agbus 212 lec4 w03

Utility Maximization: 

Utility Maximization Chapter 7 (pp. 169-187): Casavant, Infanger, and Bridges

Agenda: 

Agenda Discuss the pillars of Consumer Theory Discuss the concepts of utility, marginal utility, and the Law of Marginal Diminishing Utility Discuss indifference curves and marginal rates of substitution Discuss budget constraint

Consumer Theory: 

Consumer Theory There are two important pillars that consumer theory rests upon: The utility function The budget constraint

Utility Function: 

Utility Function A utility function is a function/process that maps a bundle of goods into satisfaction where the satisfaction can be ranked for each bundle of goods. In essence, it allows us to rank the desirability of consuming different bundles of goods. A bundle of goods is a particular set of goods you choose to consume. This is also referred to as a consumption bundle.

Mathematically Representing Utility: 

Mathematically Representing Utility In Economics, we tend to define our utility function as the following: U = u(x1, x2, …, xn) where U is the level of satisfaction you receive from consuming the bundle of goods consisting of x1, x2, …, xn u( ) is the function that maps the bundle of goods into satisfaction. (Think of this as the mathematical representation of your brain.) xi for i = 1, 2, …, n is the quantity of a particular good consumed from a bundle of goods, e.g., x1 may be 5 pancakes.

Example of Representing Utility in Mathematical Terms: 

Example of Representing Utility in Mathematical Terms Suppose you have a choice of consuming one of two different bundles of goods. Bundle 1 has a large pepperoni pizza, a two liter bottle of coke, and 20 spicy buffalo wings. Bundle 2 has a dozen doughnuts, a liter of coffee from Starbuck’s, and a half dozen blueberry muffins.

Example of Representing Utility Cont.: 

Example of Representing Utility Cont. For bundle 1 x1 = a large pepperoni pizza x2 = a two liter bottle of coke xn = x3 = 20 spicy buffalo wings For bundle 2 x1 = a dozen doughnuts x2 = a liter of coffee from Starbuck’s xn = x3 = a half dozen blueberry muffins

Example of Representing Utility Cont.: 

Example of Representing Utility Cont. Let’s assume for the moment that it’s 7 a.m. and consumption of bundle 1 makes you very happy, which we will call U1, and that consumption of bundle 2 makes you very, very, very happy, which we will call U2. What we are saying is that U2 > U1.

Example of Representing Utility Cont.: 

Example of Representing Utility Cont. We can further write the following: U1 = u(a large pepperoni pizza, a two liter bottle of coke, 20 spicy buffalo wings) U2 = u(a dozen doughnuts, a liter of coffee from Starbuck’s, a half dozen blueberry muffins) Remember u( ) is just a function that transforms bundles of goods into satisfaction. (Think of it as your brain.)

Some Notes on Utility: 

Some Notes on Utility It was once thought that utility could be broken down into units called utils. Total utility is defined as the utility you receive from consuming a particular bundle of goods. For analytical tractability, total utility can be given a number value. In our previous example we could say that U1 = 100 and U2 = 200.

Some Notes on Utility Cont.: 

Some Notes on Utility Cont. There are two views of utility: Cardinal Utility Cardinal utility is the belief that utility can be measured and compared on a unit by unit basis. E.g., A utility measure of 200 is twice as big as a utility measure of 100. Ordinal Utility Ordinal utility is where you rank bundles of goods, but cannot say how much greater one bundle is to another. Ranking is the only thing that matters when dealing with ordinal utility.

Marginal Utility: 

Marginal Utility Marginal utility is defined as the change in total utility divided by a change in the consumption of a particular good. In mathematical terms, Marginal utility of good i = Δ Total Utility/ Δ in level of consumption of one good or Marginal utility of good i = Δ U/ Δ xi for i = 1, 2, …, n Note that Δ means change

Marginal Utility Cont.: 

Marginal Utility Cont. Marginal Utility (MU) for good i is equal to: Δ U / Δ xi (U2 - U1) / (xi2 - xi1) Where U2 equals the utility received from consuming xi2, i.e., U2 = u(xi2). Where U1 equals the utility received from consuming xi1, i.e., U1 = u(xi1). xi2 and xi1 are differing levels/quantities of the same good.

Calculating Marginal Utility: 

Calculating Marginal Utility Suppose Dr. Hurley only like to consume chocolate. Also, assume that Dr. Hurley’s total utility he receives for eating differing quantities of chocolate is represented by the following: Number of chocolates per week Total Utility 1 2 3 4 5 6 10 25 38 50 59 62

Calculating Marginal Utility Cont.: 

Calculating Marginal Utility Cont. Number of chocolates per week Total Utility 1 2 3 4 5 6 10 25 38 50 59 62 Marginal Utility Δ U / Δ xi 15 = ((25-10)/(2-1)) 13 = ((38-25)/(3-2)) 12 = ((50-38)/(4-3)) 9 = ((59-50)/(5-4)) 3 = ((62-59)/(6-5))

Deriving Total and Marginal Utility from a Function: 

Deriving Total and Marginal Utility from a Function Suppose you could represent your utility function for drinking Mountain Dew as the following: U = u(x) Where u(x) = 5x – x2 What is the Total Utility, the Marginal Utility, and graph both?

