Chapter 5 lecture outline AAEC 3315, Benson 1 Pages 145-174 in Nicholson and Snyder 1. Income and substitution effects 1.1. The goal of this chapter is to use the utility maximization model to see how a consumer’s behavior (demand for goods) changes when prices or income changes 1.2. We need to understand how to derive a consumer’s demand function 1.3. This is important for understanding analysis and applications of consumer demand (like, say, marketing) 2. Demand functions 2.1. It is usually possible to solve first-order conditions of the Lagrangian for optimal levels of and λ – these are functions of prices and income 2.2. Or, if there are only two goods, we usually write it ( ) ( ) 2.2.1. The demand function, once specified, allows us to predict how much a consumer will buy 2.2.2. Prices/income are “exogenous” 2.2.2.1. “Exogenous” means originating outside 2.2.2.2. Exogenous variables originate outside an equation, and are used to determine the “endogenous” variables of the equationChapter 5 lecture outline AAEC 3315, Benson 2 2.2.2.3. Demand of the goods considered are the endogenous variables in these demand equations 2.3. We use demand equations to see what happens to demand when the exogenous variables (sometimes called “parameters”) change 2.3.1. Specifically, we look at partial derivatives of the demand functions 2.3.1.1. ⁄ describes how demand for x changes with income 2.3.1.2. ⁄ ⁄describes how demand for x changes when prices of x or y change 2.4. Homogeneity 2.4.1. What happens to demand for a good if prices and income were all doubled? 2.4.1.1. Optimal quantities of x and y wouldn’t change 2.4.1.2. What if prices and income were all multiplied by the same constant? 2.4.2. To see this, think about what happens to a budget constraint when prices and income double 2.4.3. Homogeneity is written mathematically as For any t > 0. 2.4.3.1. Functions that obey the above equation are “homogeneous of degree 0” 2.4.3.2. Demand functions are homogeneous of degree zero in all prices and income 2.4.4. Individuals’ demands are not affected by pure inflation that raises prices and income by the same factor 2.4.5. Example: suppose , and , derive the demand functions for x and y and show that they are homogeneous of degree 0 2.4.5.1. Utility maximization occurs whereChapter 5 lecture outline AAEC 3315, Benson 3 2.4.5.2. The partial derivatives of utility are 2.4.5.3. So the utility maximizing condition becomes: 2.4.5.4. Re-write with as ⁄ and ⁄ as 2.4.5.5. Substitute the above expression for y into the budget constraint ( ) ( ) ( )Chapter 5 lecture outline AAEC 3315, Benson 4 2.4.5.6. If are all increased by the same factor, t, the demand functions for x and y don’t change at all 3. Changes in income 3.1. As a person’s purchasing power increases, what happens to the quantity of each good consumed? 3.2. It is natural to think that income increases always affect quantity demanded like the diagram below 3.2.1. In the above, as I increases, x* and y* increase 3.2.1.1. That is, ⁄ and ⁄ are both positive 3.2.1.2. x and y are therefore normal goodsChapter 5 lecture outline AAEC 3315, Benson 5 3.3. For some goods, quantity consumed decreases as income increases 3.3.1. Here, the quantity of x decreases as income increases 3.3.2. ⁄ 4. Changes in a good’s price 4.1. This is more complicated 4.1.1. Changing a price of a good changes the slope of the budget line, not just the intercepts 4.1.2. So the new utility maximizing combination of x and y is not only on a new indifference curve, but at a point where MRS has changed from the initial value 4.1.3. Two different effects determine how a price change affects demand 4.1.3.1. The substitution effect: consumption adjusts to match MRS to the new price ratio 4.1.3.2. The income effect: consumption adjusts to changes in “real income” 4.2. Graphical analysis of a decrease in price 4.2.1. We can diagram the effect of a decrease in px on demand for x 4.2.2. We want to separate out the substitution and income effectsChapter 5 lecture outline AAEC 3315, Benson 6 4.2.3. At initial price consumer maximizes utility at x*, y* 4.2.4. If price decreases to , the consumer chooses a new utility maximizing combination of x and y at x** and y**, on the new indifference curve U2 4.2.5. Movement to x**, y** can be decomposed into the two effects 4.2.5.1. The change in the slope of the budget constraint would have motivated a move to point B even if income had been decreased to match the decrease in price 4.2.5.1.1. Think of the move to the dotted budget constraint as a new constraint in which income has decreased just enough to keep the consumer just as well off as before the price changed 4.2.5.1.2. The movement from x*, y* to B is a graphical representation of the substitution effect 4.2.5.2. The move from B to the new optimum is the income effectChapter 5 lecture outline AAEC 3315, Benson 7 4.2.5.2.1. In this case, x is a normal good, because the income effect is positive (more x is consumed at x** than at B 4.2.5.3. Nobody actually moves from x*, y* to B to x**, y**, this is just an illustration of the two different effects on changes in demand 4.3. Graphical representation of an increase in price 4.3.1. This is similar to a decrease in price 4.3.2. In this case, the increase in px causes the budget constraint to swing in towards the origin along the x-axis 4.3.3. To identify the substitution effect, we first draw a budget constraint parallel to the new one, but tangent to the original indifference curve 4.3.3.1. The movement from x**, y** to B represents how the consumer would react if price of x decreased, but income was increased just enough to offset the effect (on utility) of the price increaseChapter 5 lecture outline AAEC 3315, Benson 8 4.3.3.2. It is how the consumer would behave if he were constrained to keep his utility constant, but still observe the price change 4.3.4. The income effect is the movement back to x*, y* from B 4.4. Effects of price changes for inferior goods 4.4.1. For normal goods, the