Transcript Extensive Form - London School of Economics

Frank Cowell: Microeconomics
Revision Lecture
EC202
http://darp.lse.ac.uk/ec202
29th April 2010
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 4
 2008 1(b)
 2009 1(c)
Reference materials used (2)
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CfD presentations 2.9, 3.3, 8.12
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
2009 1(c)
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
2008 1(b)
Frank Cowell: Microeconomics
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Principles and model-solving
Write down the principle
Write down the basics of the model
WARP can be stated simply in terms of “affordability”
To check whether week 2’s bundle can be afforded at week 1’s
prices (etc. etc.) we need to write down the costs
Check the on-line answers for the (short) detailed reasoning…
2006 4
Frank Cowell: Microeconomics
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Straight principles can come up in long questions
Don’t ignore them in a rush to get to the model!
Compare this with CfD 8.12
CfD (from book) doesn’t have this bit, but take it seriously
There are some easy marks just writing down the definition…
…and the diagram helps you to answer part (b)
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 2.9 and 3.3 are straight model solving
Ex 8.12 incorporates some 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
How to combine
principles and
model-solving…
Doing short
questions
Planning
answers
Doing long
questions
•CfD 2.9
•CfD 3.3
•CfD 9.12
Ex 2.9(1): Question
Frank Cowell: Microeconomics
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purpose: demonstrate relationship between short and long run
method: Lagrangean approach to cost minimisation. First part can be
solved by a “trick”
Ex 2.9(1): Long-run costs
Frank Cowell: Microeconomics
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Production function is homogeneous of degree 1
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CRTS implies constant average cost
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increase all inputs by a factor t > 0 (i.e. z → tz)…
…and output increases by the same factor (i.e. q → tq)
constant returns to scale in the long run
C(w, q) / q = A (a constant)
so C(w, q) = Aq
differentiating: Cq(w, q) = A
So LRMC = LRAC = constant
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Their graphs will be an identical straight line
Ex 2.9(2): Question
Frank Cowell: Microeconomics
method:
 Standard Lagrangean approach
Ex 2.9(2): short-run Lagrangean
Frank Cowell: Microeconomics
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In the short run amount of good 3 is fixed
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z3 = `z3
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Could write the Lagrangean as
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But it is more convenient to transform the problem thus
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where
Ex 2.9(2): Isoquants
Frank Cowell: Microeconomics
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Sketch the isoquant map
z2
z1
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Isoquants do not touch the axes
So maximum problem must have an interior solution
Ex 2.9(2): short-run FOCs
Frank Cowell: Microeconomics
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Differentiating Lagrangean, the FOCS are
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This implies
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To find conditional demand function must solve for l
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use the above equations…
…and the production function
Ex 2.9(2): short-run FOCs (more)
Frank Cowell: Microeconomics
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Using FOCs and the production function:
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This implies
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where
This will give us the short-run cost function
Ex 2.9(2): short-run costs
Frank Cowell: Microeconomics
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By definition, shortrun costs are:
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This becomes
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Substituting for k:
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From this we get
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SRAC:
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SRMC:
Ex 2.9(2): short-run MC and AC
Frank Cowell: Microeconomics
marginal
cost
average
cost
q
Ex 2.9(3): Question
Frank Cowell: Microeconomics
method:
 Draw the standard supply-curve diagram
 Manipulate the relationship p = MC
Ex 2.9(3): short-run supply curve
Frank Cowell: Microeconomics
average cost curve
marginal cost curve
minimum average cost
p
supply curve
p
q
q
Ex 2.9(3): short-run supply elasticity
Frank Cowell: Microeconomics
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Use the expression for marginal cost:
Set p = MC for p ≥ p
Rearrange to get supply curve
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Differentiate last line to get supply elasticity
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Ex 2.9: Points to remember
Frank Cowell: Microeconomics
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Exploit CRTS to give you easy results
Try transforming the Lagrangean to make it easier
to manipulate
Use MC curve to derive supply curve
Overview...
Revision lecture
Frank Cowell: Microeconomics
Styles of
question
A problem with
discontinuous
supply…
Doing short
questions
Planning
answers
Doing long
questions
•CfD 2.9
•CfD 3.3
•CfD 8.12
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

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
Modelling choice
under uncertainy
Doing short
questions
Planning
answers
Doing long
questions
•CfD 2.9
•CfD 3.3
•CfD 8.12
Ex 8.12(1): Question
Frank Cowell: Microeconomics
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purpose: to develop an analysis of insurance where terms are less
than actuarially fair
method: model payoffs in each state-of-the-world under different
degrees of coverage. Find optimal insurance coverage. Show how
this responds to changes in wealth
Ex 8.12(1): model
Frank Cowell: Microeconomics
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Use the two-state model (no-loss, loss)
Consider the person’s wealth in extremes
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Suppose partial insurance is possible
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if uninsured: (y0, y0  L)
if fully insured: (y0  κ, y0  κ)
if person insures a proportion t of loss L…
…pro-rata premium is tκ
So if a proportion t is insured wealth is
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([1  t]y0 + t [y0  κ], [1  t][y0  L] + t [y0 κ])
which becomes (y0  tκ, y0  tκ + [1  t]L)
Ex 8.12(1): utility
Frank Cowell: Microeconomics
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Put payoffs (y0  tκ, y0  tκ + [1  t]L) into the
utility function
Expected utility is
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Therefore effect on utility of changing coverage is
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Could there be an optimum at t =1?
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Ex 8.12(1): full insurance?
Frank Cowell: Microeconomics
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What happens in the neighbourhood of t = 1?
We get
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Simplifying, this becomes [Lπ  κ] uy(y0  κ)
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positive MU of wealth implies uy(y0  κ) > 0
by assumption Lπ <κ
so [Lπ  κ] uy(y0  κ) < 0
In the neighbourhood of t =1 the individual could increase
expected utility by decreasing t
Therefore will not buy full insurance
Ex 8.12(2): Question
Frank Cowell: Microeconomics
Method
 Standard optimisation
 Differentiate expected utility with respect to t
Ex 8.12(2): optimum
Frank Cowell: Microeconomics
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For an interior maximum we have
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Evaluating this we get
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So the optimal t∗ is the solution to this equation
Ex 8.12(3): Question
Frank Cowell: Microeconomics
Method
 Take t* as a function of the parameter y0
 This function satisfies the FOC
 So to get impact of y0:
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Differentiate the FOC w.r.t. y0
Rearrange to get t* / y0
Ex 8.12(3): response of
*
t
to y0
Frank Cowell: Microeconomics
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Differentiate the following with respect to y0:
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This yields:
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On rearranging we get:
Ex 8.12(3): implications for coverage
Frank Cowell: Microeconomics
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Response of t* to y0 is given by
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The denominator of this must be negative:
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uyy(⋅) is negative
all the other terms are positive
The numerator is positive if DARA holds
Therefore ∂t*/∂y0 < 0
So, given DARA, an increase in wealth reduces the
demand for insurance
Ex 8.12: Points to remember
Frank Cowell: Microeconomics
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Identify the payoffs in each state of the world
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Set up the maximand
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ex-post wealth under…
…alternative assumptions about insurance coverage
expected utility
Derive FOC
Check for interior solution
Get comparative static effects from FOCs