Georg Nöldeke
Herbstsemester 2012
Advanced Economic Theory
Hints for Problem Set 2
1. Properties of the Cobb-Douglas Production function:
(a) The Cobb-Douglas production function is homogenous of degree α + β. This is
demonstrated by the following calculation:
f (tx1 , tx2 ) = A(tx1 )α (tx2 )β
β β
= Atα xα
1 t x2
β
= tα+β Axα
1 x2
= tα+β f (x1 , x2 ).
(b) To verify strict concavity, it suffices to show that the Hesse matrix is negative
definite. To obtain the Hesse matrix, we first calculate the partial derivatives
(the final equalities will serve to simplify subsequent calculations):
∂f (x1 , x2 )
= αAx1α−1 xβ2
∂x1
∂f (x1 , x2 )
β−1
= βAxα
1 x2
∂x2
α
f (x1 , x2 )
x1
β
=
f (x1 , x2 )
x2
=
and then calculate the second-order partial derivatives (the first equality in each
line just introduces a more convenient notation):
∂ 2 f (x1 , x2 )
∂x21
2
∂ f (x1 , x2 )
=
∂x22
∂ 2 f (x1 , x2 )
=
∂x1 ∂x2
∂ 2 f (x1 , x2 )
=
∂x2 ∂x1
α(α − 1)
f (x1 , x2 )
x21
β(β − 1)
f (x1 , x2 )
=
x22
αβ
=
f (x1 , x2 )
x1 x2
αβ
f (x1 , x2 )
=
x1 x2
f11 =
= α(α − 1)Ax1α−2 xβ2 =
f22
β−2
= β(β − 1)Axα
1 x2
f12
f21
= αβAx1α−1 x2β−1
= αβAx1α−1 x2β−1
For strict concavity two properties are needed:
• f11 < 0.
• f11 f22 − f12 f21 > 0.
The first condition is satisfied for 0 < α < 1. Because we have assumed that
α > 0, β > 0 and α + β < 1 holds, this condition is satisfied. To verify the second
condition, we calculate:
f (x1 , x2 ) 2
.
f11 f22 − f12 f21 = α(α − 1)β(β − 1) − α2 β 2
x1 x2
This expression is strictly positive whenever the expression in square brackets is
strictly positive. As we have
α(α − 1)β(β − 1) − α2 β 2 = αβ (1 − α − β) ,
and the right side of this equality is strictly positive when α > 0, β > 0 and
α + β < 1 holds, this finishes the proof.
1
(c) The following properties need to be checked:
• f (0, 0) = 0.
• f is strictly increasing.
• f is strictly quasiconcave.
The first of these should be obvious (f (0, 0) = A·0α ·0β = A·0·0 = 0). The second
is implied by the fact that the partial derivatives of f (·) are strictly positive (see
above).
To establish the third property, we can first observe that we have already established that every Cobb-Douglas production function with α + β < 1 is strictly
concave and, thus, strictly quasiconcave. Hence, it remains to consider the case
α + β ≥ 1. For this case observe that we can rewrite the production function as
h
iα+β+1
f (x1 , x2 ) = A f˜(x1 , x2 )
,
where
α/(α+β+1) β/((α+β+1)
f˜(x1 , x2 ) = x1
x2
.
The function f˜(·) is a Cobb-Douglas production function with exponents that sum
to less than one. Hence, it is strictly concave and, thus, strictly quasiconcave. As
f (·) is a positive monotonic transformation of f˜(·)1 it follows that f (·) is strictly
quasiconcave2 finishing the argument.
2. Cost Minimizaton and Duality:
(a) To derive the cost function for the Cobb-Douglas production function, we need
to consider the cost minimization problem. The Lagrangian for this problem is
L(x1 , x2 , λ) = w1 x1 + w2 x2 − λ [f (x1 , x2 ) − y] .
The corresponding Lagrange conditions are
w1 − λαAx1α−1 xβ2 = 0
β−1
w2 − λβAxα
=0
1 x2
β
y − Axα
1 x2 = 0.
It will prove to be a good idea to rewrite these equations as3
αλy =w1 x1
βλy =w2 x2
β
y =Axα
1 x2 .
1 See
Theorem 1.2 in the textbook for the definition of a positive monotonic transformation.
is easy enough to check from the definition of strict quasiconcavity. Alternatively, the result follows
from Theorems 1.2 and 1.3 in the textbook. Viewing f˜(·) as a utility function, the second part of Theorem 1.3
implies that f˜(·) represents a strictly convex preferences relation. As a positive monotonic transformation
of f˜(·) the function f (·) represents the same preference (Theorem 1.2). Using the other direction of the
statement in the second part of Theorem 1.3 it follows that f (·) is strictly quasiconcave.
3 To obtain the first of the following equations multiply the first equation above by x and then observe
1
β
that the term multiplying α is equal to Axα
1 x2 = y. The argument for the second of the following equations
is analogous.
2 This
2
The minimal cost of producing output y is given by w1 x1 + w2 x2 where (x1 , x2 )
(together with λ) is the solution to the Lagrange conditions. As we know that the
solution to the Lagrange condition satisfies the first two of the above equations
we know that the cost function is given by
c(w1 , w2 , y) = w1 x1 + w2 y2 = (α + β) λy.
