Lecture 03 Consumer reference Theor 1. Consumer preferences will tell us how an individual would rank (i.e. compare the desirabilit of) an two consumption bundles (or baskets), assuming the bundles were available at no cost. Of course, a consumer s actual choice will ultimatel depend on a number of factors in addition to preferences.. Assumptions about the consumer preferences a. completeness: A f B, A p B, or A B b. transitivit: If A f B and B f C, then A f C c. continuit: gradual change in preference with gradual change in consumption d. monotonicit: strong non-satiation or more is better continuous utilit function 3. Cardinal vs. Ordinal Utilit Jerem Bentham the greatest happiness of the greatest number Jevons, Menger, and Walras Marginal Revolution areto s ordinal utilit through indifference curve Hicks and R.G.D. Allen 4. Utilit Surface U = U U = 4 8 8 4 4 5. Diminishing? E) U = H + R, where H is number of hamburgers and R is number of root beer. 6. roperties of Indifference Curves a) Downward sloping. b) Conve to the origin (see MRS). c) Indifference curves cannot intersect. d) Ever consumption bundle lies on one and onl one indifference curve (everwhere dense). e) Indifference curves are not thick. f) The farther from the origin, the more utilit it has. 4
7. Marginal Rate of Substitution (MRS) 8 G C H 5 F i Δ Δ D J Δ K U 5 8 Δ d slope of I.C. = or = MRS, Δ d = U(, ). Totall differentiating to get du = d + d =0 (wh?). d = = MRS,. dmrs d d, < 0 or d > 0 (MRS is diminishing) d cf) Can ou draw an indifference curve with increasing MRS,? 8. Special Utilit Functions a) erfect Substitutes b) erfect Complements ancakes Right Shoe c) One Good and one Bad d) One Good and one neuter (neutral good) ollutant Used clothes Waffles Left Shoe Corn Sandwich 5
e) Cobb-Douglas Utilit Function (b Charles Cobb & aul Douglas) α β U = A, where A, α, and β are positive constants. Cobb-Douglass utilit function has three properties that make it of interest in the stud of consumer choice. (1) s are positive. Check it out. () Since s are all positive, the indifference curves will be downward sloping. (3) It also ehibits a diminishing MRS f) Quasi-Linear Utilit Function (imperfect substitution) It can describe preferences for a consumer who purchases the same amount of a commodit regardless of his income. E) toothpaste and coffee U(, ) = v( ) + b, where v ( )is a function that increases in and b is positive constant. The indifference curves are parallel, so for an value of, the slopes of I.C. will be the same. E) U = + 9. Budget Constraint + M From this constraint we can derive the budget line (or price line) to visualize in -D space. M = +, where is slope and M is vertical intercept. M Feasibilit Set or Opportunit Set M 10. Change in Income and Change in rice E) M 0 = $800, = $0, = $40. Draw the budget lines in each case. 1) Income rises from $800 to $1,000 ) rises to $5, holding initial income and constant. 3) Now, income rises from $800 to $1,000 and rises to $5 and rises to $50. 6
11. Optimal Choice ma U(, ): choose and to maimize utilit, subject to: + M: ependitures on and must not eceed the consumer s income If the consumer likes more of both goods, the marginal utilities of good and are both positive. At an optimal basket all income will be spent. So, the consumer will choose a basket on the budget line + = M. B At point A, d = = MRS, = d A U At point B, > or > U0 U1 1. Lagrangean Function (b Joseph-Louis Lagrange) ma U(, ), subject to: + M We define the Lagrangean ( L ) as L (,, λ) = U (, ) + λ( M ), where λ is a Lagrange multiplier. The first-order necessar condition (FOC) for an interior optimum (with > 0 and > 0 ) are L = U(, ) 0 = λ (1) L = U(, ) 0 = λ () L = 0 M λ = 0 (3) We can combine (1) and () to eliminate the Lagrange multiplier, so FOCs reduce to: = or = (4) + = M (5) From the above equations, we can derive demand function of and. 1 / / E) You are given U(, ) = + = + 1 and + = M. Now derive demand functions of and when the consumer is maimizing his utilit. 7
= 1 1 / = 1 1 1 / 1 and = =. So MRS, = = And FOC of utilit maimization is quantities demanded. * * * = =, where and are utilit-maimizing * * * B squaring both sides of above equation and rearranging terms, we find that = lugging the last epression into budget equation and simplifing, we get * * + = M + * * * * + = + = = M. So, * * M M = = =. So, 3 + + 13. Change in Income and Optimal Choice ICC ICC * = * = M (1) + M () +. Income (M) Income (M) Engel Curve Engel Curve (Normal Good, η > 0 ) (Inferior Good, η < 0 ) 8
14. Change in rice and Optimal Choice CC (or Offer Curve) cf) What would be the shape of CC if the price elasticit of demand is 1 (unitar elastic)? cf) Think about the shape of CC if the quantit is reduced with price hike. And also think about the shape of demand for that good. Demand X 15. Change in rice: Revisited * rice Effect ( E E ) ( E E ): Substitution Effect due to price change ( E E ): Income Effect In this case, good is a normal good. E E E i 1 i 0 cf) Derive substitution and income effect if good is neither a normal nor an inferior good. cf) Derive substitution and income effect if good is an inferior good. Mabe ou can think of two different cases. 0 ( M / ) ( M / ) 9
16. Slutzk Equation (Slutzk Decomposition) 0 = ( 0) + ( ) (1) Dividing both sides b, we get the following equation, 0 = 0 + () l.h.s. can be eplained Δ Δ M. And the first term on r.h.s. is equal to And the second term on r.h.s. is now rewritten as follows, Δ =, where R means real income. (3) means the change in real income w.r.t. change in price. So, = (Shepard s lemma) (4) Δ Δ U. And Δ Δ = (wh? Think about the definition of income effect!) (5) ΔM lugging (4) and (5) in (3) and then in (), we can get Δ Δ Δ = (Slutzk Equation) ΔM M U 17. Labor Suppl and Leisure (Application) H + L =4, where H is hours worked and L is amount of leisure activities. And hourl wage is given at w 0 and non-labor income is given at V 0. The utilit function is now U = U( L, M) (1) The budget equation is M = w0h + V0 = w0( 4 L) + V0 = 4w0 w0l+ V0 () Rewriting (), we can get M + w0l= 4 w0 + V0 (full-income constraint) (3) If we assume the price level of goods (composite good) is C = $1, then M on l.h.s would equal total number of goods that this consumer bus. Can ou interpret terms in equation (3)? 1) Now, can ou draw the budget line? ) Find the utilit-maimizing point. 3) Suppose wage rises. Overlap new optimal points on the original budget line. 4) Connect those optimization points to get CC. And labor suppl curve. 5) Wh do ou think is it a backward bending suppl curve? 10