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Consider the utility function U(x, y) = 7xy with the price of X equal to 55 and the price of Y equal to 48.  Income devoted to the purchase of x and y is $250.

 1) Find the equilibrium quantities of X and Y. (Use the LaGrange Method)2) Using the budget constraint and utility function graph what this looks like.3) What happens to equilibrium quantities of X and Y if income increases to $375. (Use the LaGrange Method)4) What happens to equilibrium quantities of X and Y if the price of X decreases to $45 and income remains at $375?5) Does MRS = Price Ratio?

L = 7xy + λ(250-55X-48Y)

The first order conditions are

dL/dx = 7y-55λ

or y = 55λ /7 (1)

dL/dy = 7x-48λ

x= 48λ/7 (2)

dividing (1) by (2)

y//x = 55/48

or 48Y = 55X

dL/dλ = 250-55X-48Y = 0

= 250-55X -55X =0 and = 250-48Y-48Y = 0

110X = 250 and 96Y = 250

X = 250/110 = 25/11 = 2.27 or 3 units

Y = 250/96 = 2.604 or 3 units

If income increases to 375

L = 7xy + λ(375-55X-48Y)

The first order conditions are

dL/dx = 7y-55λ

or y = 55λ /7 (1)

dL/dy = 7x-48λ

x= 48λ/7 (2)

dividing (1) by (2)

y//x = 55/48

or 48Y = 55X

dL/dλ = 350-55X-48Y = 0

= 375-55X -55X =0 and = 375-48Y-48Y = 0

110X = 375 and 96Y = 375

X = 375/110 = 3.40 or 4 units

Y = 375/96 = 3.906 or 4 units

L = 7xy + λ(375-55X-45Y)

The first order conditions are

dL/dx = 7y-55λ

or y = 55λ /7 (1)

dL/dy = 7x-45λ

x= 45λ/7 (2)

dividing (1) by (2)

y//x = 55/45

or 45Y = 55X

dL/dλ = 350-55X-45Y = 0

= 375-55X -55X =0 and = 375-45Y-45Y = 0

110X = 375 and 90Y = 375

X = 375/110 = 3.40 or 4 units

Y = 375/90 = 4.166 or 4 units