Part IIA, Paper 1 Consumer and Producer Theory

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Part IIA, Paper 1Consumer and Producer Theory

• Lecture 6
• Labour supply, Savings behaviour
• Producer Theory Technology
• Flavio Toxvaerd

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Todays Outline

• Labour supply, savings, asset pricing
• Technology
• Production
• Assumptions

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Labour Supply
Domain consumption good c leisure l
c
u3
u2
u1
Utility function u(c,l) Assume well behaved
preferences
c
Budget constraint pc (24 - l )w M
M/p
l
l
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Mathematically
How will labour supply vary with wages?
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Mathematically
be the minimum level of non-labour income
required to achieve the utility level u0
From duality, we know that
Differentiating gives
Slutsky Eqn.
(gt0, if l is normal)
(lt0)
(gt0)
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Labour Supply
Labour supply equals 24 minus leisure time
w
Conclusion Impact of increase of wages on labour
supply is ambiguous
Ls
Labour supply may be backward bending
Labour
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Savings Behaviour

• An important consumer decision is how to allocate
  resources over time - that is, how much to borrow
  and save
• Will consider a simplified two period model
  (periods 1 and 2), where the consumer decides how
  much to consume in each period (c1 and c2),
  subject to an intertemporal budget constraint
• Domain consumption in each period, c1 and c2

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Mathematically
Primal problem max u(c1,c2) s.t. p2c2 m2(m1 -
p1c1)(1r)
Lagrangian L u(c1,c2)- ?(p2c2 p1c1(1r) - m2
- m1(1r))
Euler equation
F.O.C.s
Present Value of Consumption Expenditure
Present Value of Income
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Dual Problem
min (p1c1)(1r) p2c2 s.t. u(c1,c2) u0
Solution gives
be the period 2 income required to achieve u0
From Envelope Theorem have that
Impact of increase in interest rate on savings is
ambiguous
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Savings
c2

• Consider impact of change in interest rate
• Rise in interest rate causes budget line to
  rotate around endowment point
• When c1 is a normal good, impact on rate of
  savings ambiguous

c2
m2 /p2
S.E.
c1
I.E.
c1
m1 /p1
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Evaluating Income Streams

• If the interest rate is r, then 1 invested today
  will return (1r) one year from now.
• If today you invest 1/(1r) then this will
  return 1 next year.
• We may say that 1 given to us next year is of
  equivalent value to 1/(1r) today
• Similarly, the present value of a stream of
  income m0, m1, m2, can be written

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Asset Pricing

• How much should you pay for an asset that will
  payout 1 next year.
• Answer PV of asset 1/(1r), so price of asset
  today 1/(1r)

Note Price of assets will vary inversely with
the interest rate
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Asset Pricing
What if the asset will generate 1 next year and
in every subsequent year ?
PV of asset
Alternatively rV 1 so V 1/r
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Example 1 The Magdalene Bursar

• The Story In 1600 Magdalene College leased 1
  acre around Covent Garden for 30 per annum, in
  perpetuity
• Today the College still receives 30 ground rent
  for the acre of land, and the Bursars ineptitude
  is celebrated by the disgusting statue placed on
  the side of the river by Henrys bar.
• NPV1600 30/r 600 (if r 0.05)
• NPV2004 (600 in 1600) 600(1r)404 218
  billion

How did the College spend this legacy ?!
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Producer Theory
We wish to analyse production decisions of firms
with the same rigour as that used to analyse
consumption decisions
Consumer Theory Represent preferences (Utility
function) Utility maximisation Dual problem (min
expenditure to obtain desired utility)
Producer Theory Represent technology (Production
function) Profit maximisation Cost function (min
expenditure to obtain desired output)
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Technology
What is technology?
May be represented as a mapping from inputs (X)
to outputs (Y)
such that, for any vector of inputs
denotes the set of outputs that can be obtained
using x
Note T(x) is not a function, more than one
output may be possible with the same input
x
Inputs
Outputs
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Production Plans

• A production plan is a vector of inputs and
  outputs (x,y)
• where x (x1,,xn) is a vector of inputs
• and y (y1,,ym) is a vector of outputs
•  A production plan (x , y) is feasible if y ?
  T(x) and so, technology allows the vector of
  inputs, x, to be converted into the vector of
  outputs, y

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The Production Function Erdos
A mathematician is a machine for turning coffee
into theorems.
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Production Possibility Set
The production possibility set (PPS) is the set
of all feasible production plans
Efficient production
y
PPS
x
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Assumptions on the PPS

• Assumption 1 If x 0 and (x,y)?PPS, then y 0
  
• Assumption 2 Free disposal
• if (x,y)?PPS then
• for all x?x, (x,y)?PPS
• for all y?y,(x,y)?PPS
• Assumption 3 The PPS is closed so the boundary
  of the PPS is contained in the PPS - and
  efficient production is well defined

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Input Requirement Set

• The input requirement set is the set of vectors
  of inputs required to produce a particular vector
  of outputs

x2

• The efficient boundary of IRS is an isoquant
• Assumption 4 All input requirement sets are
  convex
• (PPS is quasi-concave)

x1
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Restricted Production Set
Efficient boundary of RPS is also sometimes
called the production possibility frontier
y2
or, when there is only one output, the production
function
y1
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Summary

• Applications of consumer theory
• Technology
• Production

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Readings

• Varian, Intermediate Microeconomics, chapters
  9,10, 11 and 18