Utility Maximization
UTILITY MAXIMIZATION
We will examine the nature of consumer choices by working with a simple model in which the consumer DM chooses how to allocate their income M between two good X and Y. The kinds of choices we examine with this model can be quire general with X and Y varying from subjects as diverse as income versus leisure, consumption today versus consumption tomorrow, and different classes of consumption goods. The utility model has proven quire useful in a number of real world applications. We do not assume that DM actually "knows" a utility function that they actually maximize when making decisions. However, if DM (1) "prefers more to less", (2) can rank or choose between different good bundles, and (3) has transitive choices i.e. A B C A C , we can describe DM's choices being consistent with a person who maximizes utility. Alternatively, we can say that DM makes choices "as if" they maximize utility.
We can visualize DM's preferences between different combinations of X any Y by creating as set of contour plots of DM's implied utility function. Figure 1 plots a set of "indifference curves" assuming a utility function of U = X Y.
FIGURE 1
U=X^1Y^1
100
U= 1 U= 8100 U= 6400 U= 4900 U= 3600 U= 2500
Y
60
80
40
U= 1600 U= 900 U= 400 U= 100
0
20
0
20
40
X
1
60
80
100
Marginal Rate of Substitution The slope of the indifference curve is called DM's Marginal Rate of Substitution (MRS) and provides information with respect to tradeoff's DM is willing to make with respect to X and Y. For example if MRS = -2, DM is willing to give up approximately 2 units of Y to obtain an additional unit of X and would feel no worse off (i.e. be "indifferent") in doing so. With U=XY we can show that DM's MRS
Y . We note that the slope of the X
indifference curves (i.e. the MRS) will differ at different locations in X-Y space. The slopes tend to be steeper in the "left side" of X-Y space where we have
We will examine the nature of consumer choices by working with a simple model in which the consumer DM chooses how to allocate their income M between two good X and Y. The kinds of choices we examine with this model can be quire general with X and Y varying from subjects as diverse as income versus leisure, consumption today versus consumption tomorrow, and different classes of consumption goods. The utility model has proven quire useful in a number of real world applications. We do not assume that DM actually "knows" a utility function that they actually maximize when making decisions. However, if DM (1) "prefers more to less", (2) can rank or choose between different good bundles, and (3) has transitive choices i.e. A B C A C , we can describe DM's choices being consistent with a person who maximizes utility. Alternatively, we can say that DM makes choices "as if" they maximize utility.
We can visualize DM's preferences between different combinations of X any Y by creating as set of contour plots of DM's implied utility function. Figure 1 plots a set of "indifference curves" assuming a utility function of U = X Y.
FIGURE 1
U=X^1Y^1
100
U= 1 U= 8100 U= 6400 U= 4900 U= 3600 U= 2500
Y
60
80
40
U= 1600 U= 900 U= 400 U= 100
0
20
0
20
40
X
1
60
80
100
Marginal Rate of Substitution The slope of the indifference curve is called DM's Marginal Rate of Substitution (MRS) and provides information with respect to tradeoff's DM is willing to make with respect to X and Y. For example if MRS = -2, DM is willing to give up approximately 2 units of Y to obtain an additional unit of X and would feel no worse off (i.e. be "indifferent") in doing so. With U=XY we can show that DM's MRS
Y . We note that the slope of the X
indifference curves (i.e. the MRS) will differ at different locations in X-Y space. The slopes tend to be steeper in the "left side" of X-Y space where we have