Rob has the utility function U(x_1, x_2) = x_1(x_2 + 1). Rob\'s utility function

Question

Rob has the utility function U(x_1, x_2) = x_1(x_2 + 1). Rob's utility function is quasiconcave. What does that mean for Rob's preferences? The price of good 1 is 2 and the price of good 2 is 1. What is Rob's optimal choice when his income is 10 and what is then Rob's utility level? Write down Rob's maximization problem and make sure to state the theorem you are using, why you are using it and explain every step of your calculations. If Rob's income doubles and prices stay unchanged, will Rob's demand for both goods double? Explain carefully your answer. Thomas, Rob's friend, has the utility function. U(x_1, x_2) = squareroot [x_1(x_2 + 1)] + 4 If the price of good 1 is 2, the price of good 2 is 1 and Thomas' income is 10, what is Thomas' optimal bundle? Explain carefully your answer. What is the equation of the price offer curve for good 1 for Thomas? Will the price offer curve for good 1 for Thomas differ from Rob s? Explain your answer.

Explanation / Answer

Multiple questions asked.

First 3 are answered below.

1)

Since Rob has quasi-concave preferences, it implies that he has convex preferences, i.e. he prefers averages over extremes.

2)

Utility function: x1(x2+1) = x1x2+x1

Budget constraint: 2x1+1x2=10

At the optimal point, MRS = Price ratio

That is, (x2+1)/x1 = 2/1

Or, x2 = 2x1-1

Use this relation in the budget constraint.

That is, 2x1+1x2=10

Or, 2x1+(2x1-1)=10

4x1 = 11

X1* = 2.75

X2* = 4.5 units.

3)

When income doubles, the new constraint becomes: 2x1+x2=10

Upon solving using the old substitution method, the results come out as: x1*=5.25 and x2*=9.5

Thus, it can be seen that values of bundles have not doubled.

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