Esercitazione 2 Teoria del consumatore

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Esercitazione 2 Teoria del consumatore
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Esercizio 1
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part 2
Step 1 set up the Lagrangean
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part 2
First-order conditions
Solve to get
Offer curve is a vertical line at 10a
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part 3

• Points to watch
• Use your knowledge of the shape of the
  solution
• For example we can get the solution to type A by
  adapting part 2
• Types B-D follow by using the diagrams in Part 1

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part 3, type A
x2
20
We can use the demand function from part 2.
Income is 20 now (instead of 10r) so solution
must be
201-a
x1
Offer curve is horizontal line at 201-a
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part 3, type B
x2
x'
Solution must be on one or other axis unless rb
indifference curve
b
x1
x''
Offer curve is line segment with kink at x''
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part 3, type C
x2
x'
Solution must be on one or other axis
? g
x1
x''
Offer curve is blob at x' and line segment from
x''
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part 3, type D
Solution must lie on corner of the indifference
curve where x2dx1. Using this fact and the
budget constraint x2rx120 we have
x2
d
x1
Offer curve is line through all the corners
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Esercizio 2 Rationing
Part 1 standard demand functions
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Utility maximisation
Maximise x1x2x3x4 subject to
The Lagrangean is
Differentiating, the FOC is
which implies
Ordinary demand function
Using the budget constraint we get l 4/M. So
we have
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Derive related functions
Indirect utility function
Substitute optimal demands in utility function
Cost function
Rearrange to get M as a function of u
Compensated demand function
Differentiate
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Ordinary elasticity
Take the ordinary demand function
Take logs and differentiate with respect to log
p1 and with respect to log pj
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Compensated elasticity
Take the compensated demand function
Again take logs and differentiate with respect to
log p1 and with respect to log pj
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Consumption and rationing
Part 2 introduce a side constraint
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Modify the problem

• x4 is now fixed at A4
• Define M' M p4 A4
• Problem is equivalent to maximising x1x2x3A4
  subject to budget with adjusted income M' .
• Use results from part 1 applied to 3-good economy

Errore manca A4-1/3
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Ordinary elasticity again
Take the ordinary demand function
An income effect
Take logs and differentiate with respect to log p4
Take logs and differentiate with respect to log
p1 and with respect to log pj
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Compensated elasticity again
Take the compensated demand function
Errore manca A4-1/3
Closer to 0 than before
Again take logs and differentiate
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Consumption and rationing
Part 3 more constraints
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More constraints
Reapply the method with one fewer commodity
Errore manca un pezzo contenente A4 e A3
Closer to 0 than before
Differentiate again
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Consumption and rationing
Part 4 Interpretation

• Model illustrates the comparative statics of
  someone who is subject to a quota ration.
• But not rich enough to determine which
  commodities are consumed at a conventional
  equilibrium and which will be constrained by the
  ration.
• Parts 2 and 3 show clearly how the compensated
  demand becomes steeper the more external
  constraints are imposed

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Points to remember

• Modify the problem where appropriate to get a
  more tractable equivalent.
• Re-use the solution to one part of the problem to
  build the next.

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Esercizio 3 part 1(a)
.
These are easy parabolic contours
Even easier fixed money income

• First steps
• Sketch indifference curves
• Write down budget constraint
• Set out optimisation problem

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Slope is vertical here
We could have x20
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• Budget constraint

• Substitute this into the utility function

• We get the objective function

• FOC for an interior solution

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• FOC for an interior solution

• Therefore, if positive amounts of the two goods
  are bought

• But this requires

• Otherwise x20 and we get x1 from the budget
  constraint.

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• Demand functions

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• Maximised utility gives us the indirect utility
  function

• Otherwise

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For case where both goods are consumed

• For the cost function use the relation uV(p,M)

• and solve for M

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Esercizio 3 part 1(b,c)
.

• Method
• Use C function to write down CV
• (Equivalently use V function to write down CV)
• Check income effects

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• Compensating variation is

• But demand for good 1 has zero income effect

• So CV CS EV in this case

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Esercizio 3 part 2(a)
.

• Method
• Find monopolists AR from consumer demand using
  part 1.
• Use standard optimisation procedure

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• Aggregate demand over N consumers using part 1

• Rearrange to get AR curve

• Total Revenue is

• Profits are therefore

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MC MR

• FOC

• Optimal output

• From AR curve, price at optimum is

Price gt MC

• simplified

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Esercizio 3 part 2(b)
.

• Method
• Aggregate the CV for each consumer to define L
• Use marginal cost and monopolists equilibrium
  price to evaluate L

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• Use definition of CV with p1 c

• Evaluate L at p1 2c

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Esercizio 3 part 2(c)
.

• Method
• Add bonus B into the expression for profits
• Again use standard optimisation procedure

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• Profits are now

• Value of bonus is

• Expressed in terms of quantity

• So profits become

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• FOC

Price MC

• Resulting equilibrium

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Esercizio 3 Points to note

• The fact that indifference curves touch the axis
  does not cause any great problem
• Aggregate welfare loss is found from individual
  CV
• Regulation causes monopoly to behave like
  competitive firm

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Esercizio 4 setting

• Sketch the indifference curves shifted
  Cobb-Douglas
• k is minimum consumption requirement of other
  goods.
• a is share of budget that goes on rice after an
  amount has been set aside to buy the min
  requirement

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part 1 approach

• Work out the budget constraint.
• Use the utility function to set out the
  Langrangean
• Find the FOCs for an interior solution
• Check whether the solution makes sense
• Find the demand functions
• Use these to get household supply function

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part 1 getting the FOC

• The budget constraint is
• px1 x2 ? y
• where
• y pR1 R2

• The Lagrangean is
• a log(x1) 1a log(x2k) l y px1
  x2

• The FOC for an interior maximum are
• a
• pl 0
• x1
• 1a
• l 0
• x2k
• y px1 x2 0

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part 1 supply function

• From the FOC
• a 1a
• px1 x2 k
• l l
• Adding these and using the budget constraint, we
  have
• y k 1/l

• Eliminating l in the above
• a
• x1 y k x2 ak 1a y
• p
• Supply of good 1 is given by
• S(p) R1 x1.
• Substituting in for y, we have
• a
  
• S(p) 1a R1 R2 k
• p

Supply increases with price if R2 gt k
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part 2

• If ak 1a y ? c nothing changes from
  previous case.
• Otherwise
• px1 c y
• so that
• R2 c
  
• x1 R1
• p