These exercises are adapted from Chapter 7 of Diewert’s Barcelona Lectures, Section 4.

Consider the following period t (net) product function (e.g. GDP function):

gt (P,x) ≡ max{P · y : (y,x)ϵSt }                                  (1)

for periods t = 1, 2 ..., where St  is the available technology set (closed convex cone that exhibits free disposability), P is a vector of output prices, y is a vector of the corresponding (net) outputs. (The quantities can be positive or negative; a negative quantity implies that the good is an intermediate input.)

We have the following first order partial derivatives of the function:

yt     =   ∇P gt (Pt ,xt ),

Wt     =   ∇xgt (Pt ,xt ),

(2)

(3)

where Wt  is a vector of input prices which corresponds to the input vector xt .

With constant returns to scale, then

gt (Pt ,xt ) = Pt  · yt  = Wt  · xt                                                                       (4)

Now, for two periods, 1 and 2, we can define a family of output indexes for each reference

output price vector P :

Q(P,x1 ,x2 ) ≡ g2 (P,x2 )

g1 (P,x) .

(5)

(6)

Note that only the technology (represented by the superscript on g) is changing in going from the denominator to the numerator, thus isolating technical change. If τ(P,x) is greater than one then there has been technical progress.