1. Consider the following optimization problem consisting of two individuals Ann

Question

1. Consider the following optimization problem consisting of two individuals Ann and Bill A and B, two goods commodity X and commodity Y, and two factor inputs labor and capital, L and K: Max W W(uA, UB) subject to K-K+KY XAtXB-XK', Lx) -=Lx + LY (Endowments) YA+YB-YKY, LY) (Technologies) (Preferences) where W(UA, U is a Bergson-Samuelson social welfare function. Using the method of Lagrange multiplier, solve for the four first-order conditions for social welfare maximization. Assume that the second-order conditions are satisfied. State the 10 equations and 10 unknowns. Give an economic interpretation of each of the optimality conditions, i.e., the three economic efficiency conditions and the equity condition, that you have derived

Explanation / Answer

When you want to maximize (or minimize) a multivariable function f(x,y,…) subject to the constraint that another multivariable function equals a constant, g(x,y,…)=c follow these steps:

Step 1: Introduce a new variable ? and define a new function L as follows:

L(x,y,…,?)=f(x,y,…)??(g(x,y,…)?c)

This function Lmathcal{L}L is called the "Lagrangian", and the new variable ? is referred to as a "Lagrange multiplier"

Step 2: Set the gradient of L equal to the zero vector.

?L(x,y,…,?)=0?Zero vector

In other words, find the critical points of L.

Step 3: Consider each solution, which will look something like (x0,y0,…,?0) Plug each one into f. Or rather, first remove the ? as an input. Whichever one gives the greatest (or smallest) value is the maximum (or minimum) point your are seeking.

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