(a) Find the set of Pareto optimal allocations in this economy and illus-trate the Pareto optimal allocations in appropriate diagrams.(b) Find the competitive equilibrium prices and allocations in this econ-omy.(e) For the competitive equilibrium in part (b), let 1; (a, L) denote theequilibrium income of consumer i = A, B. CalculateI(a, L)I(a, L) + I(a, L)Q"(a, L) =and illustrate this function in an appropriate diagram. Is Q' (a, L)increasing or decreasing in a? Provide an economic interpretation foryour answer. Problem 3: Production-exchange economyConsider an economy with two consumers, A and B, two goods, X and Y,two factors of production, labor and capital, and two firms: firm X useslabor and capital to produce good X and firm Y uses labor and capital toproduce good Y.Consumer A has no initial endowments of labor or capital but owns both ofthe firms. Consumer B has an initial endowment of labor Is =L>0 andan initial endowment of capital kg = K = L, but does not own any shareof the firms. Both consumers i = A, B have preferences over consumptionbundles can be represented by the Cobb-Douglas utility functionwhere r; is the quantity of good X for consumer i, y; is the quantity ofgood Y for consumer i, and a e (0, 1) is a preference parameter.Firm X produces the output good X with a technology that can be de-scribed by the following production function:fx(lx, kx) = lx + kx,where lx is the quantity of labor and kx is the quantity of capital.Firm Y produces the output good Y with a technology that can be de-scribed by the following production funetion:fr(ly, ky) = y + Vky,where ly is the quantity of labor and ky is the quantity of capital.

Pareto Efficiency:Pareto performance, named after the Italian economist Vilfredo Pareto, is a…

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Question

(a) Find the set of Pareto optimal allocations in this economy and illus-
trate the Pareto optimal allocations in appropriate diagrams.
(b) Find the competitive equilibrium prices and allocations in this econ-
omy.
(e) For the competitive equilibrium in part (b), let 1; (a, L) denote the
equilibrium income of consumer i = A, B. Calculate
I(a, L)
I(a, L) + I(a, L)
Q"(a, L) =
and illustrate this function in an appropriate diagram. Is Q' (a, L)
increasing or decreasing in a? Provide an economic interpretation for
your answer.

Problem 3: Production-exchange economy
Consider an economy with two consumers, A and B, two goods, X and Y,
two factors of production, labor and capital, and two firms: firm X uses
labor and capital to produce good X and firm Y uses labor and capital to
produce good Y.
Consumer A has no initial endowments of labor or capital but owns both of
the firms. Consumer B has an initial endowment of labor Is =L>0 and
an initial endowment of capital kg = K = L, but does not own any share
of the firms. Both consumers i = A, B have preferences over consumption
bundles can be represented by the Cobb-Douglas utility function
where r; is the quantity of good X for consumer i, y; is the quantity of
good Y for consumer i, and a e (0, 1) is a preference parameter.
Firm X produces the output good X with a technology that can be de-
scribed by the following production function:
fx(lx, kx) = lx + kx,
where lx is the quantity of labor and kx is the quantity of capital.
Firm Y produces the output good Y with a technology that can be de-
scribed by the following production funetion:
fr(ly, ky) = y + Vky,
where ly is the quantity of labor and ky is the quantity of capital.

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