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 H) 0 s x1 .p1; p2; x / s x1 .p1; p2; x / < 0: Thus, 0 s x1 .p1; p2; x / 0 p1 s x1 .p1; p2; x / 0 < 0 for all p1 p1; p1 s @x1 .p1; p2; x / < 0: @p1 and therefore, That is, the substitution effect is always negative. Slutsky equation
s Let us calculate @x1 [email protected] 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 Cp2x2 /C .p1; p2; p1x1 Cp2x2 / x1 : @p1 @p1 @m Omitting the arguments and rearranging 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 moneytheir income mto 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
substitution effect negative @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
wage pC D M C wL N pC C w.L N N L/ D p C C w L L
number of units of the consumption good you have: M/p N leisure = R D L N N pC C wR D p C C w L
value of endowment just like ordinary budget constraint supply of labor like demand of leisure w=p is price of leisure (opportunity cost) Comparative statics
subsitution effect income effect Slutsky:
how leisure changes with wages @R N D SE C .R @w @R R/ @m Leisure is normal good. Yet, ambiguous sign. Backward bending supply
Difference between the number of hours he is endowed in a day (Rbar) and the number of hours he is actually getting (R) when labor is very low, the substituion effect dominates because he is working very few hours. He switches from leisure to consumption very rapidly. But once labor reaches a certain point, this effect dissapates and the labor supply curve "bends backwards" New topic: intertemporal choice
tradeoff between consumption today and consumption tomorrow .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/ money in period 2 + money not spent in period 1 plus interest 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. Various forms of the budget constraint .1 C r/c1 C c2 D .1 C r/m1 C m2  future value
c2 m2 c1 C 1Cr D m1 C 1Cr  present value 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? rising interest rates lead the consumer to save more and spend more tomorrow 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 c1/; c2 D m2 1 C r C .m1 p2 p2 c1/ if is rate of inflation, then p2 D .1 C /p1 1C D D
1Cr 1C 1Cr 1C is the real interest rate = lending rate/interest rate 1r Closer look at present value Recall intertemporal budget constraint: c2 m2 c1 C D m1 C 1Cr 1Cr Righthand 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 timevarying 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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 SAMUELSON
 Microeconomics, Interest Rates, Slutsky Equation

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