MICROECONOMIC THEORY I
PROBLEM SET 3
MARCIN PĘSKI
Hotelling Lemma for inputs. Let z ∈ RN be the vector of inputs, let w ∈ RL be
the vector of prices of the inputs. Let f (z) be a single-output production function.
Define the profit function
π (w) := arg max
f (z) − w · z
z
(we normalize the price of output to 1) and let z (w)be the optimal demand for inputs
given vector prices w. Show that
z (w) = −∇w π (w) .
Assume all the differentiability that you want. Compare with the Hotelling’s Lemma
from class.
Define function h (z; w) = f (z) − w · z. Apply The Envelope Theorem.
Theory of firm behavior.
(1) MWG 5.C.8.
TBA
(2) (Computations) MWG 5.C.9 and 5.C.10.We find the conditional factor demands in the case of CES technology. The cost minimization problem is
min w1 z1 + w2 z2 st. (z1ρ + z2ρ )1/ρ ≥ q.
The FOC are
wi − λρziρ−1 = 0 for each i,
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MARCIN PĘSKI
which implies that
ρ
− 1−ρ
zi = q w1
ρ
− 1−ρ
+ w2
w−i
=q 1+
wi
ρ
− 1−ρ
− ρ1
1
(wi )− 1−ρ
− ρ1
.for some constant c1 > 0 that can be derived from the constraint: c =
ρ
− 1−ρ
q w1
ρ
− 1−ρ
+ w2
−ρ
.
(3) (Revenue-maximization) MWG 5.C.13. (Hint: Start with proving a version
of Hotelling’s Lemma that is appropriate for this context.)
TBA
Facts about increasing differences. Prove the following observations
(1) Show that if f : R × T → R satisfies increasing differences, then f (., t) can
be ordered by the single crossing condition.
TBA
(2) Show that if f and g satisfy increasing differences, then f + g also.
TBA
(3) Show that any function that either does not depend on t or does not depend
on x satisfies weakly increasing differences.
TBA
(4) Show that functions f (x, t) = axt, axα tβ have increasing differences for any
positive constants a, α, β for x, t ≥ 0.
We will check the second case. We will do it directly using the definition
(instead of computing the cross-derivative, which is a simpler method in this
case). We need to show that for any x<x’ and t < t .
f (x , t ) − f (x, t ) − f (x , t) + f (x, t) > 0.
ADVANCED MICRO THEORY II
3
Indeed, notice that
f (x , t ) − f (x, t ) − f (x , t) + f (x, t)
=
Increasing differences and single crossing. Suppose that h : R → R is a strictly
increasing function.
(1) Show that if family f (., t) is ordered by single-crossing condition, then h (f (., t))
is also ordered by single-crossing condition.
Suppose that f (., t) is ordered by single-crossing condition and h (.) is
strictly increasing. Let g = h ◦ f. Then, for any x, x , t, t , we have
g (x, t) ≥ g (x , t ) iff f (x, t) ≥ f (x , t ) .
Together with increasing differences of f, the above means that
g (x , t) ≥ g (x, t) implies f (x , t) ≥ f (x, t) implies f (x , t ) ≥ f (x, t ) implies g (x , t ) ≥ f (x, t ) .
Thus, the first condition of the single-crossing is satisfied. The second condition is proven in the same way (we need to replace ≤ by < in the above
chain).
(2) Given an example of strictly increasing function h and family f (., t) that has
increasing differences, such that family h (f (., t)) does not have increasing
differences.
By the result from the class, function g = h ◦ f has increasing differences
iff and only if gxt ≥ 0. Notice that
∂2
g (x, t) = h (f (x, t, )) fx (x, t) ft (x, t) + h (f (x, t)) fxt (x, t) .
∂x∂t
If fuunction f has increasing differences, then fxt ≥ 0. If function h is strictly
increasing, then h > 0. Thus, the only chance of making the above negative
is to find increasing h with negative second-derivative, h << 0.
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MARCIN PĘSKI
For example, let f (x, t) = xt and let h (z) = −e−Az , where A > 0 is some
constant. Then,
gxt = −A2 e−Axt xt + Ae−Axt = Ae−Axt (1 − Axt) .
The above is negative for each x and t such that Axt > 1.
Comparative statics of savings decision. An agent chooses consumption in two
periods, c1 = w − s and c2 = s, where s are the savings. She has discounted separable
utility over two periods
u1 (w − s) + βu2 (s) ,
where we assume (only!) that functions ui are weakly increasing.
