Class 2 - Advanced Microeconomic Analysis
Giulio Trigilia
Warwick University
October 19, 2015
Exercise 1
The utility function of an individual is u1 (x) = x11 and his endowment is
e1 = (1, k) for k > 0. The utility function of another individual is u2 (x) =
x21 + α−1 (x22 )α for α < 1 and his endowment is e2 = (1, 1).
Compute the competitive equilibrium as the endowment parameter, k, and
the preference parameter, α vary.
First notice that we can only have strictly positive prices for both goods
in order to be able to clear the markets.
Consider individual 1. Whenever p2 > 0, he will set x1∗
2 = 0. Local-nonsatiation guarantees that the budget constraint holds with equality, hence
we get:
x1∗
1 =
p1 + kp2
p1
Now consider individual 2. At an interior solution, by local-non-satiation
I can write:
x21 = 1 +
p2
1 − x22 (A)
p1
1
Substituting (A) inside the objective function we obtain an unconstrained,
strictly concave maximisation problem (check the second derivative: that’s
why α < 1 !). Hence, the FOC is necessary and sufficient for an optimum:
x2∗
2
=
p2
p1
1/(α−1)
(B)
Substituting (B) into (A) we get:
x21
p2
=1+
−
p1
p2
p1
α/(α−1)
Finally, consider the market clearing condition for good 2. We can rewrite
it solving for the price ratio as:
p∗2
= (1 + k)α−1
∗
p1
(C)
Clearing L−1 is sufficient since Walras’ Law holds in this economy. Hence
the interior CE is given by (C) and



 x1∗


1

1∗

x
=




x1∗
 2
x∗ =

 x2∗


1

2∗

x
=


 x2∗

2
= 1 + k(1 + k)α−1
=0
= 1 − k(1 + k)α−1
= (1 + k)
Notice that non-negativity of consumption requires x2∗
1 = 1 − k(1 +
k)α−1 ≥ 0. Therefore, a necessary condition for the interior CE to exist
is:
1 ≥ k(1 + k)α−1
If the inequality is violated, we have:
2



 x1∗


1

1∗

x
=


 x1∗

 2
x∗ =

 x2∗


1

2∗

x
=


 x2∗

2
=2
=0
=0
= (1 + k)
which can be supported by a price ratio p∗1 = kp∗2 . To see this, notice that
at the corner solution above we must have:
p1 x11 = p1 + p2 k
and
p2 x22 = p1 + p2
both of which are equal to the aggregate endowment (in the left case of good
1, in the right case of good 2) at p∗1 = kp∗2 .
Show that the utility of individual 1 at equilibrium may decrease with k: the
individual may gain by destroying part of his endowment; does this contradict
the rationality of the individual when he does not ?
In this case, we can simply focus on the interior solution (for you to check
whether parameter values derived here are compatible with it).
The indirect utility of individual 1 at equilibrium is given by:
V 1 = 1 + k(1 + k)α−1
(D)
Taking the derivative of (D) with respect to k we get:
(1 + k)α−1 + k(α − 1)(1 + k)α−2
(E)
We need to show that there exists a pair of (α, k) such that (E) is negative.
Notice that (E) is negative if and only if:
1+
k(α − 1)
< 0 (F )
1+k
3
and therefore whenever α < −k −1 . The RHS is always negative, but since
α < 1 there are values of α for which (F) holds.
The result does not contradict rationality, since the demands have been
derived optimally.
Exercise 4
The utility function of an individual is u1 (x) = (x11 )2 + (x12 )2 and his endowment is e1 = (1, 2). The utility function of another individual is u2 (x) =
x21 + a(x22 ) for a > 0 and his endowment is e2 = (2, 1).
Compute the competitive equilibrium prices and allocations
Consider individual 1. His utility function is strictly convex in both arguments, hence we know we should look for corner solutions. In particular:
1∗
1. If p1 > p2 then x1∗
1 = 0 and x2 = p1 /p2 + 2;
1∗
2. If p1 < p2 then x1∗
2 = 0 and x2 = 2p2 /p1 + 1;
3. If p1 = p2 then both bundles above are optimal, and nothing else.
Now consider individual 2. As we know from the previous exercises, we
have:
2∗
1. If p2 > ap1 then x2∗
2 = 0 and x1 = 2 + p2 /p1 ;
2∗
2. If p2 < ap1 then x2∗
1 = 0 and x2 = 2p1 /p2 + 1;
3. If p2 = ap1 then every feasible bundle on the budget line is optimal.
Notice first that if p1 6= p2 there cannot be an equilibrium because of
individual 1 excess demand. Now we only have to analyse the possible cases
arising when p∗1 = p∗2 :
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1. If a < 1 then the CE is:



 x1∗


1

1∗

x
=


 x1∗

 2
x∗ =

 x2∗


1

2∗

x
=


 x2∗

2
=3
=0
=0
=3
2. If a > 1 then the CE is:



