Chapter 14.5, Problem 17E

Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Chapter
Section

Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

Utility Suppose that the utility function for two products is given by U   =   x 2 y , and the budget constraint is 2 x   +   3 y   — 120 . Find the values of x and y that maximize utility. Check by graphing the budget constraint with the indifference curve for maximum utility and with two other indifference curves.

To determine

To calculate: The values of x and y that maximize utility. Suppose that the utility function for two products is given by U=x2y and that the budget constraint is 2x+3y=120. Check by graphing the budget constraint with the indifference curve for maximum utility and with two other indifference curves.

Explanation

Given Information:

The provided utility function is U=x2y subject to the constraint 2x+3y=120.

Formula used:

According to the Lagrange multipliers method to obtain maxima or minima for a function z=f(x,y) subject to the constraint g(x,y)=0,

(1) Find the critical values of f(x,y) using the new variable Î» to form the objective function F(x,y,Î»)=f(x,y)+Î»g(x,y).

(2) The critical points of f(x,y) are the critical values of F(x,y,Î») which satisfies g(x,y)=0.

(3) The critical points of F(x,y,Î») are the points that satisfy âˆ‚Fâˆ‚x=0, âˆ‚Fâˆ‚y=0, and âˆ‚Fâˆ‚Î»=0, that is, the points which make all the partial derivatives of zero.

For a function f(x,y), the partial derivative of f with respect to y is calculated by taking the derivative of f(x,y) with respect to y and keeping the other variable x constant. The partial derivative of f with respect to y is denoted by fy.

Power of x rule for a real number n is such that, if f(x)=xn then fâ€²(x)=nxnâˆ’1.

Constant function rule for a constant c is such that, if f(x)=c then fâ€²(x)=0.

Coefficient rule for a constant c is such that, if f(x)=câ‹…u(x), where u(x) is a differentiable function of x, then fâ€²(x)=câ‹…uâ€²(x).

Calculation:

Consider the function, U=x2y.

The provided constraint is 2x+3y=120.

According to the Lagrange multipliers method,

The objective function is F(x,y,Î»)=f(x,y)+Î»g(x,y).

Thus, f(x,y)=x2y and g(x,y)=2x+3yâˆ’120.

Substitute x2y for f(x,y) and 2x+3yâˆ’120 for g(x,y) in F(x,y,Î»)=f(x,y)+Î»g(x,y).

F(x,y,Î»)=x2y+Î»(2x+3yâˆ’120)

Since, the critical points of F(x,y,Î») are the points that satisfy âˆ‚Fâˆ‚x=0, âˆ‚Fâˆ‚y=0, and âˆ‚Fâˆ‚Î»=0.

Recall that, for a function f(x,y), the partial derivative of f with respect to y is calculated by taking the derivative of f(x,y) with respect to y and keeping the other variable x constant.

Use the power of x rule for derivatives, the constant function rule and the coefficient rule.

Thus,

âˆ‚Fâˆ‚x=0(2x)y+Î»(2)=02Î»=âˆ’2xyÎ»=âˆ’xy

And,

âˆ‚Fâˆ‚y=0x2+Î»(3)=03Î»=âˆ’x2Î»=âˆ’x23

And,

âˆ‚Fâˆ‚Î»=02x+3yâˆ’120=02x+3y=120

Solve the equations Î»=âˆ’xy and Î»=âˆ’x23.

Substitute âˆ’xy for Î» in Î»=âˆ’x23.

âˆ’xy=âˆ’x23âˆ’3xy=âˆ’x2x2âˆ’3xy=0x(xâˆ’3y)=0

Simplify it further,

x=0 or xâˆ’3y=0

That is, x=0 or x=3y

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