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Homework answers / question archive / Consider the problem of maximizing the utility function u(x, y) = x'/2 + y'/2 on the budget set {(x, y) e R px + y = 1}
Consider the problem of maximizing the utility function u(x, y) = x'/2 + y'/2 on the budget set {(x, y) e R px + y = 1}. Show that if the nonnegativity constraints x > 0 and y > 0 are ignored, and the problem is written as an equality-constrained one, the resulting Lagrangean has a unique critical point. Does this critical point identify a solution to the problem? Why or why not?
Lagrangean with an equality constraint
max (x,y) √x + √y
s.t. pxx + y = 1.
L(x, y, λ) = √x + √y − λ (pxx + y − 1).
FOC
∂L/ ∂x = (1/2) * x−1/2 − λpx= 0 (1)
∂L ∂y = (1/2)*y −1/2 − λ = 0 (2)
∂L ∂λ = − (pxx + y − 1)=0 (3)
Solving equations 1 and 2
y 1/2/ x1/2= px /1
y= px2 x
substitute this into the budget constraint to get
pxx +px2 x=1
x(px2 +px) =1
x*=1/(px2 +px)
Similarly y* is
px*1/(px2 +px)+y=1
px/(px2 +px)+y=1
y*=1-px/(px2 +px)
The critical point will identify the solution of this problem. The budget constraint can be written with equality because the utility function is strictly increasing both in x and y, the consumer will never choose a point on the interior of the budget set.
