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TOPIC V: UTILITY AND DEMAND I. Preferences: A. B. Total utility (TU) Marginal utility (MU) II. Budget Constraint III. Maximizing Utility A. B. Law of diminishing marginal utility Equal marginal rule (understand the common sense meaning of this rule): MU x/P x = MUy/Py CONCEPTS AND PRINCIPLES The theory we develop for investigating how consumers make “purchasing” decisions is called optimization subject to a budget constraint. The basic idea is to choose the combination of goods that would maximize utility (satisfaction, happiness, etc), constrained by what one can afford to purchase. Thus, the theory has two basic components: (1) preferences and (2) a budget. V1 CONSUMER PREFERENCES Basically we assume that individuals know what is available and can rank “bundles” of goods in terms of a ranking system we will call utility. Don’t think of what the goods will cost to purchase, just what satisfaction (utility) they will bring. Two fundamental measurements we consider are: (1) total utility and (2) marginal utility. Total Utility (TU) refers to the total utility (satisfaction) one would receive from a bundle of goods. For example: 2 bananas, 1 apple, 3 sodas, 4 bags of chips, and 2 hotdogs might yield me 1,200 “utils.” Marginal Utility (MU) refers to the additional utility one would receive from an additional unit of some good. It is measured as “the change in TU”/”the change in number of units of the good” MUX = )TU X /)QX. For example what if my utility from adding 1 soda to the example above was an extra 50 utils - my MU from that soda would be 50 and my total utility would now be 1,250. A simple example: Q sodas 0 1 2 3 4 5 TU sodas 0 100 300 380 430 440 MU sodas 100 200 80 50 10 A standard simplifying assumption that we often make is that as more and more units of a good are consumed, the MU of additional units will eventually decrease. In the example above, this occurs at the 3rd soda and continues after that. We call this the assumption of “diminishing marginal utility.” V2 BUDGET CONSTRAINT The idea here is to determine what is feasible to purchase given existing market prices and existing income. Consider the following example. Assume I have $50 to spend on sodas and pizza. Assume the price of a soda is $1 and the price of a piece of pizza is $2. What combinations of soda and pizza would be available to me? Consider the graph shown below. Any combination of soda and pizza that would cost $50 would be on my budget constraint - for example, 50 sodas and no pizza, 25 pizza slices and no soda, 30 soda and 10 pizza slices, etc. V3 MAXIMIZING UTILITY Maximizing Utility means finding that combination of goods that is affordable and yields the highest total utility. How do we determine that combination? One way is to understand something the “marginal utility per dollar spent on a good” and something called the “equal marginal principle.” The marginal utility per dollar spent on a good X = MUX /PX The equal marginal principle states that one should buy the combination of goods where the marginal utility per dollar spent on each good is the same. Consider the following example. Assume there are two goods, X and Y. Also, assume that you are considering buying a combination of those goods where MUX/PX > MU y/Py. If this is true, this means that you are getting more additional utility per dollar spent on X than on Y. Thus, if you took a dollar from the consumption of Y and moved it to the consumption of X, your total utility would increase. Note the following. If you are buying X and Y at a point where you face diminishing MU, then as you increase the amount of X you buy, the MUX decreases and as you buy less of Y, the MU of Y increases. This tends to move you in the direction of MUX becoming equal to the MU Y. Similarly, what if you are considering buying a combination of those goods where MU X/P X < MUy/Py. If this is true, this means that you are getting less additional utility per dollar spent on X than on Y. Thus, if you took a dollar from the consumption of X and moved it to the consumption of Y, your total utility would increase. Note the following. If you are buying X and Y at a point where you face diminishing MU, then as you decrease the amount of X you buy, the MUX increases and as you buy more of Y, the MU of Y decreases. This tends to move you in the direction of MU X becoming equal to the MU Y. This leads us to the next “principle” - the equal marginal utility per dollar