By Ruey S. Tsay
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Extra info for Analysis of Financial Time Series (Wiley Series in Probability and Statistics - Applied Probability and Statistics Section Series)
It is more interesting to consider biased technical progress. Here ξ = α K, and η = β L and η = (β − α )p. The first two equations lead to the conclusion we derived earlier that the equation for virtual expansion path for biased technical progress is Lα /K β = c1 , a constant. The last and middle equation leads to the condition pα K β −α = c2 . For an almost homothetic CES, the share ratio under competition can be shown to be − dL K = R(e(1/β ) logL − e(1/α ) logK ) dK L so if capital and labor are growing at an exponential rate α and β then the share ratio will be a constant; the share ratio is the differential invariant.
12). In short, output is thought as increasing in steps from (K, L) to (K + UK · t, L + UL · t), from (K + UK · t, L + UL · t) to t2 t2 K + UK + UK 2 , L + UL · t + UL2 , and so on till it reaches (K , L ) given by 2! 2! P1 . The output increases from Y1 to Yt . Now consider the process as starting from another input combination (K2 , L2 ) given by P2 in Fig. 5; technical progress will transform it in time t to (eα t K, eα t L). If the point corresponding to this input combination, P2 lies on the isoquant Y2 , then the effect of the technological change is to renumber the isoquant Y1 to Y2 .
F −λg n1 ⎣ n1 −g1 ⎤ ⎤⎡ ∂x ⎤ ⎡ 1 . . f1n − λ g1n −g1 −h1α ∂α ⎥ ⎢ ⎢ .. ⎥ .. ⎥ .. ⎥ ⎢ . . ⎥ ⎢ .. ⎥. ⎥ ⎢ ⎥ ⎥⎢ ⎥ ⎢ ∂x ⎥ ⎢ . . fnn − λ gnn −gn ⎥ ⎦ ⎣ ∂ αn ⎦ ⎣ −hnα ⎦ ∂λ ... 8) Since in this chapter, we are interested in the derivation of the Slutsky equation, we assume that the objective function representing the utility level of the consumer is concave and that the constraint is linear. We further specialize to the case in which the objective function is not affected by the transformation. For the n + 1 parameters, p and I and for λ , the Lagrangian is: n I − ∑ p i xi .
Analysis of Financial Time Series (Wiley Series in Probability and Statistics - Applied Probability and Statistics Section Series) by Ruey S. Tsay