哈佛大学：《中级微观经济学》（英文版）Lecture 8 Financial Markets

We can sign this in two ways (1)some assumption about concavity or 2)looking at the effect of price coming from perfect insurance Assume that there is free entry in supply but it costs c to process each dollar of insurance(is this reasonable? ); then we must have that pl-TI-cl=o for any insurance level 1, or p=T+c(arbitrarily fair insurance Using this the f.o. c becomes U(r-T+cM U(Y-z+(1-x-c)) 1-丌)(兀+c)

We can sign this in two ways: (1) some assumption about concavity or (2) looking at the effect of price coming from perfect insurance. Assume that there is free entry in supply but it costs c to process each dollar of insurance (is this reasonable?); then we must have that pI  I  cI  0 for any insurance level I, or p    c (“arbitrarily fair insurance”). Using this the F.O.C becomes U Y    cI U Y  Z  1    cI  1  c 1    c

Forc=0, U(r-pD=U(r-Z+1-pl That is, consumption/income is equal or/=Zandr-pl=Y-TI=Y-Z+l- across states Hence, for c=0, people perfectly insure against the shock. ( What if c>0?

For c  0, U Y  pI  U Y  Z  I  pI or I  Z and Y  pI  Y  I  Y  Z  I  pI. That is, consumption/income is equal across states. Hence, for c  0, people perfectly insure against the shock. (What if c  0?)

The derivative of insurance w.r.t. c or p, for c=becomes -U(r-TD -(1-m)p2+m(1-p)2)U"(Y-D U(Y-TD -(p2+x-2p)"(Y-D which is negative and inversely proportional to the coefficient of absolute risk aversion at y-丌l

The derivative of insurance w.r.t. c or p, for c  0 becomes I p  U Y  I 1  p2  1  p2 UY  I  U Y  I p2    2pUY  I which is negative and inversely proportional to the coefficient of absolute risk aversion at Y  I

Alternately, go back to the denominator of the ugly expression we had before (1-m)U(Y-p)+(1-m)lU(Y-p) TU(r-Z+I-pl-1-plU(r-Z+I We know that p=丌+c>or (1-丌)>丌(1-p) so as long as UGr-pl>U(r-Z+1-pl were done- this would require what to hold?

Alternately, go back to the denominator of the ugly expression we had before: 1  U Y  pI  1  pIUY  pI U Y  Z  I  pI  1  pIUY  Z  I  We know that p    c   or 1  p  1  p so as long as UY  pI  UY  Z  I  pI we’re done– this would require what to hold?

Alternately, as long as U(r-Z+1-pl>-(1-plv(r-Z+I-pl were done -what would ensure that?

Alternately, as long as U Y  Z  I  pI  1  pIUY  Z  I  pI we’re done – what would ensure that?