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Cooleconomics.com mana-utility ©Michael Francis Williams. Authorized use is encouraged; unauthorized use is prohibited. 1 Utility Maximization Theory: A person's happiness, or total utility, depends upon the decisions that she makes throughout her life. Important decisions include: how much to work, how much to save, what/how much to buy. Assumptions often made about utility: although not necessary, it is often realistic to make the following assumptions concerning the b
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  Cooleconomics.com mana-utility ©Michael Francis Williams. Authorized use is encouraged; unauthorized use is prohibited. 1 Utility Maximization Theory: A person's happiness, or total utility , depends upon the decisions that she makes throughout her life. Important decisions include: how much to work, how much to save, what/how much to buy. Assumptions often made about utility: although not necessary, it is often realistic to make the following assumptions concerning the behavior of a person's utility: 1) Non-satiation: Higher consumption of products that people buy, holding all else equal, leads to higher total utility. 2) Logic: A person is able to rank   the utility derived from of all possible decisions that she faces (from highest to lowest). This requires that completeness of  preferences ; the individual is aware of all options available to her. It also requires transitivity ; if choice B is better than choice A, and choice C is better than choice B, then it better be true that choice C is better than choice A. 3) Diminishing Marginal Utility:  Marginal utility  measures the increase in total utility attained when an activity (such as consumption of a good) is incrementally increased. If you prefer, marginal utility = increase in total utility ÷ increase in activity Diminishing marginal utility means, for example, that eating the 2 nd  potato chip causes a larger increase in utility than eating the 3rd potato chip. (Note, however, that the 3rd potato chip does not reduce total utility; it causes total utility to rise  by less than the 2 nd  chip.)  Example : (below, we measure utility in units known as utils ) Potato chips eaten Marginal chip utility Total chip utility 0 -- 0 1 50 50 2 40 90 3 30 120 4 15 135 5 5 140  Cooleconomics.com mana-utility ©Michael Francis Williams. Authorized use is encouraged; unauthorized use is prohibited. 2 Algebraic Representation of Utility: U = f(decisions) “U” is a number of utils representing the happiness of the individual. The higher the value of U, the better off is the individual. Often to maintain tractability (and to be able to graph the individual's behavior) we sometimes limit an individual's choice to this: the individual may buy only two types of goods; how many units of each should she buy? Given this limitation, there are many different functional forms that we could use to represent an individual’s utility. Many realistic forms are quite complex mathematically and we shall omit those. Below are three less realistic, but more tractable examples of utility functions: the Cobb-Douglas utility function, the linear utility function, and the Leontief utility function. Example 1: The Cobb-Douglas  utility function represents the utility possible from consuming two goods, x and y. In general, a Cobb-Douglas function has the form U = X a Y (1-a)  where X is units of x consumed Y is units of y consumed U is utils of total utility a is a constant less than 1 Here’s a specific example of a Cobb-Douglas utility function, for an individual who consumes only beer and pizza: U = units of beer consumed .4  x units of pizza consumed .6 Example 2: The  Linear   utility function is another ( sometimes not too realistic) representation of the utility gained by consuming goods x and y. U = aX + bY where X is units of x consumed Y is units of y consumed U is utils of total utility a and b are positive constants Here’s a specific example of a linear utility function, for an individual who consumes only Coke and Pepsi: U = (10 x units of Coke consumed) + (10 x units of Pepsi consumed)  Cooleconomics.com mana-utility ©Michael Francis Williams. Authorized use is encouraged; unauthorized use is prohibited. 3 Example 3: The  Leontief   utility function is another (sometimes not too realistic) representation of the utility gained by consuming goods x and y. U = min[aX, bY] where X is units of x consumed Y is units of y consumed U is utils of total utility a and b are positive constants Here’s a specific example of a Leontief utility function, for a (pantsless) individual who consumes only Shirts and Buttons: U = minimum of [1 x # of shirts consumed, 1/6 x # of buttons consumed] Total Utility Graphed Using Indifference Curves  One can use indifference curves to graph a person's prospective total utility levels An indifference curve  represents all combinations of two goods x and y that provide the individual with an equal level of total utility. (A related concept is the marginal rate of substitution (MRS) : the amount of X that one must be given, to compensate for a loss of good Y, in order to maintain total utility at a constant level.) Example 1: If one draws an indifference curve for a Cobb-Douglas utility function then it will look as this:   Units of Y consumed Units of X consumed U = U 0    Cooleconomics.com mana-utility ©Michael Francis Williams. Authorized use is encouraged; unauthorized use is prohibited. 4  Note this about the above indifference curve: --the negative slope: This indicates that the if one takes some X from the consumer, then she must be given more Y in order to maintain her total utility at its initial level. --the convex curve: This indicates diminishing MRS Example 1 continued: Here are 4 combinations of beer and pizza that provide 100 utils of utility to an individual with this utility function: (numbers are rounded) U = B .4  Z .6 point Units of beer, B Units of pizza, Z Total utility, U (=B .4 Z .6 ) A 100 100 100 utils B 90 107.3 100 utils C 80 116 100 utils D 70 126.8 100 utils If you were to graph point A B C and D and connect them, you would have a part of this dude’s indifference curve representing U=100 utils of utility. The entire U=100 utils indifference curve constitutes ALL combinations of beer and pizza that provide 100 utils of utility. It would have a convex curvy shape similar to the indifference curve on page 3.  Note the MRS between points A and B: If one takes 10 units of beer  from this dude (reducing him from 100 units of beer to 90 units of beer), then one must compensate him with an extra 7.3 units of pizza  (increasing him from 100 to 107.3) in order to maintain his utility at 100 utils. Hence his MRS between A and B = 10 beer / 7.3 pizza = 1.3699 Verify that MRS between B and C = 1.1494 Verify that MRS between C and D = .9259 See how the MRS diminishes as one moves along the indifference curve?
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