principle. Principle 9 - Total Utility is maximized when a combination of goods (X,Y,Z) is chosen that is “on” the budget constraint and one’s preferences are such that MUx/Px = MUy/Py = MUz/Pz. In applying this principle, the idea is that individuals will make decisions in line with moving toward combinations of goods where marginal utility per dollar is equal across all goods. In practice problems (or even in the real world) the key idea is that individuals consider the marginal value per dollar spent and make purchasing decisions that reflect the best value for the dollar. V4 Applying the Marginal Utility per dollar principle Assume you have $100/month to spend. Assume the table below displays your preferences for three goods in terms of the marginal utility you receive from each good. MU is the marginal utility from each extra unit of coke, burgers, or movies. MU/$ is the marginal utility per dollar received from expenditures on each extra unit of these goods. Pcoke = $.50 Pburger = $2.00 Pmovie = $5.00 Q MUC MU/$ Q MUB MU/$ Q MUM MU/$ 1 30 60 1 400 200 1 2000 400 2 30 60 2 200 100 2 1800 360 3 30 60 3 140 70 3 1300 260 4 30 60 4 50 25 4 1000 200 5 30 60 5 20 10 5 800 160 6 30 60 6 10 5 6 200 40 7 .... .... .... 30 .... .... .... 60 .... .... .... 7 0 0 7 100 20 Using the marginal utility per dollar principle, we evaluate the extra benefit from extra expenditures on each good. Start with the good that gives the greatest MU/$. “Purchase” that good, subtract the price from income, go to the next good (or next unit of the same good) with the greatest MU/$, purchase that good, subtract the price from income, continue the process until all income is spent. This is the “essence” of the marginal utility per dollar principle. When we do this, we can show that the optimal purchase of these good would be: Quantities: QC = 138 Expenditures: $ 69 QB = 3 $6 (138x$.50) QM = 5 $25 (3x$2.00) Total= $100 (5x$5) We can now show how the choices above relate to the demand for each of these goods. The information we gain from solving the optimization problem is depicted in the graphs below. V5 Solving the problem shows us one point on each demand curve. This figure shows us what we have learned about the demand for coke when the Pburger= $2, Pmovie= $5, and the person has $100 to spend. We really only know about the single point shown. This figure shows us what we have learned about the demand for burgers when the Pcoke= $.50, Pmovie= $5, and the person has $100 to spend. We really only know about the single point shown. This figure shows us what we have learned about the demand for movies when the Pcoke= $.50, Pburger= $5, and the person has $100 to spend. We really only know about the single point shown. V6 Now, consider what happens when the price of one of the goods changes. Let: Pmovie increase to $20. Pcoke = $.50 Pburger = $2.00 Pmovie = $20.00 Q MUC MU/$ Q MUB MU/$ Q MUM MU/$ 1 30 60 1 400 200 1 2000 100 2 30 60 2 200 100 2 1800 90 3 30 60 3 140 70 3 1300 65 4 30 60 4 50 25 4 1000 50 5 30 60 5 20 10 5 800 40 6 30 60 6 10 5 6 200 10 7 30 60 7 0 0 7 100 5 Using the equal marginal principle, we evaluate the extra benefit from extra expenditures on each good. When we do this, we can show that the opimal combination of these goods would be: Quantities: Expenditures: $ 34 QC = 68 QB = 3 $6 QM = 3 $60 Note that we now have: 1) a new demand curve for cokes 2) a new demand curve for burgers 3) and we have a new point on the demand curve for movies. V7 The new demand curve shows a smaller quantity demanded (than before) at the price of $.50. In other words, when the price of movies increased - the demand for cokes decreased making cokes and movies complements. The new demand curve shows the same quantity demanded (than before) at the price of $2.00. In other words, when the price of movies increased - the demand for burgers did not change - making movies and burgers neither substitutes nor complements for this particular example. Notice that in the case of the demand for movies - we are moving along the original demand curve. That is, as the price of movies increased, the consumer moved up along the original demand (reducing the quantity demanded). V8 QUESTIONS Short Answer 1. Complete the following table: Qx 0 1 2 3 4 2. TUx _____ _____ _____ _____ _____ MUx _____ 1000 800 300 80 Complete the following table: Qx 0 1 2 3 4 TUx _____ _____ _____ _____ _____ MUx _____ 0 100 0 80 Can you think of any goods that might lead to such a strange set of preferences? 