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## Demand & Utility

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**What is Utility?**Satisfaction, happiness, benefit**Cardinal Utility vs. Ordinal Utility**Cardinal Utility: Assigning numerical values to the amount of satisfaction Ordinal Utility: Not assigning numerical values to the amount of satisfaction but indicating the order of preferences, that is, what is preferred to what**Util**A unit of measure of utility**Total Utility**The amount of satisfaction obtained by consuming specified amounts of a product per period of time.**Example: TU(X) = U(X) = 16 X – X2where X is the**amount a good that is consumed in a given period of time. 5 units of the product per period of time yields 55 utils of satisfaction**Marginal Utility**The change in total utility (TU) resulting from a one unit change in consumption (X). MU = TU/ X**Diminishing Marginal Utility**Each additional unit of a product contributes less extra utility than the previous unit.**When the changes in consumption are infinitesimally small,**marginal utility is the derivative of total utility. MU = dTU/dX**Calculating MU from a TU FunctionExample: TU(X) = 16 X –**X2 MU = dTU/dX = 16-2X**In general, the derivative of a total function is the**marginal function. The marginal function is the slope of the total function.**Graphs of Total Utility & Marginal Utility**Total Utility TU X1 is where marginal utility reaches its maximum. This is where we encounter diminishing marginal utility. The slope of TU has reached its maximum; TU has an inflection point here. X X1 X2 Marginal Utility X2 is where total utility reaches its maximum. MU is zero. This is the saturation point or satiation point. After that point, TU falls and MU is negative. MU X1 X2 X**Example: If TU = 15 X + 7X2 – (1/3) X3, find (a) the MU**function,(b) the point of diminishing marginal utility, & (c) the satiation point. • MU = dTU/dX = 15 +14X – X2 • Diminishing MU is where MU has a maximum, or the derivative of MU is zero. 0 = dMU/dx = 14 – 2X X = 7 c. Satiation is where TU reaches a maximum or its derivative (which is MU) is zero. How do we determine where MU = 15 +14X – X2 = 0 ?**Apply Quadratic Formula to –X2 + 14 X + 15 = 0 .**Since a negative amount of a product makes no sense, X must equal 15.**If the previous example were about eating free cookies at a**party, you’d eat 15 of them. That is where you become satiated. After 15 cookies, you begin to feel a bit bloated.**When you have more than one product, the marginal utility is**a partial derivative. Calculating partial derivatives is no more difficult than calculating other derivatives. You just treat all variables as constants, except the one with which you are taking the derivative. We denote the partial derivative of Z with respect to X as Z/X instead of dZ/dX. Example: Z = X2 +3XY + 5Y3 To take the partial derivative with respect to X, pretend Y is a constant (like 4). Then, Z/X = 2X + 3Y . Similarly, to take the partial derivative with respect to Y, pretend X is a constant (like 4). So, Z/Y = 3X + 15Y2 .**The Connection between Demand and Utility**Instead of thinking in terms of utils, let’s think in terms of dollars. Suppose the purchase of one unit of a good gives you $10 worth of satisfaction. In other words, the marginal utility of that first unit of the good is $10. Then you would be willing to pay up to $10 for it.**If a second unit of the good contributes $8 more of**satisfaction, the marginal utility of your second unit is $8 and you would be willing to pay up to $8 for it. If a third unit of the good contributes $6 more of satisfaction, the marginal utility of your third unit is $6 and you would be willing to pay up to $6 for it.**Remember that the demand curve tells you what people are**willing to pay for various amounts of a good or, equivalently, how many units of a good they are willing to purchase at various prices. So, since we just found that the marginal utility tells us what we are willing to pay for a good, the marginal utility provides us with information that we can use to determine our demand curve.**price**10 8 6 Demand quantity 1 2 3 If we graph that information, we get our demand curve.**Notice that by adding our MU values we can determine our**total utility at different consumption levels.**Indifference Curve**A set of combinations of goods that are viewed as equally satisfactory by the consumer.