# Chapter 10 Intertemporal choice Two periods, c 1 (amt of money spent in period 1), c 2, m 1, m 2 c 1 > m 1 : borrower, c 1 < m 1 : lender No debt or bequest.

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Chapter 10 Intertemporal choice Two periods, c 1 (amt of money spent in period 1), c 2, m 1, m 2 c 1 > m 1 : borrower, c 1 < m 1 : lender No debt or bequest left, then c 2 =m 2 +(m 1 - c 1 )(1+r), rearranging, (1+r)c 1 +c 2 =(1+r)m 1 +m 2 (future value) c 1 +c 2 /(1+r)=m 1 +m 2 /(1+r) (present value) 1 dollar today = 1+r dollars tomorrow Budget line: endowment (m 1,m 2 )

Fig. 10.2

Fig. 10.3

As before, when r increases, if before the change, a lender, after the change, still lender (better off); before the change, a borrower, after the change, could be a borrower (worse off) or lender (?) ∆c 1 /∆r = ∆c 1 s /∆r+(m 1 - c 1 )∆ c 1 m /∆m, so if a borrower, m 1 - c 1 <0, assuming normal, ∆c 1 /∆r<0 (when interest goes high, consume less). On the other hand, if a lender, TE is not clear.

Fig. 10.4

Fig. 10.5

Incorporate inflation: c 1, c 2, m 1, m 2, p 1, p 2 p 2 c 2 = p 2 m 2 + p 1 (m 1 - c 1 )(1+r), rearranging, c 1 + p 2 c 2 /(p 1 (1+r))=m 1 +p 2 m 2 /(p 1 (1+r)) Denote p 2 /p 1 =1+  Denote (1+r)/(1+  )=1+ , the budget line becomes c 1 +c 2 /(1+  )=m 1 +m 2 /(1+  ) and we call  the real interest rate If you give up one unit of c 1 today, you save p 1 and therefore you can get p 1 (1+r)/p 2 = 1+  tomorrow

Extending to 3 periods is straightforward: c 1 +c 2 /(1+r 1 )+c 3 /((1+r 1 ) (1+r 2 ))=m 1 +m 2 /(1+r 1 )+m 3 /((1+r 1 ) (1+r 2 )) An endowment with higher present value gives the consumer more consumption possibilities. An income stream (M 1,M 2 ) can be purchased by making a stream of payment (P 1,P 2 ) where M 1 +M 2 /(1+r)- (P 1 +P 2 /(1+r))>0 (net present value), then it is worth doing

A financial instrument: bond Pay a fixed coupon value x every year, at maturity date T, pay back face value F What should the price (P) of this bond be? First calculate the present value of the payment: x/(1+r)+x/(1+r) 2 +…+ F/(1+r) T If P>present value, then no one would buy the bond, on the other hand, if P<present value, then too many people will buy and hence P=present value.

( 模糊 ) In reality, there are many interest rates. Key is the interest measures the opportunity cost of funds, so you should use the interest that reflects your second best alternative of using the funds (If you don’t buy the bond, what would you do with the money left? The interest implicit behind that way of using your money is the interest you should use). Always bear in mind that we want to make the budget set as large as possible.

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