Capítulo 1: La moda como reflejo social
1.3 Vestuario de cine: lenguaje no verbal…
Inverse demand systems are widely used in economic studies to analyze the demand of fish and seafood (Barten and Bettendorf 1989; Burton 1992; Eales,
Durham, and Wessells 1997; Holt and Bishop 2002; Park, Thurman, and Easley 2004; Y. Lee and Kennedy 2008; Dedah, Keithly, and Kazmierczak 2011; M.-Y. A. Lee and Thunberg 2013). The inverse demand systems can be derived using specification of distance functions (Eales and Unnevehr 1991; Brown, Lee, and Seale Jr 1995).
Inverse demand systems are mainly used to analyze demand of perishable goods such as fruits and vegetables, meat products, and seafood. The supply of perishable products is very inelastic in the short term and therefore
fishermen/producers are price takers (Barten and Bettendorf 1989). In such case, the price of good is affected by quantity and it is reasonable to assume that the normalized price with respect to income is a function of quantity of the good available and total real expenditure of all goods considered (Barten and Bettendorf 1989; Burton 1992) as defined in inverse demand system. Moreover, looking at the policy perspective,
usually the fisheries management authorities are interested in understanding the effect of quantity harvested on price since they are interested in setting policy standards for the quantity harvested (M.-Y. A. Lee and Thunberg 2013).
In this study we used the Inverse Almost Ideal Demand System (IAIDS) model developed by Eales and Unnevehr (1991), which derives demand from distance
function2 a dual to the expenditure function. The function is assumed to have a linear-
2 Deaton (1979) and Deaton and Muellbauer (1980b) describes the use of distance function in demand
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homogenous, concave, non-decreasing in quantities, and decreasing in utility similar to the properties explained for cost function in AIDS model.
The general form of IAIDS can be written as Eales and Unnevehr (1991):
𝑤𝑤𝑖𝑖 = 𝛼𝛼𝑖𝑖 + � 𝛾𝛾𝑖𝑖𝑖𝑖ln 𝑞𝑞𝑖𝑖 + 𝛽𝛽𝑖𝑖ln 𝑄𝑄,
𝑖𝑖 (1)
where wi is the value share3 of good i, qj is the quantity of good j, Q is the quantity index,4 and α, β, and γ are parameters. The quantity index, ln Q derived by Eales and Unnevehr (1991) is as follows:
ln 𝑄𝑄 = 𝛼𝛼0+ � 𝛼𝛼𝑖𝑖ln 𝑞𝑞𝑖𝑖+ 0.5 � � 𝛾𝛾𝑖𝑖𝑖𝑖ln 𝑞𝑞𝑖𝑖ln 𝑞𝑞𝑖𝑖 𝑖𝑖
𝑖𝑖
(2)
Due to the non-linear nature of the quantity index, most researchers use a linear approximation of this quantity index in their study for the ease of computation (Moschini 1995). Stone’s Quantity index is a widely used quantity index similar to the original suggestion of Deaton and Muellbauer (1980a). The Stone’s quantity index can be written as:
ln 𝑄𝑄 = ∑ 𝑤𝑤𝑖𝑖 𝑖𝑖 ln 𝑞𝑞𝑖𝑖, (3)
substituting equation (3) into (1), and noting that our data is a time series we add a t subscript to the resulting equation
attain a particular indifference curve. Refer Deaton (1979) and Eales and Unnevehr (1991) for further details.
3 Value share of a product is the ratio of its value to the total value of product in consideration which
can be written as
4 Quantity index does not have a meaningful interpretation; it exists merely due to mathematical
derivation of equation (1). For details refer to (Eales and Unnevehr 1991).