**Indifference Map**A collection of indifference curves**Assumptions**• The consumer can rank all bundles of commodities. • If bundle A is preferred to bundle B and B is preferred to C, then A is preferred to C. (This property is called transitivity.) • More is better.**Indifference curves slope down from left to right.**Consider 2 points A & B on the same indifference curve. At point B, you have more food than at point A. If the amount of clothing you had at point B was the same as or more than at point A (like at point C), you would not be indifferent between A and B (since more is better). So A & B could not be on the same indifference curve, which goes against our initial statement that they are. So you must have less clothing at B, which means than B lies below A and the indifference curve slopes downward. clothing A C B food**clothing**E A IC1 food Indifference curves to the northeast are preferred. Point E is preferred to point A because it has more food & more clothing. Since you are indifferent between A & all points on IC1, E must be preferred to all points on IC1. Since you are indifferent between E and all points on IC2, all points on IC2 must be preferred to all points on IC1. IC2**Indifference curves can not intersect.**Suppose indifference curves could intersect. Let the intersection of IC1 & IC2 be D. Then you must be indifferent between D & any other point A on IC1. Similarly, you must be indifferent between D & any other point B on IC2. clothing By transitivity, you must be indifferent between A & B. But A & B are not on the same indifference curve, which they should be if you are indifferent between them. Then, our initial supposition that indifference curves could intersect must be wrong. D B IC2 IC1 A food**Indifference curves are convex[the slopes of IC’s fall as**we move from left to right, orwe have a diminishing marginal rate of substitution (MRS)] When we have lots of clothing & not much food (as near A & B), we are willing to give up a lot of clothing to get a little more food. When we have lots of food & not much clothing (as near C & D), we are willing to give up very little clothing to get a little more food. clothing A B C D food**Odd special cases that are not consistent with the**characteristics listed previously.**Perfect Complements**tires You need exactly 4 tires with 1 car body (ignoring the spare tire). Having more than 4 tires with 1 car body doesn’t increase utility. Also having more than 1 car body with only 4 tires doesn’t increase utility either. IC2 8 4 IC1 1 2 Car bodies**Mini-packs**10 5 IC2 IC1 1 2 Jumbo packs Perfect Substitutes Consider two packs of paper; the mini-pack has 100 sheets & the jumbo pack has 500 sheets. No matter how many mini-packs or jumbo packs you have, you are always willing to trade 5 mini-packs for 1 jumbo pack. Since the rate at which you’re willing to trade is the slope of the IC, and that rate is constant, your IC’s have a constant slope. That means they are straight lines.**production**IC2 IC1 pollution Goods versus Bads As you get more of a bad, you need more of a good to compensate you, to keep you feeling equally happy. So IC of a good & a bad slopes upward.**Neutral good**IC1 IC2 IC3 Desired good “Neutral” Good Your utility is unaffected by consumption of a neutral good.**Addict**Substance 1 The more substance 1 the addict has the more he/she is willing to give up of substance 2 to get a little more of 1 (& vice versa). So the IC’s are concave instead of convex. Substance 2**The slope of the indifference curve is the rate at which you**are willing to trade off one good to get another good. It is called the marginal rate of substitution or MRS.**Clothing C**A B D Food F What is the MRS or slope of the IC? Suppose points A & B are on the same indifference curve & therefore have the same utility level. Let’s break up the move from A to B into 2 parts. AD: TU = C (MUC) DB: TU = F (MUF) AB: 0 = TU = C (MUC) + F (MUF) C (MUC) = – F (MUF) C/F = – MUF / MUC IC2 IC1 So along an indifference curve, the slope or MRS is the negative of the ratio of the marginal utilities (with the MU of the good on the horizontal axis in the numerator). MRS = – MUX / MUY**For example,**Clothing C Suppose IC1 is the 90-util indifference curve & IC2 is the 96-util indifference curve. Point A is 7 units of food & 6 of clothing. B is 9 units of