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lecture06_slides - Econ 121. Intermediate Microeconomics....

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Unformatted text preview: Econ 121. Intermediate Microeconomics. Eduardo Faingold Yale University Lecture 6 Outline of the course I. Introduction II. Individual choice III. Competitive markets IV. Market failure Outline of the course I. Introduction II. Individual choice         Budget constraint (Ch. 2) Preferences (Ch. 3) Utility (Ch. 4) Consumer problem (Ch. 5) Revealed preference (Ch. 7) Slutsky equation (Ch. 8) Endowment income effect Intertemporal choice III. Competitive markets IV. Market failure Slutsky equation  Recall that x1 .p1; p2; m/ designates the demand for good 1 under prices p1, p2 and income m. Define the compensated demand for good 1 given prices p1, p2 and a bundle x as follows:  s x1 .p1; p2; x/ D x1 .p1; p2; p1x1 C p2x2/; s i.e. x1 .p1; p2; x/ is the optimal consumption bundle given the budget line that has slope p1=p2 and passes through bundle x . 0 This means that when the price of good 1 changes from p1 to p1, the consumer’s income is adjusted by 0 m D .p1 p1/x1 : Slutsky equation Fix p and m and write x  D x .p1; p2; m/. By the revealed preference argument we saw in the previous lecture, 0 p1 > p1 Thus, H) 0 s x1 .p1; p2; x / 0 p1 and therefore, 0 s x1 .p1; p2; x / s x1 .p1; p2; x / < 0: s x1 .p1; p2; x / 0 < 0 for all p1 ¤ p1; p1 s @x1 .p1; p2; x / < 0: @p1 That is, the substitution effect is always negative. Slutsky equation s Let us calculate @x1 =@p1 using the definition    s x1 .p1; p2; x / D x1 .p1; p2; p1x1 C p2x2 /: By the chain rule of calculus, s   @x1 @x1 @x1       .p1; p2; x / D .p1; p2; p1x1 C p2x2 / C .p1; p2; p1x1 C p2x2 / x1 : @p1 @p1 @m Omitting the arguments and re-arranging yields  s @x1 @x1 D @p1 @p1  @x1  x; @m 1 that is, Total effect D Subst. effect Income effect: So far....  ... we have assumed that ppl have some exogenous amount of money—their income m—to exchange for goods.  In reality, ppl sell things they own (e.g. labor hrs) to acquire goods. Want to model this idea. Net and gross demands  endowment: .!1; !2/ - what you have before you enter the market  gross demands: .x1 ; x2/ - what you end up consuming  net demands: .x1 sell (if negative) !1; x2 !2/ - what you actually buy (if positive) or Budget Constraint  value of what you consume = value of what you have  p1x1 C p2x2 D p1!1 C p2!2  See picture. Endowment belongs to the budget line.  with 2 goods, always net demander on one good and net supplier of the other Comparative Statics  Changing the endowment – normal vs inferior – consumer always better off when value of endowment increases. Different from increasing value of consumption bundle.  Changing prices – if price of a good consumer is selling goes down and consumer remains seller, welfare goes down. See figure. – if consumer is a net buyer and price goes down, consumer will remain a net buyer. See figure. Slutsky Equation  when prices change, we now have 3 effects – ordinary substitution effect – ordinary income effect – endowment income effect - change in the value of endowment affects demand – Slutsky equation s @x1 N @x1 .p1; p2; !/ D .p1; p2; x/ C .!1 N @p1 @p1  @x1 x1/ N .p1; p2; p1!1 C p2!2/ @m – Proof: follows from the regular Slutsky eq. and the definition  x1.p; !/ D x1 .p; p1!1 C p2!2/: N Labor supply  consumption C  labor L  money M Budget constraint for labor supply  pC D M C wL N  pC C w.L N N L/ D p C C w L N  leisure = R D L L N N  pC C wR D p C C w L  just like ordinary budget constraint  supply of labor like demand of leisure  w=p is price of leisure (opportunity cost) Comparative statics  Slutsky: @R N D SE C .R @w @R R/ @m  Leisure is normal good. Yet, ambiguous sign.  Backward bending supply New topic: intertemporal choice  .m1; m2/ money in each time period is endowment  allow the consumer to borrow and lend at rate r  c2 D m2 C .1 C r/.m1 c1/  note that this works for both borrowing and lending, as long as it is at the same interest rate  if borrowing and lending interest rates are different, then budget line has a kink. See picture. Budget Constraint  .m1; m2/ money in each time period is endowment  allow the consumer to borrow and lend at rate r  c2 D m2 C .1 C r/.m1 c1/  note that this works for both borrowing and lending, as long as it is at the same interest rate Various forms of the budget constraint  .1 C r/c1 C c2 D .1 C r/m1 C m2 - future value  c1 C 1c2r D m1 C 1m2r - present value C C  choice of numeraire  See picture  preferences - monotonocity and convexity are very natural Comparative Statics  If consumer is initially a lender and interest rate increases, he remains a lender, by revealed preference. See figure.  A borrower is made worse off by an increase in interest rate, by revealed preference. See figure.  Slutsky allows us to look at the effect of increasing the price of today’s consumption (increasing the interest rate) – change in consumption today when interest rate increases equals SE C .m1 c1/  IE – assuming normality, an increase in interest rate lowers current consumption for a borrower, and has an ambiguous effect for a lender  Intuition? Inflation  Previous analysis implicitly assumed price of “consumption good” remains unchanged in period 2. What if there is inflation?  put in prices p1 D 1 and p2  budget constraint takes the form p2c2 D m2 C .1 C r/.m1 or c2 D m2 1 C r C .m1 p2 p2  if  is rate of inflation, then p2 D .1 C /p1  1C D  D 1 Cr 1 C 1 Cr 1 C is the real interest rate 1r  c1/; c1/ Closer look at present value  Recall intertemporal budget constraint: c2 m2 c1 C D m1 C 1Cr 1Cr  Right-hand side expresses the present value of endowment  Recall that if the value of the endowment increases, consumer is better off.  Hence, provided the consumer can borrow and lend freely at a constant interest rate, consumer always prefers a pattern of income with a higher present value to a pattern with lower present value.  Present value works for any number of periods c2 m2 c3 m3 c1 C D m1 C C C 1 C r .1 C r/2 1 C r .1 C r/2  also works for time-varying interest rates c1 C c2 c3 m2 m3 C D m1 C C 1 C r1 .1 C r1/.1 C r2/ 1 C r .1 C r1/.1 C r2/ ...
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This note was uploaded on 03/01/2012 for the course ECON 121 taught by Professor Samuelson during the Spring '09 term at Yale.

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