 x1∗


1

1∗

x
=


 x1∗

 2
x∗ =

 x2∗


1

2∗

x
=


 x2∗

2
=0
=3
=3
=0
3. Finally, if a = 1 both cases above are CE, and nothing else.
Compute the Pareto optimal allocations.
Notice that because of the preferences of individual 1, interior points
cannot be a PO. Once again, the PO set depends crucially on a:
1. If a < 1, then the PO set is given by the lower-right axes;
2. If a > 1, then the PO set is given by the upper-left axes;
3. if a = 1, then the PO set is given by all axes.
Do the two welfare theorems hold ?
The FWT holds because preferences are locally-non-satiated and the consumption set is convex.
The SWT does not hold because the preference relation of individual 1 is
not convex.
Compute the core allocations.
See the figure at the board.
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Exercise 3
The utility function of an individual is u1 (x) = x11 +ln x12 and his endowment is
e1 = (3, 0). The utility function of another individual is u2 (x) = min{x21 , x22 }
and his endowment is e2 = (0, 3).
Compute the Competitive Equilibrium
Consider agent one. I can rewrite the budget constraint as
x11 = 3 −
p2 1
x
p1 2
Plugging it inside the objective function (which is strictly concave) I get the
following (necessary and sufficient) FOC: x1∗
2 = p1 /p2 . Therefore, we know
that x1∗
1 = 2.
Now consider agent two. At any optimum we know that
x12∗ = x2∗
2 =
3p2
p1 + p2
From the market clearing condition for good one I get p∗1 = 2p∗2 .
Summing up, the CE of the economy is given by every price vector where
2∗
p∗1 = 2p∗2 > 0, which support the allocation where x2∗
1 = x2 = 1 and
1∗
1
=
x1∗
2 = 2.
The economy is modified to allow for production. A firm produces commodity 2 using the commodity 1 as input according to the technology:
{y : y2 = a(−y1 ), y1 ≤ 0, a > 0}
Compute a Competitive Equilibrium
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Let’s consider the profit maximisation problem for the firm first. As the
technology is linear (CRS - Constant Return to Scale), we have to analyse
cases:
1. If ap2 > p1 , then y1∗ → ∞, hence markets cannot clear, and a CE
cannot exist;
2. If ap2 < p1 , then y1∗ = 0, and the presence of production does not
matter: it is too costly to be used. There can be an equilibrium without
production if and only if a ∈ (0, 2), and it is identical to the one derived
above;
3. Finally, if ap2 = p1 , then y1∗ ≤ 0. Consider the market clearing condition for good one:
1 + y1 =
3p2
p1 + p2
y1 ≤ 0
Subsituting the inequality into the market clearing condition I get:
3p2
≤1
p1 + p2
which can be simplified to 2p2 ≤ p1 . As a result, an equilibrium with
production exists if and only if a ≥ 2.
Does the consumption sector play a role in the determination of equilibrium
prices ?
In the equilibrium without prodution, clearly the prices are fully determined by the demand schedules of the consumption sector.
However, this is not so in the equilibrium with production. Because of
the linearity of the production function, the price ratio is constrained to be
equal to a not to have an excess demand for the input. As a result, the price
ratio is given to the consumers.
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allocations ?
In this case, the consumption sector plays a role, while the production
sector does not.
does the answer generalize to other economies with production ?
It does not generalise beyond CRS production functions.
Exercise 4
Commodities are a consumption good, x, and labor, l. The endowment of
each individual consists of one unit of labour. The utility function of an
individual is u1 (x, l) = x(1 − l), x ≥ 0 and l ∈ [0, 1]. The utility function
of another individual is u2 (x) = x. The technology of a firm is {x : x ≤
√
−l, l ≤ 0}. Individual 2 owns the firm.
Write the profit of the firm and the budget constraint of each individual.
1. The profits of the firm are given by π ≡ px x + pl l;
2. The budget constraint of individual one is px x1 ≤ pl l1 ;
3. The budget constraint of individual two is px x2 ≤ pl l2 + π.
Write the supply and demand or excess demand functions of each individual
as well as the firm.
Let me start with the firm. Plugging into the profit function the tech√
nological constraint I get π = px −l + pl l. By strict concavity, the FOC is
necessary and sufficient for an optimum. I get:
px
(−l)1/2 + pl = 0
2
I can solve for l and obtain:
px
l =−
2pl
∗
8
2
which implies that
x∗ =
px
2pl
Now consider agent one. Because of LNS, I can apply the substitution
method and, by strict concavity of the objective function, I can solve for the
maximum simply considering the FOC:
pl
(1 − 2l1 ) = 0
px
which implies that l1∗ = 0.5, and plugging back into the budget constraint
x1∗ =
pl
2px
Finally, consider agent two. We know that he is going to supply labour
inelastically, i.e. l2∗ = 1. This is because he does not get disutility from
labour, and he owns the firm. Hence, the budget constraint can be rewritten
as
x
2∗
pl + π ∗
=
px
where
π∗ =
pl p2
p2
p2x
− 2x = x
2pl
4pl
4pl
Combining the latter with the budget constraint I get:
x2∗ =
pl
px
+
px 4pl
Prove that there exists a unique competitive equilibrium price.
Consider the market clearing condition for good x:
pl
pl
px
px
+
+
=
2px px 4pl
2pl
√
Solving for the price ratio I get p∗x /p∗l = 6.
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For the other market clearing (either for labour or for leisure) refer to the
discussion we had in class. If unclear do let me know!
Does equilibrium in the market for the consumption good guarantee equilibrium in the labor market as well ?
We do not need to clear the other market because of Walras’ Law.
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