Mark scheme
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40 marks for question 1 (8 marks for each of the five parts)
20 marks for each of the other three questions
multipart questions: except where it’s obvious, roughly equal
marks across parts
Question style – three types
Frank Cowell: Microeconomics
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1 Principles
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2 Model solving
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a standard framework
you just turn the wheels
3 Model building
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reason on standard results and arguments
can use verbal and/or mathematical reasoning
usually get guidance in the question
longer question sometimes easier?
Examples
from past
question 1
One type not necessarily “easier” or “harder” than another
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part A (question 1) usually gets you to do both types 1 and 2
type 3 is usually only in parts B and C of paper
Overview...
Revision lecture
Frank Cowell: Microeconomics
Styles of
question
How to tackle the
main types of
question
Doing short
questions
Planning
answers
Doing long
questions
2006 1(d)
Frank Cowell: Microeconomics
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Straightforward
“principles” question
Just say what you
need to say
2005 1(a)
Frank Cowell: Microeconomics
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Straight “principles”
Note the contrast
between firm and
consumer
Be sure to give your
reasons
2006 1(a)
Frank Cowell: Microeconomics
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Principles again
But format of question
gives you a hint…
…write out
decomposition formula
Then read off results
2007 1(d)
Frank Cowell: Microeconomics
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(i)
o
o
(ii)
o
o
(iii)
o
o
o
Principles and modelsolving
Jot down the principle
(as note for self)
Write in the elements
of the model
Principle: vNM is of the form p0u(y0)+p1u(y1) (or transform of this)
Model: p0 is a/[a+b]; p1 is b/[a+b]: just need g=d to get the above form
Principle: c-e income x implicitly defined by u(x) = p0u(y0)+p1u(y1)
Model: plug y values into −x−g = −a y0−g −b y1−g
Principle: risk premium defined by Ey − x
Model: plug a,b,g,d values into answer to part (ii)
[Check the online solutions for full detail on this one]
2008 5
Frank Cowell: Microeconomics
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(a)
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Straight principles can come
up in long questions
Don’t ignore them in a rush
to get to the model!
Could be some easy marks…
State concept of PE
In absence of externalities
PE requires MRS=MRT
But monopolist forces p>MC
 Four Marks (Yay!!)
(b)
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See CfD 4.12 (below)
(c)
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See on-line answers
Overview...
Revision lecture
Frank Cowell: Microeconomics
Styles of
question
How to do well in
exams
Doing short
questions
Planning
answers
Doing long
questions
Planning Answers
Frank Cowell: Microeconomics
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What’s the point?
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See the big picture
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take a moment or two..
…make notes to yourself
what is the main point of the question?
and the subpoints?
balance out the answer
imagine that you’re drawing a picture
if pressed for time, don’t rush to put in extra detail…
…you can go back
Be an economist with your own time
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don’t solve things twice!
reuse results
answer the right number of questions!!!
Frank Cowell: Microeconomics
Tips
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Follow the leads
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Pix
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help you to see the solution
help you to explain your solution to examiner
What should the answer be?
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examiners may be on your side!
so if you’re pointed in the right direction, follow it…
take a moment before each part of the question
check the “shape” of the problem
use your intuition
Does it make sense?
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again take a moment to check after each part
we all make silly slips
Frank Cowell: Microeconomics
Long questions
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Let’s look at three examples
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Illustrates two types of question
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taken from exercises in the book
but of “exam type” difficulty
covered in CfD
Ex 3.3, 4.12 straight model solving
Ex 7.8 incorporates a little bit of model building
Look out for tips
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In all three questions, use pictures to clarify solution
following hints in 3.3 [The “Explain carefully…” bits]
Overview...
Revision lecture
Frank Cowell: Microeconomics
Styles of
question
A problem with
discontinuous
supply…
Doing short
questions
Planning
answers
Doing long
questions
•CfD 3.3
•CfD 4.12
•CfD 7.8
Ex 3.3(1) Question
Frank Cowell: Microeconomics
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purpose: to derive competitive supply function
method: derive AC, MC
Ex 3.3(1) Costs
Frank Cowell: Microeconomics
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Total cost is: F0 + ½ aqi2
Marginal cost: aqi
Average cost: F0/qi + ½ aqi
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Therefore MC intersects AC where:
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This is at output level q where:
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At this point AC is at a minimum p where:
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For q below q there is IRTS and vice versa
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Ex 3.3(1) Supply
Frank Cowell: Microeconomics
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If p > p the firm supplies an amount of output such that
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If p < p the firm supplies zero output
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otherwise the firm would make a loss
If p = p the firm is indifferent between supplying 0 or q
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p = MC
in either case firm makes zero profits
To summarise the supply curve consists of :
Ex 3.3(1): Supply by a single firm
Frank Cowell: Microeconomics
Average cost
p
Marginal cost
Supply of output
q
qi
Ex 3.3(2) Question
Frank Cowell: Microeconomics
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purpose: to demonstrate possible absence of equilibrium
method: examine discontinuity in supply relationship
Ex 3.3(2): Equilibrium?
