Transcript The Firm: Optimisation - London School of Economics

LECTURE EXAMPLES
EC202
http://darp.lse.ac.uk/ec202
Additional examples provided during lectures in 2014
Frank Cowell
8 Dec 2014
Frank Cowell: Lecture Examples
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Example – single technique
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Frank Cowell: Lecture Examples
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Example – two techniques
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Frank Cowell: Lecture Examples
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Example – multiple techniques
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Frank Cowell: Lecture Examples
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Example:
• Use spreadsheet to find (z1, z2) such
that log 2 = 0.25 log z1+ 0.75log z2)
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z2
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• Plot on graph
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• Z(2) = {z: f (z)  2}
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Example
z2
• Isoquant q = 2 (as before)
• Isoquant q = 1
• Isoquant q = 3
• Equation of isoquant
• Homotheticity
• Check HD 1 from original equation
• double inputs → double output
z1
8 Dec 2014
Frank Cowell: Lecture Examples
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Example
• Production function
• Keep input 2 constant
• Marginal product of good 1
8 Dec 2014
Frank Cowell: Lecture Examples
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8 Dec 2014
Frank Cowell: Lecture Examples
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Example – cost-min, single technique
z2
z2
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z
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z
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z1
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Example – cost-min, two techniques
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z2
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Frank Cowell: Lecture Examples
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Example
z2
• Isoquant (as before)
• does not touch either axis
• Constraint set for given q
• Cost minimisation must have interior solution
z1
8 Dec 2014
Frank Cowell: Lecture Examples
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Example
• Lagrangean for cost minimisation
z2
• Necessary and sufficient for minimum:
• Evaluate first-order conditions

z*
z1
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Frank Cowell: Lecture Examples
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Example
• First-order conditions for cost-min:
• Rearrange the first two of these:
• Substitute back into the third FOC:
• Rearrange to get the optimised Lagrange multiplier
8 Dec 2014
Frank Cowell: Lecture Examples
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Example
• From first-order conditions:
• Rearrange to get cost-min inputs:
• By definition minimised cost is:
• In this case the expression just becomes l*
• So cost function is
8 Dec 2014
Frank Cowell: Lecture Examples
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8 Dec 2014
Frank Cowell: Lecture Examples
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Example
• First-order conditions for cost-min:
• Rearrange the first two of these:
• Substitute back into the third FOC:
• Rearrange to get the optimised Lagrange multiplier
8 Dec 2014
Frank Cowell: Lecture Examples
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Example
• From last lecture, cost function is
• Differentiate w.r.t. w1 and w2
• Slope of conditional demand functions
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Frank Cowell: Lecture Examples
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8 Dec 2014
Frank Cowell: Lecture Examples
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Example
x2
•
indiff curve u = log 1
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indiff curve u = log 2
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indiff curve u = log 3
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From the equation
•
Equation of IC is
•
Transformed utility function
x1
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Frank Cowell: Lecture Examples
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8 Dec 2014
Frank Cowell: Lecture Examples
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Example
x2
• Indifference curve (as before)
• does not touch either axis
• Constraint set for given u
• Cost minimisation must have interior solution
x1
8 Dec 2014
Frank Cowell: Lecture Examples
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Example
• Lagrangean for cost minimisation
x2
• For a minimum:
• Evaluate first-order conditions

x*
x1
8 Dec 2014
Frank Cowell: Lecture Examples
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Example
• First-order conditions for cost-min:
• Rearrange the first two of these:
• Substitute back into the third FOC:
• Rearrange to get the optimised Lagrange multiplier
8 Dec 2014
Frank Cowell: Lecture Examples
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Example
• From first-order conditions:
• Rearrange to get cost-min inputs:
• By definition minimised cost is:
• In this case the expression just becomes l*
• So cost function is
8 Dec 2014
Frank Cowell: Lecture Examples
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Example
• Lagrangean for utility maximisation
x2
• Evaluate first-order conditions

x*
x1
8 Dec 2014
Frank Cowell: Lecture Examples
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Example
• Optimal demands are
x2
• So at the optimum

