Let U(x,y)= Ax^ay^b be the utility function of an individual. The individual has x hour leisure time per day and consumers y units of other goods. The individual works and is paid w $ per hour. The average price of the other goods is p $. We assume that the individual spends his/her total income i.e.
py = w(24 - x)
Use the Lagrange method to determine how many hours this individual works per day to maximize the utility.© BrainMass Inc. brainmass.com October 10, 2019, 3:36 am ad1c9bdddf
To use the Lagrange Multiplier, we first set up the Lagrange equation.
Λ = f(x,y) + λ[g(x,y) - c]. where f(x,y) is the function we want to maximize (or minimize), and g(x,y) = c is the constraint. λ is a free parameter.
In our case, the function f(x,y) = A(x^a)(y^b), and the constraint is py = 24w - wx, or wx + py = 24w
Λ = A(x^a)(y^b) + ...
This solution explains how to maximize utility with the Lagrange method.