Example Continued: 

Example Continued U = u(x) = 5x – x2 Total Utility (U) Marginal Utility Amount Consumed (x) 0 1 2 3 4 5 6 0 4 6 6 4 0 -6 4 = ((4-0)/(1-0)) 2 = ((6-4)/(2-1)) 0 = ((6-6)/(3-2)) -2 = ((4-6)/(4-3)) -4 = ((0-4)/(5-4)) -6 = ((-6-0)/(6-5))

Lessons Learned from Example: 

Lessons Learned from Example As long as marginal utility is positive, total utility will increase. Total and marginal utility can be negative. Total Utility follows marginal utility. When marginal utility is zero, total utility is maximized. This graph demonstrated the Law of Diminishing Marginal Utility.

Law of Diminishing Marginal Utility: 

Law of Diminishing Marginal Utility This law states that as you consume more units or a particular good during a set time, at some point your marginal utility will decrease as your consumption increases. Note: Beware of laws in economics, they are not like physical laws. Experiment: Go home and try consuming as much of a particular good as you can. How do you feel after eating each additional unit?

Isoutility: 

Isoutility To this point we have examined utility based on the consumption of one good. Realistically, our consumption bundles have more than one item in them. When more than one good exist in our consumption bundle and utility function, it becomes important for us to examine the idea of isoutility.

Isoutility: 

Isoutility Isoutility is a concept where differing bundles of goods provide the same level of utility. From the idea of isoutility comes the graphical representation--an indifference curve. An indifference curve is the graphical representation of differing bundles of goods giving the same level of utility.

Demonstrating an Indifference Curve: 

Demonstrating an Indifference Curve Suppose we have two goods in our utility function—sandwiches (S) and corn dogs (C). To obtain an indifference curve, we want to hold our utility constant and look at the different bundles of consumption of each good that provide the same utility. Suppose we know that we have the following utility function U = u(S, C) = S*C

Demonstrating an Indifference Curve Cont.: 

Demonstrating an Indifference Curve Cont. To derive the indifference curve, we want to keep U constant. In this case we will assume U = 10 What bundles of goods will give us this level of satisfaction? What if U = 20?

Slide25: 

Increasing Utility

Observations: 

Observations The two utility curves do not cross each other. Why? Utility is increasing as you move away from the x and y-axis. As you consume more of one of the goods, the consumption of the other good has relatively more value to you. Any other observations?

Marginal Rate of Substitution (MRS): 

Marginal Rate of Substitution (MRS) When examining indifference curves, it is important to look at the tradeoff between consuming one good versus the other. This is called examining the marginal rate of substitution. The marginal rate of substitution can be defined as the rate at which the consumer is willing to trade one good for another. Why is this important to examine?

Mathematical Representation of MRS: 

Mathematical Representation of MRS MRS of good xi for good xj =Δ xj /Δ xi = (xj2 – xj1)/(xi2 - xi1) Where xj1 and xj2 are differing levels of consumption good j E.g., sandwiches Where xi1 and xi2 are differing levels of consumption good i E.g., corn dogs

Example of Calculating MRS: 

Example of Calculating MRS From our previous example, we had the following utility function: U = u(S,C) = S*C Setting U = 10 we saw that one of our consumption bundles (1,10) and another was (2,5) What is the MRS of corn dogs for sandwiches going from one sandwich to two sandwiches?

Example of Calculating MRS Cont.: 

Example of Calculating MRS Cont. MRS = Δ xj /Δ xi = (xj2 – xj1)/(xi2 - xi1) From our previous information, we know that xi2 = 5 corn dogs, xi1 = 10 corn dogs, xj2 = 2 sandwiches, and xj1 = 1 sandwiches This implies MRS = (2-1)/(5-10) = 1/-5 = -1/5 This says we are willing to give up one sandwich to get five corn dogs or we are willing to give up five corn dogs to get one sandwich.

Implication of Indifference Curve: 

Implication of Indifference Curve Due to indifference curves, we have the following relationship for differing bundles of goods: (Δ xj / Δ xi) = (MUxi /MUxj) Which implies (Δ xj * MUxj) = (MUxi * Δ xi) This implies that the loss in utility from consuming less of good xj is just matched by the gain in utility from consuming more of xi.

Budget Constraint: 

Budget Constraint This constraint defines the feasible set of consumption bundles you are able to consume. It is dependent on three things: The prices of the goods in your consumption bundle. The quantity of each good in your consumption bundle. Your income.

Budget Constraint Cont.: 

Budget Constraint Cont. The budget constraint for two goods is defined as the following: I = p1*x1 + p2 * x2 I = income p1 = the price of x1 p2 = the price of x2 x1 = the quantity consumed of good 1 x2 = the quantity consumed of good 2

Budget Constraint Cont.: 

Budget Constraint Cont. Assume that we want to graph are budget constraint and choose x2 as the good we want to put on the y-axis. I = p1*x1 + p2 * x2 can be rewritten as: x2 = (I/p2) – (p1/p2) * x1 With this equation we can draw the budget constraint on a graph. Note that the slope of the budget constraint is equal to the negative of the price of good 1 divided by the price of good 2.

Budget Constraint Example: 

Budget Constraint Example Suppose you have $100 to spend on chips and soda. You know that the price of chips is $1 and the price of soda is $2. Draw your budget curve assuming that chips will be on the y-axis.

Budget Constraint Example: 

Budget Constraint Example Budget equation: $100 = $2 *x1 + $1 *x2 x2 = 100 – 2*x1

Budget Constraint Cont.: 

Budget Constraint Cont. # of Chips # of Sodas 50 100 x2 = 100 – 2*x1 Feasible Set Not Feasible