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Indifference Curve, Budget Line & Optimal Utility

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A household's utility over two consumption goods x and y is U= U(x,y) = xy.

1. Describe the household's indifference curve for U = 1 for values of x and y less than 3 (ie. the curve containing all combinations of x and y such that U(x,y)=1.

Now assume that the household's wealth is w=4 and that the prices of the goods are px = 2 and py = 2.
2. How many units of x can the household consume at most if it does not consume any y?
3. Describe the household's budget line and its relationship to the indifference curve.
4. What is the household's optimal consumption bundle?
5. What is the Marginal Rate of Substitution (MRS) between the two goods?

Note the budget constraint can be written as 2x + 2y = 4

6. Solve this budget constraint for y and substitute it into the utility function to obtain an expression for utility that depends on x only.
7. Maximize this utility to obtain the optimal amount of x. (Take the derivative of this expression with respect to x, set the derivative equal to zero, solve for x)
8, Find the optimal amount of y by using the result for x.
9. How does this optimal consumption bundle compare to the one found graphically.

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Solution Preview

1. The curve will be azymptotic to both axes. It will pass through the points (1/3,3), (1/2,2), (1,1), (2,1/2), (3,1/3), etc.

2. 2 units of x

3. It will connect the points (0,2) and (2,0). It will be tangent to the ...

Solution Summary

This solution shows how to find a household's optimal consumption bundle algebraically and graphically.

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5. Use consumer theory (i.e. indifference curves and budget constraints), where the usual assumptions apply, to illustrate the following:

Assume the individualâ??s utility is an increasing function of medical goods (m) and all other goods (X). That is,

Utility =U (m,x) where delta u/delta m >0, delta u/delta x >0 , and H = f(m) where delta H/deltam > 0.

That is, health or health status increases utility and health is an increasing function of medical goods.

a. Illustrate the individualâ??s utility-maximizing choice of .
b. Next, on the same graph, show the effects of the individual â??getting sick.â? When doing this, assume that the illness harms the individualâ??s ability to earn as much income as before the illness. Label the new optimum .
c. Next, suppose instead, that the individual has sick-loss insurance or sick leave from work and this would replace any income lost from missing work (do I hear a duck? Aflac!). Label this outcome .
d. This exercise is an illustration that illness has caused three potentially observable things to happen. What are they?

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