ES20011 Whole group problems - 2
1. Compute the MRS for the following utility functions
1
1
(i) U(x, y) = x 2 y 2
3
1
(ii) U (x, y) = x 4 y 4
1
1
(iii) U (x, y) = (x 2 + y 2 )2
(iv)U(x, y) = ln x + ln y
2. Let U (x, y) = xy, M = 40, Px = 1 and Py = 2
(i)
The SLUTSKY EQUATION
Duality tells us that (see slide 10 of the duality slides)
x(U , Px , Py ) = x( M , Px , Py )
Differentiating this with respect to Px gives us:
x(U , Px , Py )
Px
=
x( M , Px , Py ) M
M
Px
+
x( M , Px , Py )
Px
1
This equation i
The Marshallian, Hicksian
and Slutsky Demand
Curves
Graphical Derivation
1
An aside: before we look at how price changes affect demand
we should look at how income changes affect the demand for
a good.
y
income expansion path
x
In this example, as income
ES20011 Problem Set 5 (for completion by end of week 9)
1.
(a) Explain attitude towards risk using a utility function U (c) = c.
(b) How can revealed preference analysis help us learn about an individuals preferences
without knowing their utility functon?
ES20011 Problem Set 4 (for completion by end of week 7)
1. Suppose a consumers preferences are represented by the following utility function
U (x, y ) =
x + b y,
where b > 0, and faces a budget constraint
Px x + Py y = M,
where Px and Py are the prices of
ES20011 Problem Set 3 (for completion by end of week 5)
1. An individual has a utility function dened over two goods
1
1
U (C1 , C2 ) = C12 + C22
and faces a budget constraint
Pc1 C1 + Pc2 C2 = M,
where Pc1 and Pc2 are the prices of the two goods and M is
ES20011 Problem Set 2 (for completion by end of week 3)
1.
An individual has income M. Clearly marking the intercepts, draw the individuals budget
constraint in (x, y ) space when
(i)Px = Py = 1.
(ii) Show on your diagram what happens when income and pric
ES20011: Coursework 1 to be completed by end of week 1
1. An individual has income . Draw their budget curves in ( ) space when
(i) = = 1
(ii) = 1 = 05
(iii) = 1 = 05 and the individual is prevented from consuming more than
08 units of
(iv) = 1 = 05 and
A couple of examples of production type questions.
1. A competitive industry consists of 1000 identical rms, each with a production technology
given by
1
1
q (K, L) = 6K 2 L 2
where K and L represent capital and labour respectively. Assume that the price
d)
Questions for decision-making under uncertainty wholegroup session
1. Suppose a houseowner has a house worth £250,000 faces a 5% chance in any given
year that there will be a ood, in which case, he/she suffers damages reducing the
value of the house
Production
Technology: y = f (x)
1
y=f(x)
y
Production set
x
2
Technology: y = f (x1, x2, x3, . xn)
For simplicity consider the case of 2 inputs
e.g. labour and capital, L and K
y = f (K, L)
3
y
y=f(K,L)
y1
y0
K
K1
K0
y1
y0
L0
L1
L
4
Properties of Tec
The Effect of Taxation:
an application of
Expected Utility Approach
in the context of investment
Suppose Consumer has some wealth
She can invest an amount
in a risky asset with
returns rg (>0) in good state with Prob
returns rb (<0) in bad state with Pro
Mean-Variance Analysis
an alternative approach to the
Expected Utility Approach
Suppose W = 100 and bet 50
on flip of a coin
Probability
0.5
Outcomes
50
150
EU=0.5U(50) + 0.5 U(150)
More outcomes => More complexity
E.g W = 30, Bet 30 on throw of dice
P
Choice
under
Uncertainty
1
Note that were maximising utility before we
know what the circumstances will be
were maximising ex-ante welfare
which does not necessarily mean that ex-post
welfare is actually maximised
e.g. if our expectations turn out to be w
Intertemporal Choice
What if we allow consumers consume at different
points in time?
We can still deal with this problem by thinking of
each good at each point in time as a separate
commodity.
This requires the existence of futures markets.
We assume
Consumer Surplus
- attaches a monetary measure to changes in welfare
resulting from some policy change (e.g. change in tax
rate)
- 2 questions we could ask:
- How much are you willing to pay for a price reduction?
- How much are you willing to accept as c
Revealed Preference Approach
Rather than go from preferences to demand, can
we get from demand to the underlying
preferences?
This approach is based on the simple idea that if
we choose bundle q1 when q0 was affordable, then
q1 is revealed to be preferr
ES20011 Whole group problems
1
1. If Q 200P 2 what is the price elasticity of demand, QP ?
2. If Q 400 10P
(i) what is the price elasticity of demand, QP at P 30 and P 10?
(ii) at what price is the price elasticity of demand, QP equal to -1?
3. A linear d
Duality
Instead of maximising utility subject to a
budget constraint (UMP), we could
minimise the cost of achieving a certain
level of utility (CMP)
1
So, instead of
1
Max U = x 2 y
1
2
s.t. Px x + Py y = M
we could
1
Min M = Px x + Py y s.t. x y
2
1
2
=U
Choice: solving for x and y
Now we are ready to combine preferences
with the budget constraint to determine the
consumers optimal bundle
1
y
optimal bundle is where
the slopes are equal
x
2
So, we can find the slope of the budget constraint
y
Px x + Py y
Preferences
Now we look at the other determinant of
consumer choice preferences.
Axioms of consumer choice utility
functions
Once we have a utility function we can
maximise this function subject to a budget
constraint to determine consumer choice.
1
Some
Relationship between the
indifference curve & utility
function
1
Note that restrictions on preferences will
impose restrictions on the utility function
Monotonicity of preferences implies that if
x1 > x2 x1 f x2
and we already know that
if
x1 f x2 U ( x1
ES20011 Intermediate
Microeconomics
- Introduction
This course sits best between ES10005 and ES20012
These slides are based on notes originally devised by Dr. Cillian Ryan
Key elements include:
Constrained optimisation
Equilibrium
Comparative statics
Endo
ES20013
Intermediate Macroeconomics 1
Topic 5
Aggregate supply
Based on Gartner chapter 6
Plan of the lecture
1. Meaning of potential output.
2. Wages and unemployment determination in the
labour market.
3. Labour market equilibrium and involuntary
unempl
ES20013
Intermediate Macroeconomics 1
Topic 5
Aggregate supply
Based on Gartner chapter 6
Plan of the lecture
1. Meaning of potenCal output.
2. Wages and unemployment determinaCon in the
labour market.
3. Labour ma