Frank Cowell: Microeconomics
AC,MC and supply of firm
p
Demand, low value of b
Demand, med value of b
Demand, high value of b
Solution for high
value of b is where
Supply = Demand

AC

MC
qi
Ex 3.3(2) Equilibrium
Frank Cowell: Microeconomics
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Outcome for supply by a single price-taking firm
High demand: unique equilibrium on upper part of supply curve
2. Low demand: equilibrium with zero output
3. In between: no equilibrium
1.
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Given case 1 “Supply = Demand” implies
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This implies:
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But for case 1 we need p ≥ p
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from the above this implies
Ex 3.3(3) Question
Frank Cowell: Microeconomics
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purpose: to demonstrate effect of averaging
method: appeal to a continuity argument
Ex 3.3(3) Average supply, N firms
Frank Cowell: Microeconomics
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Define average output
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Set of possible values for
average output:
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Therefore the average supply
function is
Ex 3.3(3) Average supply, limit case
Frank Cowell: Microeconomics
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As N the set J(q) becomes dense in [0, q]
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So, in the limit, if p = p average output can take
any value in [0, q]
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Therefore the average supply function is
Ex 3.3(3): Average supply by N firms
Frank Cowell: Microeconomics
Average cost (for each firm)
Marginal cost (for each firm)
p
Supply of output for averaged
firms
q

q
Ex 3.3(4) Question
Frank Cowell: Microeconomics
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purpose: to find equilibrium in large-numbers case
method: re-examine small-numbers case
Ex 3.3(4) Equilibrium
Frank Cowell: Microeconomics
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Equilibrium depends on where demand curve is located
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High demand
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characterise in terms of (price, average output)
equilibrium is at (p, p/a) where p = aA / [a+b]
Medium demand
equilibrium is at (p, [A – p]/b)
 equivalent to (p, bq) where b := a[A – p] / [bp]
 Achieve this with a proportion b at q and 1–b at 0
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Low demand
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equilibrium is at (p, 0)
Ex 3.3(4): Eqm (medium demand)
Frank Cowell: Microeconomics
AC and MC (for each firm)
Supply of output (averaged)
Demand
p
Equilibrium
Equilibrium
achieved by
mixing firms at 0
and at q
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b here
1b here
q*
q

q
Ex 3.4: Points to remember
Frank Cowell: Microeconomics
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Model discontinuity carefully
Averaging may eliminate discontinuity problem in
a large economy
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depends whether individual agents are small.
Equilibrium in averaged model may involve
identical firms doing different things
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equilibrium depends on the right mixture
Overview...
Revision lecture
Frank Cowell: Microeconomics
Styles of
question
Close to 2008 5
Doing short
questions
Planning
answers
Doing long
questions
•CfD 3.3
•CfD 4.12
•CfD 7.8
Ex 4.12(1) Question
Frank Cowell: Microeconomics
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purpose: to derive solution and response functions for quasilinear
preferences
method: substitution of budget constraint into utility function and then
simple maximisation
Ex 4.12(1) Preliminary
Frank Cowell: Microeconomics
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First steps are as follows:
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Sketch indifference curves
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Write down budget constraint
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Straightforward – parabolic contours
Straightforward – fixed-income case
Set out optimisation problem
Ex 4.12(1) Indifference curves
Frank Cowell: Microeconomics
x2
Slope is
vertical here
Could have
x2 = 0
x1
0
0
1
2
Ex 4.12(1) Budget constraint, FOC
Frank Cowell: Microeconomics
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Budget constraint:
Substitute this into the utility
function:
We get the objective function:
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FOC for an interior solution:
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Ex 4.12(1) Using the FOC
Frank Cowell: Microeconomics
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Remember that person might consume zero of commodity 2
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consider two cases
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Case 1: x2* > 0
From the FOC:
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But, to make sense this case requires:
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Case 2: x2* = 0
We get x1* from the budget constraint
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x1* = y / p1
Ex 4.12(1) Demand functions
Frank Cowell: Microeconomics
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We can summarise the optimal demands for
the two goods thus
Ex 4.12(1) Indirect utility function
Frank Cowell: Microeconomics
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Get maximised utility by substituting x* into the utility
function
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V(p1, p2, y) = U(x1*, x2*)
= U(D1(p1, p2, y), D2(p1, p2, y))
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Case 1: p1 >`p1
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Case 2: p1 ≤`p1
Ex 4.12(1) Cost function
Frank Cowell: Microeconomics
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Get cost function (expenditure function) from the indirect
utility function