(1) Consider function
f (s; β) = u1 (w − s) + βu2 (s) .
Show that family f (.; β) is ordered by a single crossing condition. In the
lecture notes, we show that this implies that the savings increase with the
discount rate.
In class.
(2) Conclude that savnings increase with discount factor.
In class.
(3) (Harder) Consider a multi-period version of the consumuer problem
max
c1 +...+ct =w
u1 (c1 ) + βu2 (c2 ) + ... + β t−1 ut (ct ) .
Can you extend the above result, i.e., can you show that the first-period
savings increase with the discount factor?
We are going to use two intermediate results.
Lemma 1. Suppose that function φ (s, β) is increasing in s and it has increasing differences in s and β (or, equivalently, it is supermodular in (s, β)).
Then, function βφ (s, β) has increasing differences in s and β.
ADVANCED MICRO THEORY II
5
Proof. Notice that for each s < s and β < β ,
[β φ (s , β ) − β φ (s, β )] − [βφ (s , β) − βφ (s, β)]
=β (φ (s , β ) − φ (s, β )) − β (φ (s , β) − φ (s, β))
= (β − β) (φ (s , β ) − φ (s, β ))
+ β ([φ (s , β ) − φ (s, β )] − (φ (s , β) − φ (s, β))) ≥ 0.
The first term after the last equality is positive because β −β > 0 and because
φ (s, β) is increasing in s. The last term is positive because φ has increasing
differences.
Lemma 2. Suppose that function function φ (s, β) is increasing in s and it
has increasing differences in s and β. Suppose that u (s) isw an increasing
function. Define function
Φu,φ (w, β) = max u (w − s) + βφ (s, β) .
s
Then, Φu,φ is increasing in w and it has increasing differences in w and β.
Proof. Let
s (w, β) = arg max
u (w − s) + βφ (s, β) .
s
To see that Φu,φ (w, β) is increasing in w, take s ∈ s (w, β) and w < w . Then,
Φu,φ (w , β) ≥ u (w − s) + βφ (s, β) ≥ u (w − s) + βφ (s, β) ,
where the last inequality follows from the fact that u (.) is increasing.
Next, we show that Φu,φ (w, β) has increasing differences. Take any w < w
and β < β and let s ∈ s (w , β) and s ∈ s (w, β ). Observe that
Φu,φ (w, β) ≥ u (w − s) + βφ (s, β) ,
Φu,φ (w, β ) = u (w − s ) + β φ (s , β ) ,
Φu,φ (w , β) = u (w − s) + βφ (s, β) ,
Φu,φ (w , β ) ≥ u (w − s ) + β φ (s , β ) ,
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MARCIN PĘSKI
Then,
Φu,φ (w , β ) − Φu,φ (w, β ) − [Φu,φ (w , β) − Φu,φ (w, β)]
=Φu,φ (w , β ) + Φu,φ (w, β) − Φu,φ (w, β ) − Φu,φ (w , β)
≥u (w − s) + β φ (s , β ) + u (w − s) + βφ (s, β)
− u (w − s ) − β φ (s , β ) − u (w − s) + βφ (s, β)
≥β φ (s , β ) − β φ (s , β )
otice that function f (x, θ) = −u (x + θ) is supermodular in (x, θ) (by Exercise ??).
Lemma 3. Suppose that function u is concave. Then, function f (x, y) =
u (x − y) is supermodular in (x, y).
Proof. It is enough to check that f (x, y) has increasing differences in x and
y. Fix x < x and for each y, define
Dx,x (y =) f (x , y) − f (x, y) = u (x − y) − u (x − y) = u (x − x + x − y) − u (x − y) .
Then, for each y < y ,
Dx,x (y )−Dx,x (y) = [u (x − x + x − y − (y − y)) − u (x − y − (y − y))]−[u (x − x + x − y) − u (x −
Define function
f (s1 , ..., sn−1 ; β) = u1 (w − s1 ) + βu1 (s1 − s2 ) + ...β n−1 un (sn−1 ) .
By the Lemma and Exercise 3, for each β, f is a sum of supermodular functions on (n − 1)-dimensional lattice {(s1 , ..., sn−1 ) : si ≥ 0}. Thus, f is also
supermodular by Exercise 2. It is easy to check that f has increasing differences in ((s1 , ..., sn−1 ) ; β).
Our multi-variate comparative statics result shows that the optimal choice
of savings (s1 , ..., sn−1 ) is increasing in β.
I am still not convinced that we need to assume concavity of ui for this
result.
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