STOCHASTIC

204 GORDON PYE
porations will be coaxed into issuing noncallables
and some of the less risk-averse
long-term lenders will be coaxed into holding
callables. Equilibrium will be reached when
the differential has risen far enough above
expected value.
An interesting fact to explain is why most
corporate bonds are callable and most
public bonds are not. To public officials
callable bonds mean higher taxes now with
the possibility of tax savings in the future.
The officials might be expected to discount
the future savings by the probability that
they will no longer be in office when these
savings accrue. The effect would be much
the same as if public officials had lower expectations
of future bond prices than most
private borrowers and lenders in the model
discussed above. In equilibrium most private
bonds would be callable and most
public bonds would not.
The phenomenon might also be explained
by risk aversion. For state and local governments
there seems to be no particular reason
to suspect that the tax savings would offset
any tax losses or increased expenditures. For
the federal government the tax savings
would offset higher interest expenditures
To prove the proposition stated in the text,
the following notation will be used. Let Q be
the n X n probability matrix [ 0, then (a) the
bond should be called in any period if and only
if p is less than some critical value for that
period; (b) these critical values will be non-decreasing
with time.
PROOF: (a) For proof it suffices to show that
the vector pt — ct — Vt is monotone-decreasing
for all /. Assume pt+\ — c