Representative consumer Buffy has y today = 50, no financial wealth today, f tod

Question

Representative consumer Buffy has ytoday = 50, no financial wealth today, ftoday = 0 and future income, yfuture = 50. Buffy lives for two periods and has the following preferences: u(ctoday, cfuture) = log(ctoday) + log(cfuture)

a. Write down the intertemporal budget constraint.

b. Write down the Euler equation.

c. Suppose = 1 and R = 0 Calculate optimal ctoday, ffuture and cfuture.

d. Suppose = 0.8 and R = 0 Calculate optimal ctoday, ffuture and cfuture.

e. Explain why Buffy chooses different amounts for ctoday and cfuture when = 0.8, even though the interest rate is 0 and ytoday = yfuture

f. Suppose = 0.8 Solve for the value for R such that Buffy consumes the same amount in both periods.

g. Continue to assume that = 0.8, is Buffy better off when R = 0 or when R is the value you calculated in part f

Explanation / Answer

ytoday = 50,
no financial wealth today, ftoday = 0
future income, yfuture = 50
u(ctoday, cfuture) = log(ctoday) + log(cfuture)
a) intertemporal budget constraint will be
ctoday + cfuture/(1+R)=ytoday + yfuture /(1+R)
ctoday + cfuture/(1+R)=50 + 50 /(1+R)
b) Euler equation

c)Suppose = 1 and R = 0
then
u(ctoday, cfuture) = log(ctoday) + log(cfuture)
ctoday + cfuture=100
Optimizing this using derivatives
cfuture/ctoday =1
hence 2ctoday=100
ctoday=50=cfuture
ffuture=ytoday-ctoday=50
d) Suppose = 0.8 and R = 0 then
u(ctoday, cfuture) = log(ctoday) + 0.8log(cfuture)
ctoday + cfuture=100
optimizing gives

cfuture/0.8ctoday=1
So 1.8Ctoday=100
ctoday=55.55
cfuture=44.44

ffuture=ytoday-ctoday=44.44 as R=0

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