Value share ( ) i , i j j Value w Value = ∑
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𝑤𝑤𝑖𝑖𝑖𝑖 = 𝛼𝛼𝑖𝑖𝑖𝑖+ � 𝛾𝛾𝑖𝑖𝑖𝑖ln 𝑞𝑞𝑖𝑖𝑖𝑖+ 𝛽𝛽𝑖𝑖� 𝑤𝑤𝑖𝑖ln 𝑞𝑞𝑖𝑖, 𝑖𝑖
𝑖𝑖 (4)
The intention of the demand model was to measure the relationship between the price and quantity of good, but not just to find out how the budget share of a good is influenced by different factors. We can estimate the relationship by taking the marginal derivative of the budget share with respect to either quantity or price depending on the type of demand model we used. In direct demand models such as AIDS, price elasticity would be estimated to measure such relationships. However, in inverse demand systems price flexibility5 will be calculated to analyze the effect of quantity harvested on price of the good. Price flexibility can be derived from the estimated parameters of IAIDS model by taking derivative of the share equation with respect to the log of quantity. We can write the equation as follows:
𝜕𝜕𝑤𝑤𝑖𝑖⁄𝜕𝜕 ln 𝑞𝑞𝑖𝑖 = 𝛾𝛾𝑖𝑖𝑖𝑖 + 𝛽𝛽𝑖𝑖�𝜕𝜕 ln 𝑄𝑄 𝜕𝜕 ln 𝑞𝑞⁄ 𝑖𝑖�, (5)
which can be rewritten as:
𝜕𝜕𝑤𝑤𝑖𝑖⁄𝜕𝜕 ln 𝑞𝑞𝑖𝑖 = 𝛾𝛾𝑖𝑖𝑖𝑖+ 𝛽𝛽𝑖𝑖�𝛼𝛼𝑖𝑖 + � 𝛾𝛾𝑘𝑘𝑖𝑖ln 𝑞𝑞𝑖𝑖 𝑘𝑘
� (6)
But, from equation (4), 𝛼𝛼𝑖𝑖+ ∑ 𝛾𝛾𝑘𝑘 𝑘𝑘𝑖𝑖ln 𝑞𝑞𝑖𝑖 = 𝑤𝑤𝑖𝑖− 𝛽𝛽𝑖𝑖ln 𝑄𝑄
Therefore equation (6) can be rewritten as:
𝜕𝜕𝑤𝑤𝑖𝑖⁄𝜕𝜕 ln 𝑞𝑞𝑖𝑖 = 𝛾𝛾𝑖𝑖𝑖𝑖+ 𝛽𝛽𝑖𝑖(𝑤𝑤𝑖𝑖− 𝛽𝛽𝑖𝑖ln 𝑄𝑄) (7)
5 While mathematically price flexibility is an inverse of price elasticity, previous studies have shown
that price flexibility is best estimated using the proper demand model that does not require computing price elasticity as an intermediate step (Huang 2005).
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However, the derivation of the right hand side will yield,
𝜕𝜕𝑤𝑤𝑖𝑖⁄𝜕𝜕 ln 𝑞𝑞𝑖𝑖 = 𝜑𝜑𝑖𝑖𝑖𝑖𝑤𝑤𝑖𝑖 (8)
Thus, after adjusting the terms on either sides, we can write own-price flexibility of good i, denoted as φi, as
(9)
and cross-price flexibility between goods i and j, denoted φij, as
(10)
A scale flexibility can be estimated using the homogeneity aggregation relation (Eales and Unnevehr 1991).
𝑓𝑓𝑖𝑖 = −1 +𝑤𝑤𝛽𝛽𝑖𝑖 𝑖𝑖
(11)
The price flexibility is interpreted similar as that of price elasticity. A good is price inflexible if the absolute value of own-price flexibility (equation 9) is less than 1. This means that the price changes less than proportionally to the unit change in
quantity. The sign of cross-price flexibility will determine the substitutability and complementarities of two goods. If the cross-price flexibility is negative, then the two products in comparison are substitutes and if the measure is positive, then the products is complement to each other. Scale flexibility explains the change in price resulting from the expansion of total expenditure. Thus, if the scale flexibility is less than -1, the
(
ln)
1 ii i i i i i w Q w γ β β ϕ = − + + −(
ln)
ij i j j ij i w Q w γ β β ϕ = + −21
good is considered as necessary goods and if greater than -1, the good is considered as luxury goods (Park and Thurman 1999).