x*
x1
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8 Dec 2014
Frank Cowell: Lecture Examples
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Example
• Results from cost minimisation:
• Differentiate to get compensated demand:
• Results from utility maximisation:
8 Dec 2014
Frank Cowell: Lecture Examples
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Example
• Ordinary and compensated demand for good 1:
• Response to changes in y and p1:
• Use cost function to write last term in y rather than u:
• Slutsky equation:
• In this case:
8 Dec 2014
Frank Cowell: Lecture Examples
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Example
• Take a case where income is endogenous:
• Ordinary demand for good 1:
• Response to changes in y and p1:
• Modified Slutsky equation:
• In this case:
8 Dec 2014
Frank Cowell: Lecture Examples
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8 Dec 2014
Frank Cowell: Lecture Examples
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Example
• Cost function:
• Indirect utility function:
• If p1 falls to tp1 (where t < 1) then utility rises from u to u′:
• So CV of change is:
• And the EV is:
8 Dec 2014
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8 Dec 2014
Frank Cowell: Lecture Examples
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Example
• Rearranged production function:
q2
high R3
• Three goods
• goods 1 and 2 are outputs (+)
• good 3 is an input ()
• If all of resource 3 used as input:
• Attainable set
low R3
q1
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8 Dec 2014
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Example
• Suppose property distribution is:
• Incomes are
• Given Cobb-Douglas preferences demands are
• So, total demand for good 1 is
• From materials-balance condition
• Which can only hold if
• So, equilibrium consumption of a is
• Therefore equilibrium consumption of b is
8 Dec 2014
Frank Cowell: Lecture Examples
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8 Dec 2014
Frank Cowell: Lecture Examples
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Example
• Suppose property distribution is:
• Reservation utility
• Incomes are
• Demands by a and b (offer curves):
• Equilibrium where
8 Dec 2014
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Example
• Marginal Rate of Substitution:
• Assume that total endowment is (12,12)
• Contract curve is
• Which implies:
8 Dec 2014
Frank Cowell: Lecture Examples
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8 Dec 2014
Frank Cowell: Lecture Examples
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Example
• Suppose property distribution is:
• Incomes are
• Demands by a and b :
• Excess demands:
• Walras’ Law
• Equilibrium price:
• Equilibrium allocation
8 Dec 2014
Frank Cowell: Lecture Examples
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8 Dec 2014
Frank Cowell: Lecture Examples
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Example
xBLUE
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indifference curves
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Implied probabilities
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Marginal rate of substitution
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A prospect
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The mean
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Find the certainty equivalent
 P0
xRED
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Example
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A prospect
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Certainty equivalent
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Risk premium: 1.75 – 1.414 = 0.346
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Felicity function
xBLUE
 P0
xRED
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2014
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Frank Cowell: Lecture Examples
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Frank Cowell: Lecture Examples
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Example
• Suppose, if you win return is r = W, if you lose return is r = L
• Expected rate of return is
• If you invest b, then expected utility is
• FOC
• Optimal investment
• Do rich people invest more?
8 Dec 2014
Frank Cowell: Lecture Examples
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8 Dec 2014
Frank Cowell: Lecture Examples
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Example: Cycles and aggregation
• What happens if Right-handers vote?
• What happens if Left-handers vote?
• What happens if there’s a combined vote?
8 Dec 2014
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Example: IID
• Suppose, Alf, Bill and Charlie have the following rankings
•
•
•
•
Everyone allocates 1 vote to the worst, 2 to the second worst,…
Votes over the four states are [8,7,7,8]
What if we exclude states 2 and 3?
If focus just on states 1 and 4 votes are [4,5]
8 Dec 2014
Frank Cowell: Lecture Examples
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8 Dec 2014
Frank Cowell: Lecture Examples
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Example: envy
• Utility functions for a and b:
• Suppose the allocation is
• Is this envy free?
• Now suppose the allocation is
• Is this envy free?
8 Dec 2014
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8 Dec 2014
Frank Cowell: Lecture Examples
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Example
• Suppose we have an exchange economy where stocks of the goods are (12, 12).
• To find efficient points, max b’s utility keeping a’s utility constant
• Lagrangean is
• First-order conditions are:
• Rearranging:
• So efficient points are characterised by:
8 Dec 2014
Frank Cowell: Lecture Examples
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8 Dec 2014
Frank Cowell: Lecture Examples
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Example
• Suppose property distribution is:
• Incomes are
• Demands by a and b :
• Materials balance:
• Equilibrium price:
• Incomes in equilibrium allocation
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Frank Cowell: Lecture Examples
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Example
• Property distribution is:
• Incomes in equilibrium allocation:
• Extreme cases:
• Income-possibility set
yb
ya
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