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Finding the optimal allocation of budget

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Suppose a firm is considering two different activities, activities X and Y, which yield
the following schedule of total benefits shown below. The price of X is $2 per unit and the price of Y is $10 per unit.

Total benefit of Total benefit of
Level of Activity Activity X Activity Y
_________________________________________________________
0 0 0
1 $30 $100
2 54 190
3 72 270
4 84 340
5 92 400
6 98 450

A. The firm places a budget constraint of $26 on expenditures on activities X and
Y. What is the level of activity that maximizes the total benefit subject tot he budget constraint? What is the total benefit if your solution is implemented?

B. Suppose that the budget increases to $58. What is the optimal level of activity
X and Y now? What is the total benefit when the budget is $58?

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Solution Preview

Please refer attached file for tables.

A. The firm places a budget constraint of $26 on expenditures on activities X and Y. What is the level of activity that maximizes the total benefit subject tot he budget constraint? What is the total benefit if your solution is implemented?

Marginal benefit per dollar spent on X (MBx/Px) is $15 for activity level 1, which is higher than that of Y i.e. $10.
So, first activity level of X should be chosen first.

Now let us see activity level 2 for X. In this case marginal benefit per dollar spent is 12, which is higher than marginal benefit per dollar
spent for first activity level of Y. So, second activity level of X should be chosen next.

Now let us see activity level 3 for X. In this case marginal benefit per dollar spent is 9, which is lower than marginal benefit per dollar
spent ...

Solution Summary

Solution describes the steps to calculate optimal allocation of expenditure on the given activities.

$2.19
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Utility Maximization Exchange Economy

This question is about Walrasian equilibrium in an exchange economy with 2 goods and 2 consumers. Taxes are introduced in the question to solve for the equilibrium and allocation under Pareto theorem.
Question (2)
Consider an exchange economy with 2 goods and 2 consumers . Consumer 1's initial endowment is and consumer 2's endowment is . In each case , the first entry in the endowment vector denotes the initial endowment of good 1, and the second entry the initial endowment of good 2 . Both consumers have the same consumption set: , and the same utility function: Suppose that consumer 1's expenditure on good 1 is taxed at a rate of 50%, and that the revenue from this tax is paid as a lump-sum transfer to consumer 2.
Find the Walrasian equilibrium of this economy . Then find another allocation which Pareto-dominates the equilibrium allocation.

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