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					                 Poverty Trap Model from the “Investment and Poverty” Lecture

This is the model discussed in Banerjee's paper “The two poverties”

The main point of this model is to show how poverty traps can arise from a model with:
   1.) Occupational choice
   2.) Imperfect credit markets (it is possible to default and escape)

The set-up of the model is as given in pages 5 and 6, and certain parts of the model are discussed
until page 11. I start here from page 12.

Occupational Choice

Here individuals can either work or engage in entrepreneurship. When choosing occupation,
agents choose in order to maximise utility.

                V[work]               =       W + er

                V[Entrepreneur]       =       A(R-r) + er -E
                                      =       be(R-r) + er – E     using A=be from page 11

Choose occupation in order to maximise V:

                       = Work                 if       V[Work]     >      V[Entrepreneur]
                                              =>       W + er      >      A(R-r) + er -E
       Choice                                 =>       W           >      A(R-r) -E
                       = Entrepreneur         if       V[Work]     <      V[Entrepreneur]
                                              =>       W + er      <      A(R-r) + er -E
                                              =>       W           <      A(R-r) -E


We now have the utilities as a function of a set of constants and e, initial endowment.

Defining    e

Define e to be the level of endowment such that the individual is indifferent between the two
occupations:

                V[work]         =     V[Entrepreneur]
                W + er          =     be(R-r) + er – E
                W+E             =     be(R-r)
                 e              =     (W + E) / [b(R-r)]

Q: Why do we care about e ?
A: Because, from looking at the two equations V[work] and V[Entrepreneur], it can be seen that the
occupational choice is determined by considering whether or not endowment is greater than or less
than e :

       Choice                   = Work                       if e< e
                                = Entrepreneur               if e>= e

This in turns implies a relationship between utility, V, and e.

                                = W + er                     if e< e
       Utility (V)              = be(R-r) + er – E           if e>= e and e< A
                                = A (R-r) + er – E           if e > A
N.B. The third line is due to the assumption on page 6 that the investment yields a return of R only
up to the point A . There is no main intuition to this assumption; it is just needed to close the
model, so don't worry about it.

End-of-period income - M

What we need to think about now is individuals end-of-period income, which I'm going to denote as
M.

Q: Why
A: Individuals leave a fraction of this income (and not their initial endowment) to their child. This
fraction is denoted as h in the lecture notes, and is assumed constant for everyone.

M will be closely related to the utilities above (as we have a linear utility function), and will depend
on e.
                                         = W + er                    if e< e
  End-of-period income (M)               = be(R-r) + er              if e>= e and e< A
                                         = A (R-r) + er              if e > A

The only difference between V and M is that for entrepreneurs, the cost of entrepreneurship is
absent from end-of-period income,as this is a utility cost of effort, but not financial.

Below, I've drawn a graph of M against e:


               M




              B
                                 E
               A




                                                                                    e
                                      e           A



Q: How do we know the vertical length of the discontinuity is E?
A: We can work it out from the following

At point A:              M            =       W+    e r
At point B:              M            =         e b[R-r] + e r


                         B-A          = ( e b[R-r] + e r) - (W +        e r)
                                      = e b[R-r] - W

From the definition of     e (see above) we know that:
                         W+E          =        e b(R-r)
                         E            =        e b(R-r) – W
Use this to look at B-A:

                       B-A                = e b[R-r] - W
                                          =E

This is really important. It is this E, the disutility of entrepreneurship effort that causes the
discontinuity to occur at point e . Without this E, everyone would engage in some
entrepreneurial activity due to the fact that R>r.

N.B. In question 2 of class 3, in a very different setting (i.e. no occupational choice, just imperfect
credit markets coupled with an indivisible project of size K) we generated a discontinuity by
assuming that the project was non-divisible. In parts d.) and e.) of question 2, when the project is
perfectly divisible up to K, there is no longer a discontinuity.

Bequests

We assume all individuals have a single child, and bequeath a fraction h (where h must be
between 0 and 1) of their end-of-period income, to this child. This is what will give us the diagrams
on page 14 and 15. We can represent the link between parental endowment, e t and child
endowment, e t1 as follows:


                                                                  45°



 e t1




                                                                                            et
                             e*       e      e**    A/ h

This is the diagram we see on page 14, where the two values e* and e** represent the lower and
upper stationary equilibria of the models. The stationary points will be reached over time. We can
write this in terms of limits (as we do with probability limits):

 lim et        =       e*                      for initial values of e in range [0,   e )
  t∞
               =       e**                     for initial values of e in range [ e , e_maximum]

Q: why do we see the kink occur at A/ h and not at A as we saw in the diagram before?
A: if the parent gives an amount   h e t > A , the child will only get the higher return on the
first A of this endowment. Divide both sides of the inequality by h to get e t > A/ h . Note
that as h<1, A/ h A
To get to the stationary points:

We can consider the model as a sequence of steps.:

              Period t
Step 1:       Individual is endowed with amount e t
Step 2:       Occupational choice is made based on whether or not    e t is greater or less than
               e
Step 3:       The period finishes, with the individual having an amount M. As we saw above M is
              a function of e t .
Step 4:       Individual leaves a fraction h of this income to child.

              Period t+1
Step 5:       Individual is endowed with amount e t1 = hM = hM( e t ).

Repeat stages 2-5 until the stationary points are reached.