Unformatted text preview: PADP 6950: Founda1ons of Policy Analysis Intertemporal Choice & Uncertainty Angela Fer1g, PhD Special cases: Different kinds of goods to choose between Before: food and other goods, etc. Intertemporal Choice: cc now vs. cc future Uncertainty: cc if state 1 (e.g. sick) vs. cc if state 2 (e.g. healthy) 1 Intertemporal Budget Constraint c 2 = m2 + (m1 - c1 ) + r(m1 - c1 ) c 2 = m2 + (1 + r)(m1 - c1 ) c 2 + (1 + r)c1 = m2 + (1 + r)m1 c2 m2 c1 + = m1 + (1 + r) (1 + r)
p1 = 1
Budget constraint in present value terms: p2 = 1 (1 + r) Intertemporal Budget Line c1 + c2 m2 = m1 + (1 + r) (1 + r) Y-intercept (c2): assume c1=0 c 2 = (1 + r)m1 + m2 c1 = m1 + m2 (1 + r) X-intercept (c1): assume c2=0 Slope: transform to form y=b+mx c 2 = (1 + r)m1 + m2 - (1 + r)c1 2 Graph How does line change when interest rate changes? If r goes from 0.05 to 0.1: slope=-(1+r) goes from -1.05 to -1.1 (steeper) c2-intercept shiYs up c1-intercept shiYs back Must pivot around endowment point. 3 Intertemporal Indifference Curve If straight line (perfect subs1tutes): consumer doesn't care whether they consume today or tomorrow If L-shaped (perfect complements): consumer wants equal amounts today and tomorrow More realis1c: convex, well-behaved indifference curve What happens to a lender when r goes up? 4 What happens to a borrower when r goes up? Discount rate Discount rate () = the rate at which we discount the value of the future We usually just let =r (interest rate) A dollar today could be invested in an interest-bearing instrument (bond) and then it would be worth $1*(1+r) in the next period So, $1 in the future is worth $1/(1+r) today However, some people are more impa1ent and have a higher discount rate than r and some are very pa1ent and for them, <r Discount rates are a very important determinant of people's intertemporal choices: Risky behavior decisions Investment decisions Some say just as important as IQ in determining child's academic success 5 Net Present Value (or Discoun1ng) General formula: NPV = t =T Vt t t =1 (1 + r) Example 1: What is $100 given to me in 1 year worth to me today (assume r=0.05)? $100 NPV = = $95.24 1.05 Example 2: What is $300 given to me in 3 installments over the next 3 years worth to me today? NPV = $100 $100 $100 + + = $95.24 + $90.70 + $86.38 = $272.32 1.05 1.05 2 1.05 3 Policy Applica1on: Benefit-Cost Analysis Discoun1ng is used for comparisons of different streams of benefits and/or costs over a number of periods/years. Assume discount rate=20%. Example: Investment A: returns $20 at end of Year 1 and $20 at end of Year 2 Investment B: returns $12 at end of Year 1 and $29 at the end of Year 2 A provides a total of $40 and B provides a total $41 so should we choose B? 6 Choice under uncertainty When deciding how much insurance to buy, really deciding between consump1on in 2 states: no loss/no illness vs. loss/illness If no insurance available, then there is no choice: (mg,mb) If insurance becomes available, then you can make a choice about how much to get. Your budget line will depend on the premium charged. Assume insurance coverage K costs K (premium): If have loss, have: If no loss, have: mb + K - K = mb + (1- )K m g - K Choice under uncertainty noins ins m g - m g + K rise c g - c g - slope = = noins = = ins run c b - c b mb - mb - (1- )K (1- ) Assume indifference curve is convex as usual. If endowment isn't at the tangent point, then buy insurance. 7 U1lity under uncertainty Another factor that must be considered (besides the premium) is: What is the probability that the bad event will occur? It is taken into considera1on through an expected u#lity func#on: U = i u(c i )
i=1 i=N Expected U1lity vs. Value If just 2 possible states: U=u(c1)+(1-)u(c2) Note this is different from expected value: EV=c1+(1-)c2 8 Risk aversion We assume that the u1lity func1on with respect to wealth/money for most is upward sloping and concave: Happier if richer, but not twice as happy if twice as rich Diminishing marginal u1lity of wealth Implica1on of this concavity: having money with certainty makes you happier than having it with some uncertainty people don't like risk Example to demonstrate the connec1on between concavity of U & risk aversion W=$20; If sick (5% chance), then W=$10, U=ln(W) Expected value: E(W ) = (1- )W well + W sick = 0.95 * $20 + 0.05 * $10 = $19.50 Expected u#lity without risk: U(E(W )) = U($19.50) = ln(19.50) = 2.97 Point on the U(wealth) curve 9 Example con1nued In reality, you don't get $19.50, you get $10 with a 5% probability and $20 with a 95% probability. Expected u#lity with risk: E(U) = (1- )U(W well ) + U(W sick ) = 0.95 * ln($20) + 0.05 * ln($10) = 2.965 Point below the U(wealth) curve 10 Risk premium Risk premium is the amount people are willing to pay to avoid risk: Risk premium = E(W) w/ risk that gives U1 - W w/ certainty that gives U1 = E(W)-WE(U) In this example, E(W)=$19.50 and WE(U)=eln(W)=e2.965=$19.39, so risk premium=11 cents 11 ...
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- Spring '11
- c1 + c2, Angela Fer1g