代写 ECON 301 Microeconomic Theory

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  • 代写 ECON 301 Microeconomic Theory

    Assignment 5
    ECON 301 Microeconomic Theory 2
    Jean Guillaume Forand ∗
    Winter 2016, Waterloo
    1. Consider a two-consumer economy in which ω A = (1,1), ω B = (2,3), u A (x A
    1 ,x A 2 ) = x A 1
    +x A
    2
    and u B (x B
    1 ,x B 2 ) = 2x B 1
    + x B
    2 .
    (a) Derive fair allocations for this economy.
    (b) Derive a competitive equilibrium for this economy that generates the allocations you
    found in (a).
    2. Consider a two-consumer economy in which ω A = (2,2), ω B = (2,2), u A (x A
    1 ,x A 2 ) = x
    A 1
    2
    1
    x A
    1
    2
    1
    and u B (x B
    1 ,x B 2 ) = x B 1
    + x B
    2 .
    (a) Derive fair allocations for this economy.
    (b) Derive a competitive equilibrium for this economy that generates the allocations you
    found in (a).
    3. Consider a two-consumer economy in which ω A = (1,1), ω B = (1,1), u A (x A
    1 ,x A 2 ) = x A 1
    +x A
    2
    and u B (x B
    1 ,x A 2 ) = x B 1
    − 3x A
    2 .
    (a) Illustrate this economy in an Edgeworth box. Label your axes carefully.
    (b) Suppose that there are only two markets: one for good 1 and one for good 2. Show
    that there exists a competitive equilibrium with p ∗
    1
    = p ∗
    2
    = 1.
    (c) Show that the equilibrium allocations from (a) are not Pareto-efficient.
    (d) Complete the missing market from the equilibrium in (a) by introducing a market for
    the externality generated by the consumption of good 2 by consumer A. Suppose that
    the allocation of rights is such that consumer A has all the rights over the consumption
    ∗ Room 131, Department of Economics, University of Waterloo, Hagey Hall of Humanities, Waterloo, Ontario,
    Canada N2L 3G1. Office phone: 519-888-4567 x. 33635. Email: jgforand@uwaterloo.ca. Website: http://arts.
    uwaterloo.ca/ ~ jgforand
    1
    of good 2, i.e., such that (ω A
    R ,ω
    B
    R ) = (2,0).
    Show that there exists a competitive
    equilibrium with p ∗
    2
    = 0 and p ∗
    1
    = p ∗
    R
    = 1.
    (e) Show that the equilibrium allocations from (d) are Pareto-efficient.
    (f) Find a competitive equilibrium for this economy if the allocation of rights is such
    that consumer B has all the rights over the consumption of good 2, i.e., such that
    (ω A
    R ,ω
    B
    R ) = (0,2). Show that the allocations in this equilibrium are Pareto-efficient.
    (g) Now suppose that the government expropriates all units of good 2 in the economy,
    and establishes a permit system: consumption of one unit of good 2 requires a permit
    which costs c > 0. All government revenue from permits is returned in equal shares
    to the two consumers, and there is a competitive market that determines the price of
    good 1. Define a competitive equilibrium for this economy.
    (h) Derive a competitive equilibrium for the economy in (g).
    (i) For what values of the permit price c are the equilibrium allocations from (h) Pareto-
    efficient?
    4. Consider a two-consumer, two-period economy, in which the consumption of consumer J in
    period i = 1,2 is c J
    i
    ≥ 0. In each period, there is 1 unit of the consumption good available,
    which is entirely owned by consumer A in period 1 and entirely owned by consumer B in
    period 2. The utility of consumer A from consumption bundles (c A
    1 ,c A 2 ) and (c B 1 ,c B 2 ) is
    q
    c A
    1
    + β
    q
    c A
    2 , while that of consumer B is
    代写 ECON 301 Microeconomic Theory
     
    q
    c B
    1
    + β
    q
    1 − c A
    2 .
    (a) Illustrate this economy in an Edgeworth box. Label your axes carefully.
    (b) Suppose that there are only two markets: one for the consumption good in period 1
    and one for the consumption good in period 2. Derive a competitive equilibrium.
    (c) Show that the equilibrium allocations from (a) are not Pareto-efficient.
    (d) Complete the missing market from the equilibrium in (a) by introducing a market
    for the externality generated by consumption in period 2 by consumer A. Suppose
    that the allocation of rights is such that both consumer have rights over
    1 / 2 units of
    consumption in the second period, i.e., such that (ω A
    R ,ω
    B
    R ) = ( 1 / 2 ,
    1 / 2 ). Show that
    there exists a competitive equilibrium with p ∗
    2
    = 0.
    (e) Show that the equilibrium allocations from (d) are Pareto-efficient.
    (f) Now suppose that the government expropriates all units of the consumption good in
    period 2, and establishes a permit system: consumption of one unit of the good in
    period 2 requires a permit which costs c > 0. All government revenue from permits is
    returned in equal shares to the two consumers, and there is a competitive market that
    determines the price of consumption in period 1. Define a competitive equilibrium for
    this economy.
    2
    (g) Derive a competitive equilibrium for the economy in (f).
    (h) For what values of the permit price c are the equilibrium allocations from (g) Pareto-
    efficient?
    3

    代写 ECON 301 Microeconomic Theory