Ch. #5 – “Demand: the Benefit Side of the Market”: • Whatever the good, consumer “purchasing decisions” are influenced by many factors: “own price,” “other prices,” “income,” “preferences,” etc. • Our aim is to develop a model of consumer choice that can be used to analyze the purchasing decisions of individuals. Subsequently (once “the best consumption bundle” is determined) we can analyze how purchasing decisions will change as a result of changes in economic factors such as “own price,” “other prices,” “income,” etc. • Let ()21, xxX = denote a “consumption bundle” (a certain amount of “commodity one” and a certain amount of “commodity two”). • “The Question” is “How do consumers go about choosing the best ()21, xxX = ?” Consumer’s Problem: • The reason for purchasing any good (or service) is to get a benefit from consuming the item. • Suppose that different consumption bundles can be ordered in terms of their relative desirability (Preferences). • Typically, goods are not “acquired for free” (that is, not “given away”) but rather must be “purchased.” • One of the “two basic functions of currency” is to act as a medium of exchange; thus, in a market economy, good are typically acquired by way of by purchasing them in exchange for money. • Given “prices” and “income,” some consumption bundles are affordable while others are not affordable (Budget Set). • The “Consumer’s Problem” could be stated as one of attempting to: Purchase the affordable consumption bundle that gives you the greatest amount of happiness. • To address this issue, we must first discuss the two separate issues of “Preferences” (a summary of what the consumer likes/dislikes) and the “Budget Set” (a summary of what is feasible or affordable for the consumer). Once we have developed these two notions separately, we will bring them together in order to solve the problem posed above. Competitive Budget Set: • Consider an individual with income of I, purchasing 1x and 2x at constant per unit prices of 1p and 2p respectively. • A consumption bundle ()21, xxX= is “affordable” so long as the expenditure necessary to purchase the bundle does not exceed the income of the consumer. That is, the inequality Ixpxp≤+2211 describes the “set of affordable consumption bundles” or “budget set.” • The equation Ixpxp=+2211 specifies the consumption bundles on the “budget line,” those affordable bundles for which expenditure is exactly equal to income.• Writing the equation for the budget line as 21212pIxppx +−=, we see that this is a straight line with: slope of 21pp−, vertical intercept of 2pI, and horizontal intercept of 1pI. The interpretation of these three values is rather intuitive. • Additionally, it is fairly straightforward to see how this line changes for many different types of changes in 1p , 2p , and/or I. Preferences: • Provide a ranking of the relative desirability of consumption bundles. • Preferences can be summarized completely independent of what a consumer can or cannot afford (that is, “prices and/or income” will never enter into a description of “preferences”) • Focus on a consumer with preferences that are: i. “complete” – any two consumption bundles can be ranked ii. “transitive” – considering three bundles, A, B, and C , if BA f and CB f , then CA f iii. “monotonic” – “more is better,” in that when comparing ()21,aaA = and ()21,bbB= such that 11ba ≥ and 22ba ≥ (with one of the inequalities holding “strictly”), it must be that BA f . iv. “convex” – “averages are better than extremes,” in that when considering ()21,aaA= and ()21,bbB= for which BA~, then BAC ~f for any ()21,ccC= such that 111)1( bacαα−+= and 222)1( bacαα−+= (for any 10<<α) • If preferences are complete and transitive, then they can be summarized by way of a “utility function,” which provides a numerical measure of “happiness” or “satisfaction.” • For a function ()21, xxU to be a valid utility function, it must be that ()( )2121,, yyUzzU > if and only if YZf and ()( )2121,, yyUzzU = if and only if YZ~. • The utility function provides a measure of “total happiness”; marginal utility provides a measure of how happiness changes as consumption changes. Specifically: “marginal utility for good one” measures the change in total utility brought about by an increase in consumption of good one, with consumption of good two fixed; “marginal utility for good two” measures the change in total utility brought about by an increase in consumption of good two, with consumption of good one fixed: 11)(xXUMU∆∆= and 22)(xXUMU∆∆=• Indifference Curve – collection of all consumption bundles resulting in a common level of utility. • Slope of indifference curve? Think about a change in the level of both goods, which maintains an initial level of utility: ()0))(())((2211=∆+∆=∆ xMUxMUXU ⇔))(())((1122xMUxMU∆−=∆ ⇔⎟⎟⎠⎞⎜⎜⎝⎛−=∆∆2112MUMUxx From here, we see that the slope of an indifference curve is equal to minus the ratio of 1MU to 2MU. • Define the “Marginal Rate of Substitution” as 212,1MUMUMRS =. Therefore, “slope of indifference curve” is equal to “minus 2,1MRS .” If preferences are “convex” (in that “averages are better than extremes”), then the Marginal Rate of Substitution should be “diminishing,” in that 212,1MUMUMRS = gets closer to zero as we move along any indifference curve by way of increasing 1x and decreasing 2x . • Properties of Indifference Curves: i. downward sloping ii. non-intersecting iii. “convex” to the origin Optimal Choice – “Budget Set and Preferences together” • Recall the Consumer’s Problem: Purchase the affordable consumption bundle that gives you the greatest amount of happiness. • If preferences are monotonic, then the optimal bundle must satisfy: Ixpxp =+2211. • Additionally, at any “interior solution” (a solution with 01>x and 02>x ), it must be that “the slope of the budget line” is equal to “the slope of the indifference curve.” That is: 2121MUMUpp−=− ⇔2121MUMUpp= ⇔2211pMUpMU= • The final representation of this condition states that (for all goods purchased in positive quantities), “the ratio of marginal