food & 5 of clothing. Since an additional unit of clothing gives you 6 more utils of satisfaction, the MU of clothing must be 6. Since an additional 2 units of food also give you 6 more utils of satisfaction, the MU of food must be 3. So, MRS = – MUF / MUC =-3/6 = -0.5 . You’d give up 2 units of food to get 1 units of clothing. IC1 =90 IC2 = 96 A 6 5 B D 7 9 Food F**Budget Constraint or Budget Line**This equation tells you what you can buy. For example, suppose you have $24, & there are two goods. The price of the first good is $3 per unit & the price of the second good is $4 per unit. So, if you buy X units of the first good for $3 each, you spend 3X on that good. Similarly, if you buy Y units of the second good, you spend 4Y on that good. Your total spending is 3X+4Y. If you spend all 24 dollars that you have, 3X+4Y=24. That equation is your budget constraint.**Y**(0,6) 0 (8,0) X Example: Budget constraint for $24 of income, and $3 & $4 for the prices of the two goods. If you spent all $24 on the 1st good, you could buy 8 units. If you spent all $24 on the 2nd good, you could buy 6 units. So we have the intercepts of the budget constraint. The slope of the line connecting these two points is Y/X = – 6/8 = – 3/4 = – 0.75 .**Y**(0,6) 0 (8,0) X Let’s generalize. Keep in mind that income was $24 and the prices of the goods were $3 & $4. The equation of the budget constraint in our example was 3X + 4Y = 24. So the budget constraint is p1X + p2Y = I Solving for Y in terms of X, p2Y = I – p1X, or Y = I /p2 – (p1/p2)X So from our slope-intercept form, we see that the intercept is I /p2, and the slope is –p1/p2 . The intercept is income divided by the price of the good on the vertical axis. The slope is the negative of the ratio of the prices, with the price of the good on the horizontal axis in the numerator.**Y**(0,9) (0,6) (8,0) (12,0) 0 X We have the intercept is I /p2, & the slope is –p1/p2 . What if income increased? The slope would stay the same & the budget constraint would shift out parallel to the original one. Suppose in our example with income of 24 & prices of 3 & 4, income increased to 36. Our new y-intercept will be 36/4 =9 & the new X-intercept will be 36/3=12.**Y**(0,6) (8,0) 0 (6,0) X Suppose the price of the good on the X-axis increased. If we bought only the good whose price increased, we could afford less of it. If we bought only the other good, our purchases would be unchanged. So the budget constraint would pivot inward about the Y-intercept. For example, if the price increased from $3 to $4, our $24 would only buy 6 units.**Similarly, if the price of the good on the Y-axis increased,**the budget constraint would pivot in about the X-intercept. Suppose the price of the 2nd good increased from $4 to $6. If you bought only that good, with your $24, your $24 would only buy 4 units of it. Y (0,6) (0,4) (8,0) 0 X**Let’s combine our indifference curves & budget constraint**to determine our utility maximizing point. Point A doesn’t maximize our utility & it doesn’t spend all our income. (It’s below the budget constraint.) IC3 Y IC2 IC1 A 0 X**Points B & C spend all our income but they don’t maximize**our utility. We can reach a higher indifference curve. IC3 Y IC2 IC1 B C 0 X**Point D is unattainable. We can’t reach it with our**budget. IC3 Y IC2 IC1 D 0 X**IC3**Y IC2 IC1 E 0 X Point E is our utility-maximizing point. We can’t do any better than at E. Notice that our utility is maximized at the point of tangency between the budget constraint & the indifference curve.**Recall from Principles of Microeconomics, to maximize your**utility, you should purchase goods so that the marginal utility per dollar is the same for all goods. If there were just two goods, that means that MU1/P1 = MU2/P2 Multiplying both sides by P1/MU2, we have MU1/MU2 = P1/P2 . The expression on the right is the negative of the slope of the budget constraint. The expression on the left is the negative of the slope of the indifference curve. So the slope of the indifference curve must be equal to the slope of the budget constraint. If at a particular point, two functions have the same slope, they are tangent to each other. That means your utility-maximizing consumption levels are where your indifference curve is tangent to the budget constraint. This is the same conclusion we reached using our graph.

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