11/13/2009

=Some thoughts on risk-neutrality=
Do you want money today, or money a month from now
col1=money today, col2=money in a month
50x  50
50  60x
50  70x
50  80x
50  90x


col1=money today, col2=col2 or 0 w/ p=.5
50x  50
50x  60
50x  100
50x  110
50  120x
50  130x
50  150x

(x=my choice)

So this shows us how savings, certainty, and future discounting apply to people
  taking on income.

If utility of x is linear (u(x) = ax+b), then for 50%/50% change of getting x,
  then .5*u(0) + .5*(100) = a*0/2 + a*100/2 = a*50 = u(50) [add the b in as well]
  
But people aren't risk-neutral.  They are risk-averse, so they want some payment
  for their risk.  So the u(x) function is convex:
   
       .'               ,  
     .'       not      ,
   .'              , 
   
So in that case, .5u(0) + .5u(100) \leq u(.5*0 + .5*100)
That's called jenson's inequality: the value of the average is less than the
  average of the values for "convex" people.

Borrowing money is like having someone give you money so that you can borrow
  from your own future.  If you can't pay youself/them back, they won't loan
  you the money.

So credit/savings lets people insure themselves.

=Microcredit/Microinsurance=
Assumptions from last time about borrowing in developing nations:
  - low likelihood of default
  - perfect competition
  - high interest
  - high variance in interest rates across people
  - rich people receive lower interest rate, and can borrow more (credit rationed
    by rate/ammount according to wealth)
  - borrowing seems to be productive

Definitions
  - \rho is cost of capital---money lenders have to walk to big town, borrow from
    big bank, etc.
  - k is capital you need to invest to get income of F(k)
  - w is your current wealth, so you need to borrow k-w
  - borrower must repay (k-w)*r
  - the borrower can shirk (run away) at a cost of \eta per dollar loaned.
  - borrower repays when the profit is greater than cost of shirking
      F(k) - (k-w)r >= F(k) - \eta*k [incentive compatibility between borrower and bank]
      rearrange: r/(r-\eta) >= k/w
      so if hold r constant, the only large enough wealth will be compatible

Supply side
  revenue = cost = cost of capital + misc.
  since lending is productive and competitive, cost = cost of borrowing.
  r(k-w) = \rho(k-w) + misc.
  let's consider different cases for misc. cost that would make the stuff above
    be true about borrowing
  -> no misc. costs -> r = \rho.  interest is the cost of going to the bank.
  -> misc. costs are linear w/ borrowing: \phi*(k-w) -> r = \rho + \phi
     incentive compatibility check is now:
       (\rho + \phi)/(\rho + \phi - \eta) >= k/w
     so for incentive compatibility, we need to include cost of monitoring that
     the misc. costs are good enough.
  -> misc. costs are linear in invested ammount: \phi*\eta*k, so
     IC is r(k-w) = \rho(k-w) + \phi*\eta*k
           wr = kr - \eta*k -> \eta*k = r(k-w)
           r = \rho/(1-\phi)
     But then interest rate is relatively fixed, and not too high!
  -> there's a fixed cost \phi that's pretty high, in order to have a loan officer
     go to the village.
     r(k-w) = \rho(k-w) + \phi
     \eta*k - \phi = r(k-w) - \phi = \rho(k-w) -> r = \rho + \phi/(k-w)
     r = \rho + (\phi*(\rho - \eta))/(\eta*w-\phi)
  so as w goes up, r goes down
  there's a high range/variance to rates

Under this model, people pooling money together and police themselves, their
  wealth hits the bump over which banks require to loan, and their risk of running
  away is lower.  Thus, from the fixed cost thing, microcredit can be plausible!