# Theoretical Basis for Measuring Social Costs Essay

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

*…show more content…*

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,