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maximised utility is u = V(p1, p2, y)
invert this to get y = C(p1, p2, u)
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Case 1: p1 >`p1
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Case 2: p1 ≤`p1
Ex 4.12(2) Question
Frank Cowell: Microeconomics
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purpose: to derive standard welfare concept
method: use part 1 and manipulate the indirect utility function
Ex 4.12(2) Compute CV
Frank Cowell: Microeconomics
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Get compensating variation (1) from indirect utility function
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before price change: u = V(p1, p2, y)
after price change: u = V(p1', p2, y − CV)
Equivalently (2) could use cost function directly
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CV = C(p1, p2, u) − C(p1', p2, u)
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In Case 1 above we have
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Rearranging, we find:
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Equivalently
Ex 4.12(3)
Frank Cowell: Microeconomics
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In case 1 we have x1* = [½ a p2 / p1]2
So demand for good 1 has zero income effect
Therefore, in this case CV = CS = EV
Ex 4.12: Points to remember
Frank Cowell: Microeconomics
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It’s always a good idea to sketch the indifference
curves
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in this case the sketch is revealing…
…because of the possible corner solution
A corner solution can sometimes just be handled
as two separate cases
There’s often more than one way of getting to a
solution
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in this case two equivalent derivations of CV
Overview...
Revision lecture
Frank Cowell: Microeconomics
Styles of
question
A standard GE
exercise… with a
little bit of a twist
Doing short
questions
Planning
answers
Doing long
questions
•CfD 3.3
•CfD 4.12
•CfD 7.8
Ex 7.8: Question
Frank Cowell: Microeconomics
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purpose: to show how to find equilibrium allocation in a GE model
method: standard construction and solution of excess demand functions.
Ex 7.8: approach
Frank Cowell: Microeconomics
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Step 1: model behaviour of each type as a price taker
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Step 2: get excess demand function for one of the goods
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Write down budget constraint for the unknown price p
Set up Lagrangean for each type
Find the FOCs
Get demand functions from the FOCs
Use the demand functions for each type from step 1
Other EDF follows by Walras’ law
Step 3: find equilibrium price(s) as root(s) of EDF
Ex 7.8: type-a problem
Frank Cowell: Microeconomics
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The endowment for type a is R1
Let price of good 1 in terms of good 2 be p
The income of type a is then pR1
The utility function is:
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So the Lagrangean of type a is:
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Ex 7.8: type-a demand
Frank Cowell: Microeconomics
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Given the Lagrangean for a:
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FOCs for interior maximum:
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Rearrange and use the budget
constraint:
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Demand by a for good 2:
Ex 7.8: type-b problem
Frank Cowell: Microeconomics
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The endowment for type b is R2
Recall that values are measured in terms of good 2
So the income of type b is just R2
The utility function is:
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So the Lagrangean of type b is:
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Ex 7.8: type-b demand
Frank Cowell: Microeconomics
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Given the Lagrangean for b:
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FOCs for interior maximum:
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Rearrange and use the budget
constraint:
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Demand by b for good 2:
Ex 7.8: excess demand
Frank Cowell: Microeconomics
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Demand by the two types
for good 2:
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Excess demand for good 2
is defined as x21 + x22  R2
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So the excess demand
function for good 2 is:
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Letting q:= 2R1/R2 excess
demand is zero where
Ex 7.8: how many equilibria?
Frank Cowell: Microeconomics
Graph of p2/3
pq  1
p2/3
Graph of pq  1
Equilibrium
 Excess demand is zero
where p2/3 = pq  1
p*
p
 There is clearly only one
equilibrium p*
Ex 7.8: the equilibrium
Frank Cowell: Microeconomics
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To find the equilibrium we need the resource values
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Equilibrium price must satisfy p2/3 = (5/8) p  1
Use trial and error to find solution
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R1 = 5
R2 = 16
So q := 2R1/R2 = 5/8
check whether there is excess demand/supply at certain prices
try easy numbers that have integer cube roots: p = 1? 8? 27? …
Clearly p = 1 is too low and p = 27 is too high
Try p = 8
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LHS: p2/3 = 4
RHS: (5/8) p  1 = 4
so this is the equilibrium
Ex 7.8: Points to note
Frank Cowell: Microeconomics
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Step by step approach gets you very close to the solution
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work out individual demands
set excess demand to zero
get a condition to determine equilibrium price
Graphical intuition helps you get the form of the solution
Don’t get fazed by awkward numbers
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trial-and-error method quickly gives you the answer