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Finding the optimal consumption bundle

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Two consumers, Mamoon and Kader, consume only two goods, good 1 and good 2, with the quantities of each consumed denoted by x1 and x2 respectively. The price of x1 is $6/unit, and the price of x2 is $5/unit. The consumer has a fixed income of $64.

Mamoon has a utility function: U(x1,x2 )=2x1^3 x2^5?
Kader's utility function is: U(x1,x2 )=x1+2x2

a. Solve Kader's utility maximization problem.
b. Show the on a diagram with the horizontal axis represents x1and the vertical axis represents x2 the way to find the optimal consumption bundle of Kader graphically, label the optimal bundle.

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Solution Preview

Please refer attached file for graph.

a. Solve Kader's utility maximization problem.
Marginal utility of x1 can be found by differentiating with respect to x1.

Marginal utility of x2 can be found by differentiating with respect to x2.

Marginal Utility of x1 per dollar spent=MU(x1)/Px=1/6=0.167
Marginal Utility of x2 per dollar ...

Solution Summary

The solution describes the steps to solve the given utilization problem.

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Optimal Consumption Bundle using the Lagrangian Method

Kaidu consumes two goods, x_1 and x_2, with respective prices p_1 and p_2. His utility function is given by U_Kaidu (x_1,x_2 )=x_1^(1/4) x_2^(3/4) . Let p_1=5 and p_2=4. Kaidu's income is 80.

- Suppose the government imposes a $1 tax each unit consumption of x_2, solve for the new optimal consumption bundle.

How do i obtain this? Is the Lagrangian Method used and if so, then how?

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