POSTING PROBLEM 7 FOR REFERENCE Problem 9. Solve a Cobb-Douglas Utility Maximiza

Question

POSTING PROBLEM 7 FOR REFERENCE

Problem 9. Solve a Cobb-Douglas Utility Maximization Problem by Directly Applying the Properties of Cobb Douglas Function (4 points) Given that we have proved that if a consumer has a Cobb Douglas utility function and a budget constraint PxX+PyY- I. This implies that at his utility maximization Xayb solution (X, Y) Use this relationship directly to solve for the consumption bundle in Problem 7 that maximizes Jackie's satisfaction. Verify that MRSxy=Px/Py at the optimal solution.

Explanation / Answer

Answer 9

As consumer is spending fixed proportion of his income on on digital music (X) and CDs (Y) with 80% and 20% respectively Which clearly means that utility function is a cobb douglas function i.e.,

U = XaYb. To calculate we have to solve a/(a+b) = 0.8 and b/(a+b) = 0 using property of cobb douglas function.

Hence a = 0.8 and b = 0.2

Hence U = X0.8Y0.2

Because it is a property of cobb dauglas function that consumer spend fix proportion of income on the goods when they have Cobb - Douglas preference.

Now lets solve for X and Y

PXX =  a/(a+b)*I = 0.8*40 and PX = 1

Hence X = 32 and

PYY = b/(a+b)*I = 0.2*40 and PY = 8

Hence, Y = 1

Hence X = 32 and Y = 1 is an optimal combination

MUX = partial differentiation of U wrt X

= 0.8*(Y/X)0.2

Similarly, MUY = 0.2*(X/Y)0.8

MRSXY = MUx/MUy =  0.8*(Y/X)0.2/ 0.2*(X/Y)0.8 = 4Y/X

= 4*1/32 = 1/8 = 0.15

PX/PY = 1/8 = 0.15

Hence MRSXY = PX/PY  

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