9.5 The Lagrange formalism for the example of another utility function

We now solve the well-known problem of the household optimum for the utility function U x,y = α ln x + βy for a given budget B, when the prices of goods are px and py, i.e.

max x,yα ln x + βy under the condition that xpx + ypy = B.

We form the Lagrange function

𝕃 x,y,λ = α ln x + βy + λ xpx + ypy B

and the first order conditions:

d dx α ln x + βy + λpx = α x + λpx = 0 (9.14) d dy α ln x + βy + λpy = β 1 2y + λpy = 0 (9.15) xpx + ypy B = 0 (9.16)

The FOC 3 represents the second order condition. We transform the other two by adding λpx and λpy, respectively

α x = λpx (9.17) β 2y = λpy (9.18)

and then dividing the equations with each other. This cancels λ.

α x β 2y = px py (9.19)

The resulting equation represents the central point of the solution. It represents a relationship between the marginal utility ratio and the price ratio. We now summarize this equation with the constraint, where we have slightly transformed both:

xpx + ypy = B 2αpy β y = xpx

We insert and obtain

B = 2αpy β y + ypy

and therefore the only positive solution (Note: Consider this as a quadratic equation in y!)

y = α β + α β2 + B py 2

x then results as

x = 1 px 2αpy β y = py px 2α β α β + α β2 + B py .

(c) by Christian Bauer
Prof. Dr. Christian Bauer
Chair of monetary economics
Trier University
D-54296 Trier
Tel.: +49 (0)651/201-2743
E-mail: Bauer@uni-trier.de
URL: https://www.cbauer.de