Utility Maximization Paper PDF

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Utility Maximization Paper covers chapters 3-5 in microeconomics ...

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Dalley, Jasmine Econ 202: Price Theory Jennifer Roff, Ph. D. Utility Maximization Utility maximization is a theory in economics that consumers look to maximize their utility, or satisfaction, within the constraint of their budget or financial capabilities. Utility maximizaton is expressed as a graph that highlights the relationship between a consumers choice and the constraint of their budget. The utility maximization graph combines various indifference curves, reflective of different satisfaction levels, with a budget constraint graph, which shows the maximum spending capability a consumer has available. An example of a utility maximization graph is shown below:

Utility maimization is the end result to a chain of economical theories and mathematical derivations that begins with a utility function. A utility function is a mathematical expression used in economics that represents a consumers consumption rate and the satisfaction that a consumer gets from the product that they’re consuming.

It can also be understood as being a mathematical expression of preference (i.e. a consumer has a greater utility funcion for good A than good B, therefore the consumer prefers good A). It is important to note that utility is a unit, meaning that a higher amount equates to a greater value. Utility is not quantifiable, such as a number which has a universal value and is always measurable; utility is a made up value used to measure satisfaction and cannot, in theory, be measured or quantified outside of counting the amount (example: 3 units of x). The most basic way one can measure utility is using indifference curves. Indifference curves show tradeoffs that consumers are willing to make between goods and helps to identify a consumers preferred good. The numerical values deduced to create the slope of indifference curves are solved for using the Marginal Rate of Substitution (MRS) formula. The Marginal Rate

of Substitution formula is as follows, MRS= -

ΔY , and describes the rate at which one is ΔX

willing to tradeoff one unit of good X for one unit of good Y. The MRS either is increasing at a diminishing rate or is a constant. When utility increases by 1 unit more, it is called the marginal utility (MU), which will come into play later in the essay. An indifference curve is always convex to the origin and on a graph with multiple indifference curves; the indifference curves never cross. An example of an indifference curve graph can look as follows:

The curve of an indifference curve is usually a convex curve, however there are certain kinds of indifference curves which produce unusual shapes. Perfect substitutes, where two goods are similar enough that a consumer can trade one for another and have the same level of utility, and perfect compliments, two goods where utility of each good is contingent on the availability of each good (think hot dogs and hot dog buns, do you want a hot dog with no bun?) have unique indifference curve shapes. The indifference curve for perfect substitutes are linear (they do not curve) and can be solved using the utility function formula U=ax +by, where a is the MU of consuming one more unit of X and b is the MU of consuming one more unit of good Y. The indifference curve for perfect compliments forms an L shape, and the utility function is U=min{aX, bY}, where a consumer reaches a given utility level by consuming a minimum amount of good x and good y (there is a balance between the two). Marginal utility is a unit that is either a constant or, more often than not, diminishing as x units increase. Remember, MU is a change in utility from 1 more unit of x. The more of x you have, the less scarce it will become and thus the less utility it will give. 1 slice of pizza will have a higher utility than 10 slices (because that is too much pizza)! Consumers want a balance between utility and quantity, and to solve for this requires the combination of the marginal rate of

substitution and marginal utility. This formula is called the marginal rate of substitution and

marginal utility (MRSxy), and is as follows MRSxy =

MUx MUy

and is the derivative of the

formula for the marginal rate of substitution. MRSxy expresses the balance struck between a consumers willingness to tradeoff one unit of good x for good y (MRS) with a consumers desire to have a balance between the availability of good x (MU) without it becoming ovwerhelming and thus, decrease utility of the good. The first portion of utility maximization is understanding indifference curves and the mathematics behind graphing them. The second portion of a utility maximizing function is income and budgeting. The budget constraint is a line (on a graph) that represents all possible spending bundles within a consumers available income. The budget constraint is an important because it marries utility, an immeasurable and nonquantifiable unit, with currency, a real and tangible unit. Income, I, is calculated as follows: I= PxQx + PyQy, where income equals to the price of x multiplied by the quantity of x plus price of y multiplied by the quantity of y. To

calculate for the slope of the budget constraint graph, solve for Qy, Qy=

Income Px Qx. Py Py

The graph of a budget constraint looks as follows:

Whatever falls under the slope of the line are feasible bundles, meaning they are things within

budget. Whatever is outside of the slope of the line (to the right) are infeasible bundles, meaning they are outside of the consumers budget. Utility Maximization is the marriage of all of these theories and mathematical expressions . Utility is maximized when the indifference curve is just touching the budget constraint (see point A on the graph below). At this point, the indifference curve and the budget constraint have equal derivatives, or share the same slope, being mathematically expressed as (MRS)

MUx = (Slope of Budget Constraint) MUy

Px . The graph expressing utility Py

maximization showsthe point at which a consumer can maximize their utility within their budget, striking a perfect balance between the quantity of goods in their bundle....