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}

Lagrangean with an equality constraint

max (x,y) √x + √y

s.t. p_xx + y = 1.

L(x, y, λ) = √x + √y − λ (p_xx + y − 1).

FOC

∂L/ ∂x = (1/2) * x^−1/2 − λp_x= 0 (1)

∂L ∂y = (1/2)*y ^−1/2 − λ = 0 (2)

∂L ∂λ = − (p_xx + y − 1)=0 (3)

Solving equations 1 and 2

y ^1/2/ x^1/2= p_x /1

y= p_x² x

substitute this into the budget constraint to get

p_xx +p_x² x=1

x(p_x² +p_x) =1

x*=1/(p_x² +p_x)

Similarly y* is

p_x*1/(p_x² +p_x)+y=1

p_x/(p_x² +p_x)+y=1

y*=1-p_x/(p_x² +p_x)

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.