Transcript Extensive Form - London School of Economics

Frank Cowell: Microeconomics
April 2007
Revision Lecture
EC202: Microeconomic Principles II
Frank Cowell
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)
2003 1(c)
2004 1(c)
2005 1(a)
2006 1(a)
Reference materials used (2)
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CfD presentations 2.9 9.6
Both related to past exam questions
Principles
Frank Cowell: Microeconomics
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Scope of exam material
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Resit
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a separate paper for anyone doing this second time around
Structure and format of paper
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what’s covered in the lectures…
… is definitive for the exam
follows that of last two years
check out the rubric from, say 2005 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 usually only in parts B and C of paper
2004 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
2003 1(c)
Frank Cowell: Microeconomics
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A model-solving
question
(i) just set E(r) = 0
and twiddle
(ii) check what
happens to E if you
change r
(iii) draw diagram
and reason
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 two 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 is mainly model solving
Ex 9.6 incorporates model building
Look out for tips
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Use of pictures in both questions
following hints in 9.6
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
Ex 9.6(1): Question
Frank Cowell: Microeconomics
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purpose: to derive equilibrium prices and incomes as a function of
endowment. To show the limits to redistribution within the GE model for
a alternative SWFs
method: find price-taking optimising demands for each of the two types,
use these to compute the excess demand function and solve for r
Ex 9.6(1): budget constraints
Frank Cowell: Microeconomics
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Use commodity 2 as numéraire
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Evaluate incomes for the two types, given their resources:
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price of good 1 is r
price of good 2 is 1
type a has endowment (30, k)
therefore ya = 30r + k
type b has endowment (60, 210  k)
therefore yb = 60r + [210  k]
Budget constraints for the two types are therefore:
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rx1a + x2a ≤ 30r + k
rx1b + x2b ≤ 60r + [210  k]
Ex 9.6(1): optimisation
Frank Cowell: Microeconomics
Jump to
“equilibrium
price”
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We could jump straight to a solution
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Cobb-Douglas preferences imply
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utility functions are simple…
…so we can draw on known results
indifference curves do not touch the origin…
…so we need consider only interior solutions
also demand functions for the two commodities exhibit constant
expenditure shares
In this case (for type a)
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coefficients of Cobb-Douglas are 2 and 1
so expenditure shares are ⅔ and ⅓
(and for b they will be ⅓ and ⅔ )
gives the optimal demands immediately…
Ex 9.6(1): optimisation, type a
Frank Cowell: Microeconomics
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The Lagrangean is:
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FOC for an interior solution
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2log x1a + log x2a + na[ya  rx1a  x2a ]
where na is the Lagrange multiplier
and ya is 30r + k
2/x1a  nar = 0
1/x2a  na = 0
ya  rx1a  x2a= 0
Eliminating na from these three equations, demands are
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x1a = ⅔ ya / r
x2a = ⅓ ya
Ex 9.6(1): optimisation, type b
Frank Cowell: Microeconomics
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The Lagrangean is:
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FOC for an interior solution
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log x1b + 2log x2b + nb[yb  rx1b  x2b ]
where nb is the Lagrange multiplier
and yb is 60r + 210  k
1/x1b  nbr = 0
2/x2b  nb = 0
yb  rx1b  x2b= 0
Eliminating nb from these three equations, demands are
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x1b = ⅓ yb / r
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x2b = ⅔yb
Ex 9.6(1): equilibrium price
Frank Cowell: Microeconomics
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Take demand equations for the two types
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substitute in the values for income
type-a demand becomes
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type-b demand becomes
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Excess demand for commodity 2:
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[10r + ⅓k]+[40r +140 − ⅔k] − 210
which simplifies to 50r − ⅓k − 70
Set excess demand to 0 for equilibrium:
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equilibrium price must be:
r = [210 + k] / 150
Ex 9.6(2): Question and solution
Frank Cowell: Microeconomics
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Incomes for the two types are resources:
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The equilibrium price is:
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r = [210 + k] / 150
So we can solve for incomes as:
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ya = 30r + k
yb = 60r + [210  k]
ya = [210 + 6k] / 5
yb = [1470  3k] / 5
Equivalently we can write ya and yb in terms of r as
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ya = 180r  210
yb = 420  90r
Ex 9.6(3): Question
Frank Cowell: Microeconomics
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purpose: to use the outcome of the GE model to plot the “incomepossibility” set
method: plot incomes corresponding to extremes of allocating commodity
2, namely k = 0 and k = 210. Then fill in the gaps.
Income possibility set
Frank Cowell: Microeconomics
incomes for k = 0
yb
incomes for k = 210
incomes for intermediate values of k
300
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attainable set if income can be thrown away
(42, 294)
 yb = 315  ½ya
200
• (294, 168)
100
ya
0
100
200
300
Ex 9.6(4): Question
Frank Cowell: Microeconomics
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purpose: find a welfare optimum subject to the “income-possibility” set
method: plot contours for the function W on the previous diagram.
Welfare optimum: first case
Frank Cowell: Microeconomics
income possibility set
yb
Contours of W = log ya + log yb
Maximisation of W over incomepossibility set
300
 W is maximised at
200
corner
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 incomes are (294, 168)
 here k = 210
100
 so optimum is where all
of resource 2 is
allocated to type a
ya
0
100
200
300
Ex 9.6(5): Question
Frank Cowell: Microeconomics
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purpose: as in part 4
method: as in part 4
Welfare optimum: second case
Frank Cowell: Microeconomics
income possibility set
yb
Contours of W = ya + yb
Maximisation of W over incomepossibility set
300
 again W is maximised
200
at corner
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 …where k = 210
 so optimum is where all
100
of resource 2 is
allocated to type a
ya
0
100
200
300
Ex 9.6: Points to note
Frank Cowell: Microeconomics
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Applying GE methods gives the feasible set
Limits to redistribution
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natural bounds on k
asymmetric attainable set
Must take account of corners
Get the same W-maximising solution
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where society is averse to inequality
where society is indifferent to inequality