1 A Recursive Dynamic Computable General Equilibrium Model For ...

The second condition is that the aggregator function (A.2) must be linear
homogeneous with respect to each of its inputs. It can be shown that the output price
composite of a linearly homogeneous function is linearly homogeneous in input prices.
Thus, the aggregate quantity and price indices are equal to the sum of the prices and
quantities of the inputs derived in each nest:
n
RX = ∑
i=
1
r
i xi
53
(A.7)
A basic property of linear homogeneous functions is that first order derivatives (i.e.
marginal products/utilities) are homogeneous of degree zero. To demonstrate this
property, take the case of a linearly homogeneous Cobb-Douglas production
function. Hence for a two input production function, MP1 is given as:
∂Y
α −1
β
MP1
= = αAx1 x2
(A.8)
∂x
1
Multiplying each of the inputs by a scalar, λ, yields:
∂Y
MP1
= = αA(
λx
)
∂x
∂Y
MP1
= = λ
∂x
1
1
1
α −1+
β
α −1
1
αAx
∂Y
0
MP1
= = λ [ αAx
∂x
α −1
1
( λx
α −1
1
x
x
β
2
2
]
)
β
β
2
(A.9)
Thus, multiplying both inputs by λ leaves the marginal product of x 1 unchanged. In
other words the marginal products are zero degree homogeneous in inputs. The same
outcome can be proved for input x2. Since the MRTS is the ratio of MPs, then proportional
increases in both inputs by the scalar value λ (implying higher isoquant levels) have no
affect on the MRS. Thus, a ray from the origin must cut all isoquants (indifference) curves
at points of equal slope. Green (1971) states that the isoquants (indifference) curves are
therefore ‘homothetic with respect to the origin’ (pp141).
As a result of this property, Allanson (1989) notes that,
‘optimal factor (commodity) allocations are independent of the level of (aggregate) output
(income)’ (pp.1).