# Utility Functions and Indifference Curves: Getting Into the Consumers Head A utility function indicates how a consumer orders different bundles of goods:

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Utility Functions and Indifference Curves: Getting Into the Consumers Head A utility function indicates how a consumer orders different bundles of goods: bread, eggs, wine. Suppose U(b,x,y) = 3b + 2x + 1y then U(1,4,6) = 3 + 8 + 6 = 17 U(2,1,3) = 6 + 2 + 3 = 11 The first bundle is preferred to the second The values of U(b,x,y) = 17 and 11 dont mean anything by themselves. –They just indicate the order of preferences.

Getting into the Consumers Head You cant argue with a consumers valuations of different bundles of goods, no matter how bizarre it seems. –If U(b,x,y) = b 3 x 2 y 1 This consumer prefers any bundle that contains a bit of each good to a bundle with lots of two of the goods but none of the other (Man cannot live by bread alone … gotta have eggs and wine too) This consumer is indifferent between a bundle with lots of bread and eggs but no wine and a bundle with just a little bread and eggs but no wine: U(B,X,0) = U(b,x,0) = 0

Getting Into the Consumers Head In theory, we can tease out the consumers utility function by observing her choices when offered different combinations of goods, her revealed preferences. In practice, we only have to accept the idea that utility functions and indifference maps exist. –We can then get some powerful insights into consumer behavior.

Getting Into the Consumers Head An Indifference Map Wine(y) u 2 u 1 Eggs(x) All combinations on U 2 are equally preferred All combinations on U 2 are preferred to combinations on U 1 MORE BEATS LESS!!!

Getting Into the Consumers Head Slope of the indifference curve = dy/dx = Rate at which wine must be substituted for eggs to keep the consumer at the same level of indifference The fewer eggs the consumer has, the more wine she needs to compensate for taking away yet another egg. Indifference curves are convex to the origin The marginal utility of eggs increases as the number of eggs the consumer has decreases: MU x when x

Getting Into the Consumers Head The Consumers Budget Line (Budget = \$B) Wine(y) B/p y u 2 u 1 Eggs(x) B/p x Slope of Budget Line = - (B/p y )/(B/p x ) = - p x /p y dy/dx = - p x /p y

Getting Into the Consumers Head At point of highest attainable indifference –the budget line just touches the highest indifferent curve –the budget line is tangent to the highest indifference curve –the slope of the budget line equals the slope of the indifference curve The consumers willingness to substitute y for x along her highest attainable indifference curve equals her ability to substitute y for x along her budget line.

Getting Into the Consumers Head An important insight: At the point of highest attainable indifference, the consumers willingness to substitute y for x along her indifference curve equals her ability to substitute y for x along her budget line – her ability to substitute y for x is the price ratio - p x /p y … the slope of the budget line Since everyone faces the same price ratio, dy/dx = - p x /p y, the slope of everyones budget line is the same –everyones relative valuation of eggs and wine (x and y) is the same.

Getting Into the Consumers Head Along an indifference curve, - MU x dx = MU y dy Reduced utility from less x=Increased utility from more y So on indifference curve, dy/dx = -MU x /MU y Consumer is in equilibrium when slope of her indifference curve equals slope of budget line - MU x /MU y = dy/dx = - p x /p y or MU x / p x = MU y / p y = … MU z / p z Get equal bang per buck from all goods!

Utility and Demand Consumer gets equal bang per buck from all goods: apples, bread, eggs, wine, zolo toys MU a /p a = MU b /p b = … MU x /p x = MU y /p y = … MU z /p z If p x, MU x must fall Since MU x when x, p x x LAW OF DEMAND! More eggs are bought when the price of eggs declines.

Demand and Consumer Surplus More eggs are bought when the price of eggs declines. Demand curve reflects price, p x, consumer is willing to pay for each quantity of x Consumer is willing to pay more for the first egg than for subsequent eggs … the first egg will be used most productively, e.g., to bake a cake. But consumer need only pay the same market price for each egg she buys. –This is less than she is willing to pay for all but the last egg she buys shed never pay more than shes willing to pay. Consumer realizes consumer surplus in market

Consumer Surplus in Pictures Price(p x ) Price consumer is willing to pay for each quantity of x C.S. Market price consumer has to pay for each eggshe buys Eggs(x)