3. If total utility from good X were to reach a maximum and then start to decline for additional units of good X, what would this imply about the marginal utility from additional units of good X? Make up a numerical example to show this. 4. Assume Z is a complement to good X, and Y is a substitute for good X. Also assume Joe Brown is currently maximizing utility with respect to consumption of each of these goods. What might we expect to happen to Joe Brown' s purchases of X, Y, and Z if the price of good X falls? Why? 5. Assume Z is a complement to good X, and Y is a substitute for good X. Also assume Joe Brown is currently maximizing utility with respect to consumption of each of these goods. What might we expect to happen to Joe Brown' s purchases of X, Y, and Z if Joe' s income increases from $1,000 per month to $1,500 per month? Why? State precisely what assumptions your answer depends upon. V9 Multiple Choice 1. The equal marginal principle of consumer behavior suggests that consumers will make purchase decisions based on comparing the: A. B. C. D. 2. total benefits of all goods. additional utility per dollar of each good. rate of diminishing total utility from successive units of a given good. all of the above Use the information below to answer the next question. Qx MUx 1 2 3 50 20 0 The total utility from 3 units of good X would be: A. B. C. D. 3. Assume: (1) that the price of cola is $2 and the price of tacos is $1; (2) that at your current consumption levels the MU of cola is 50 and the MU of tacos is 30; and (3) that you are spending all of your income. The equal marginal principle would imply that you should: A. B. 4. 0 20 70 none of the above buy additional units of tacos and less cola. buy additional units of cola and less tacos. Think through this problem carefully - use equal marginal value per dollar. Assume you have $5,000 to invest. Assume stock in IBM costs $50 per share and pays a yearly dividend of $10 per unit of stock. Assume stock in TI costs $20 per share and pays a yearly dividend of $5 per unit of stock. (Ignoring all other factors) to maximize your investment return you should: A. B. C. D. invest $5,000 in IBM. invest $5,000 in TI. split your investment so that the ratio of your shares of TI relative to IBM is 2.5 to 1. split your investment so that the ratio of your shares of IBM relative to TI is 2.5 to 1. V 10 5. Assume you are throwing a party and have $150 to spend. You are considering buying 10 cases of coke and 10 boxes of chips. The cost of coke is $5 per case and the cost of chips is $10 per box. The MU from the 10th case of coke is 132 and the MU from the 10th box of chips is 200. To maximize the value (utility) from your purchases, you should - NOTE: assume you can buy partial cases: A. B. C. D. 6. The equal marginal principle implies that: A. B. C. D. 7. demand curves will be downward sloping (the law of demand). consumers will consume all goods at a point where the extra satisfaction from additional units of each good is the same. the marginal utility per dollar spent on each good is equal. all of the above If the consumer is maximizing utility and the marginal utility per dollar spent on good X is 4 and the marginal utility per dollar spent on good Y is 2, we know that A. B. C. D. 8. buy more coke and less chips. buy more chips and less coke. buy more coke and more chips. the answer is indeterminate without knowing the total utility received from coke and chips. the price of Y must be twice the price of X. the MU of X must equal that of Y. total utility can be increased by increasing the consumption of X and decreasing the consumption of Y. all of the above must be true To maximize utility: A. B. C. D. the marginal utility is the same for all goods. the marginal utility per dollar spent is the same for all goods. the marginal cost per dollar spent is the same for all goods. all of the above V 11 Use the following table for the next two questions: 9. Number of Colas Total Utility 1 Total Utility 6 1 10 2 10 2 14 3 13 3 15 4 15 4 15 5 16 5 7 MU/$ MUB MU/$ Which of the two goods demonstrates the law of diminishing marginal utility? A. B. C. D. 10. Number of Burgers MUC cola only burgers only both cola and burgers neither cola nor burgers If the consumer has only $8 to spend, the price of cola is $1 each, and burgers cost $2 each, what would be the optimal consumption of both goods for this consumer? A. B. C. D. 4 colas and 5 colas and 2 colas and 4 colas and 2 burgers 1 burger 3 burgers 4 burgers V 12