(1)
Hence, to determine the cost function we only need to determine the value of λ.
Solving the first two of the rewritten Lagrange conditions for
x1 =
βλy
αλy
and x2 =
w1
w2
and substituting the result into the third of these conditions yields:
y=A
αλy
w1
α βλy
w2
β
Solving this equation for λ yields:4
λ=
w α/(α+β) w β/(α+β)
2
1
α
β
A−(1/(α+β)) y (1−α−β)/(α+β) .
Substituting this value for λ into (1) we obtain the cost function as
c(w1 , w2 , y) = Bw1γ w2δ y where
= 1/(α + β), γ =
α
β
, δ=
α+β
α+β
(2)
(3)
and
B = A−1/(α+β) (α + β)α−α/(α+β) β −β/(α+β) .
(4)
(b) As a function of (w1 , w2 ) the cost function in (2) has the same functional form as
the Cobb-Douglas production function. Because the parameters γ and δ satisfy
γ > 0, δ > 0 and γ + δ = 1, the calculations from Problem 1 (b) show that the
Hesse matrix of the second order partial derivatives with respect to (w1 , w2 ) is
negative semi-definite. This implies that the cost function is concave in (w1 , w2 ).
(c) Given a cost function as in (2), we can use (3) to determine the parameters of
the underlying Cobb-Douglas production function as
α = γ/ and β = δ/.
4 Write
the above equation as
y = Ãλα+β y α+β ,
where
à = A
α
w1
α β
w2
β
.
Dividing both sides by Ãy α+β yields
λα+β = Ã−1 y 1−α−β ,
implying
λ = Ã−1/(α+β) y (1−α−β)/(α+β) .
Replacing à by the expression from above yields the following equation.
3
Once α and β are known, for any given value of B we can solve (4) to detemine
A.5
3. Profit Maximization and Duality:
(a) To simplify notation when considering the profit maximization problem (in which
the focus is on the optimal choice of y) write the cost function as
c(w1 , w2 , y) = c(w1 , w2 , 1)y ,
where
c(w1 , w2 , 1) = Bw1γ w2δ .
The first order condition for the profit maximization problem6 is
p = c(w1 , w2 , 1)y −1 .
(5)
Solving this condition for y yields the output supply function
1/(1−)
y(p, w1 , w2 ) = (c(w1 , w2 , 1))
p1/(−1) .
The profit function is given by
π(p, w1 , w2 ) = py(p, w1 , w2 ) − c(w1 , w2 , 1) (y(p, w1 , w2 )) .
(6)
Rather than first substituting y(p, w1 , w2 ) into this expression and then simplifying, a better approach is to observe that we can use (5) to rewrite (6) as follows:
π(p, w1 , w2 ) = c(w1 , w2 , 1) (y(p, w1 , w2 )) − c(w1 , w2 , 1) (y(p, w1 , w2 ))
= ( − 1) c(w1 , w2 , 1) (y(p, w1 , w2 ))
1/(1−)
Substituting y(p, w1 , w2 ) = (c(w1 , w2 , 1))
simplifying yields:
p1/(−1) into this expression and
π(p, w1 , w2 ) = ( − 1) /(1−) c(w1 , w2 , 1)1+/(1−) p/(−1)
= ( − 1) /(1−) c(w1 , w2 , 1)1/(1−) p/(−1)
(b) Hotelling’s lemma states the the input demand functions are given by
xi (p, w1 , w2 ) = −
∂π(p, w1 , w2 )
.
∂wi
Using the expression for the profit function obtained in the previous problem, the
relevant partial derivative of the profit function can be determined (use the chain
rule!) as
∂π(p, w1 , w2 )
∂c(w1 , w2 , 1)
= −/(1−) c(w1 , w2 , 1)/(1−) p/(−1)
∂wi
∂wi
5 I don’t give the formula for A here as it is not important. What is important is the idea that the
parameters of the production technology can be determined from the parameters of the cost function.
6 The assumption α + β < 1 ensures that objective function in the profit maximization problem is strictly
concave in y, implying that the solution to the first order condition is indeed the profit maximizing output.
4
Recalling that c(w1 , w2 , 1) = Bw1γ w2δ we can calculate
∂c(w1 , w2 , 1)
= γBw1γ−1 w2δ
∂w1
∂c(w1 , w2 , 1)
= δBw1γ w2δ−1
∂w2
to obtain the input demand functions as
x1 (p, w1 , w2 ) = /(1−) c(w1 , w2 , 1)/(1−) p/(−1) γBw1γ−1 w2δ
x2 (p, w1 , w2 ) = /(1−) c(w1 , w2 , 1)/(1−) p/(−1) δBw1γ w2δ−1 .
(c) Recalling the functional form of the profit function,
π(p, w1 , w2 ) = ( − 1) /(1−) c(w1 , w2 , 1)1/(1−) p/(−1) ,
it should be clear that can be determined by considering how the profit changes
with the output price. Once is determined, the unit cost function c(w1 , w2 , 1)
can be determined by considering how the profit changes with w1 and w2 . Hence,
we can infer the parameters A, , γ, and δ of the cost function from the profit
function. The parameters of the production function can then be recovered from
the parameters of the cost function.
5