Advanced Microeconomics II: Problem Set 1 PDF

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Advanced Microeconomics II First Tutorial Nicolas K. Scholtes February 22, 2013 Exercise 15.B.1 Consider an Edgeworth Box economy in which two consumers have locally nonsatiated pref- erences. Let x`i (p) be consumer i’s demand for good ` at prices p = (p1 , p2 ). ! ! X X (a) Show that p1 x1i (p) − ...

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Advanced Microeconomics II First Tutorial Nicolas K. Scholtes February 22, 2013

Exercise 15.B.1 Consider an Edgeworth Box economy in which two consumers have locally nonsatiated preferences. Let x`i (p) be consumer i’s demand for good ` at prices p = (p1 , p2 ). ! (a) Show that p1

X

x1i (p) − ω ¯1

! X

+ p2

i

x2i (p) − ω ¯2

=0

∀p

i

The competitive budget set Bi (p) = {x ∈ R2+ |p · x ≤ p · ωi } in vector form for consumers i = 1, 2 is given by:

p1 p2

!



· x1i

x2i



p1 (x1i − ω1i )

≤

p1 p2

!

 · ω1i

ω2i



+ p2 (x2i − ω2i ) ≤ 0

Recall that the (Walrasian) demand function xi (p, p · ωi ) must satisfy Walras’s Law. That is, p1 (x1i − ω1i ) + p2 (x2i − ω2i ) = 0 Proof. 1 : Suppose not, ∃xi ∈ Bi (p)|p · xi 0, we get

! X

x2i

# − ω2 = 0 which is the market-clearing condition for good 2.

i

Exercise 15.B.2 Consider and Edgeworth box economy in which the consumers have the Cobb-Douglas util1−α ity functions u1 (x11 , x21 ) = xα and u2 (x12 , x22 ) = xβ12 x1−β 11 x21 22 . Consumer i’s endowments are (ω1i , ω2i )  0. Solve for the equilibrium price ratio and allocation. How do these change with a differential change in ω11 ? (I) Compute offer curves for each consumer, OCi (p) = (x?1i , x?2i ) Agent 1 UMP:

max

{x11 ,x21 }

u1 (x11 , x21 ) s.t

1−α = xα 11 x21 n p1 x11 + p2 x21 ≤ p1 ω11 + p2 ω21

For simplicity, let R1 = p1 ω11 + p2 ω21 . Recall that when preferences are convex, the optimal consumption can computed by equating the marginal rate of substitution with the price ratio:2 ∂u1/∂x11 ∂u1/∂x21

2 Tangency

=

p1 α x21 p1 ⇒ = p2 1 − α x11 p2

between the budget line and the indifference curve is a necessary and sufficient condition for optimality under convexity of preferences

Homework I Solutions

3

Substituting the budget constraint allows us to isolate x11 as a function of the exogenous parameters

 1+

α 1−α

x11

=

x11

=

  p2 α R1 − x11 p1 1 − α p2 αR1 (1 − α)p1



Simplifying terms, we arrive at the Walrasian demand of consumer 1 for good 1, x?11 . Substituting this result into the budget constraint gives us x?21 . The two demand functions constitute 1’s Offer curve:

OC1 (p) = (x?11 , x?21 ) =



αR1 (1 − α)R1 , p1 p2



The offer curve of agent 2 follows naturally given the symmetry of the utility functions3

OC2 (p) =

(x?12 , x?22 )

 =

βR2 (1 − β)R2 , p1 p2



(II) Apply the market clearing condition for good 1:4

x?11 αR1 p1

+ x?22 = ω11 + ω12 βR2 + = ω11 + ω12 p1 α β [p1 ω11 + p2 ω21 ] + [p1 ω12 + p2 ω22 ] = ω11 + ω12 p1 p1 p?2 (αω21 + βω22 ) = (1 − α)ω11 + (1 − β)ω12 p?1 Thus, our equilibrium price vector is p?1 αω21 + βω22 = ? p2 (1 − α)ω11 + (1 − β)ω12 This price vector should induce market clearing for good 2 (i.e. we can use this as a check that our calculations were done correctly)

x21 (p?1 , p?2 ) + x22 (p?1 , p?2 )

=

(1 − α)R1 (1 − β)R2 + p?2 p?2 p?1 [(1 − α)ω11 + (1 − β)ω12 ] + (1 − α)ω21 + (1 − β)ω22 p?2 αω21 + βω22 [(1 − α)ω11 + (1 − β)ω12 ] + (1 − α)ω21 + (1 − β)ω22 (1 − α)ω11 + (1 − β)ω12 αω21 + βω22 + (1 − α)ω21 + (1 − β)ω22

=

ω21 + ω22

= = =

3 Note

that for agent 2, we have R2 = p1 ω12 + p2 ω22 this necessarily implies market clearing for good 2

4 Recall:

4

Nicolas K. Scholtes

Having established the equilibrium price vector, substituting this back into the offer curves gives us each consumer’s demand of each good at equilibrium

x11 (p∗ )

p?2 ω21 p?1   (1 − α)ω11 + (1 − β)ω12 αω11 + α ω21 αω21 + βω22 α [αω21 + βω22 ] + αω21 [(1 − α)ω11 + (1 − β)ω12 ] αω21 + βω22 α(ω11 ω21 + ω21 ω12 ) + αβ(ω11 ω22 − ω21 ω12 ) αω21 + βω22 p∗1 (1 − α) ∗ ω11 + (1 − α)ω21 p2   αω21 + βω22 (1 − α)w11 + (1 − α)ω21 (1 − α)ω11 + (1 − β)ω12 (1 − α)ω11 [αω21 + βω22 ] + (1 − α)ω21 [(1 − α)ω11 + (1 − β)ω12 ] (1 − α)ω11 + (1 − β)ω12 (1 − α) (ω11 ω21 + βω11 ω22 ) + (1 − α)(1 − β) (ω21 ω12 ) (1 − α)ω11 + (1 − β)ω12

=

αω11 + α

= = = x21 (p∗ )

= = = =

By symmetry of the utility functions, agent 2’s consumption at equilibrium is as follows

x12 (p∗ )

=

x22 (p∗ )

=

β(ω12 ω22 + ω11 ω22 ) + αβ(ω12 ω21 − ω11 ω22 ) αω21 + βω22 (1 − β) (ω12 ω22 + αω12 ω21 ) + (1 − α)(1 − β) (ω11 ω22 ) (1 − α)ω11 + (1 − β)ω12

Taking the first derivative w.r.t ω11 of the equilibrium values for price and consumption allows us to ascertain the effect of a differential change in ω11 ∂p∗ ∂ω11 ∂x11 (p∗ ) ∂ω11 ∂x21 (p∗ ) ∂ω11 ∂x12 (p∗ ) ∂ω11 ∂x22 (p∗ ) ∂ω11

= = = = =

(α − 1) (αω21 + βω22 )

2 < 0 since α ∈ (0, 1) [(1 − α)ω11 + (1 − β)ω12 ] αω21 + αβω22 >0 αω21 + βω22 (1 − α)(1 − β)ω12 (αω21 + βω22 ) >0 2 [(1 − α)ω11 + (1 − β)ω12 ] (1 − α)βω22 >0 αω21 + βω22 (1 − α)(1 − β)ω12 (αω12 + βω22 ) −...