Theoretical Basis for Measuring Social Costs Essay

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Appendix B. Theoretical basis for measuring social costs

In order to analyze cases where a change in environmental quality (q) affects individual preference and demand, a basic individual utility maximization problem is considered i.e., maximizing a strictly concave utility function subject to the compact (closed and bounded) budget set (Freeman 2003):

Maximize┬(x_j )⁡〖u(x,q)〗 subject to y = ∑_(j=1)^n▒p_j x_j (B-1)

where u is individual’s utility, x is a vector of private goods quantities x = (x_1, x_2, …, x_n), and q is the level of environmental quality, which is a scalar fixed exogenously. The variable y is income, and p is the price vector of private goods p = (p, p_2, …, p_n). This
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Suppose a decrease in environmental quality (CS < 0 and ES < 0). CS is WTA for enduring the loss, the compensation which would keep the loser at an initial welfare level. ES is the loser’s WTP for avoiding the loss which would place the loser at his subsequent welfare level (Carson and Hanemann 2005).

Figure B. 1. Summary of Surplus Measures Changes in Quantity and Utility WTP WTA
Quantity Increase q^0 to 〖 q〗^1 (u^1≥ u^0) CS ES
Quantity Decrease q^0 to 〖 q〗^1 (u^1≤ u^0) ES CS
Source: Pearce (2002)
The associated dual problem to (B-1) is minimizing total consumer expenditures needed to maintain a given level of utility (Hanemann 1991):

Minimize┬(x_j )⁡〖∑_(j=1)^n▒p_j x_j 〗 subject to u=u (x,q) (B-3)

where p_j is the price vector of private good j, x_j is a vector of quantities of private good j, and ∑_(j=1)^n▒p_j x_j is the total consumer expenditures given utility u. This solution yields a set of Hicksian demand functions, h_j = h_j(p, q, u) for j = 1,2,…,n, and expenditure functions, e(p, q, u) ≡ ∑_(j=1)^n▒p_j h_j (p, q, u). In terms of this expenditure function, CS and ES are defined by (Freeman 2003; Carson and Hanemann 2005):

CS = e (p, q^0, u^0)- e (p, q^1,〖 u〗^0) = y – e (p, q^1,〖 u〗^0 ) (B-4a)

ES = e (p, q^0,u^1)- e (p,

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