**APPENDIX I**

**A Simple Model of Recurrent Cost Disequilibrium**

Let

V~t~ = Quantity of variable input at time t into public production
(e.g., labor, materials, etc.)

K~t~ = Quantity of fixed Input at time t into public production (e.g.,
building, roads, etc.)

Q~t~ = Quantity of government services produced at time t (e.g., \# of
student educated).

R~t~ = Total government revenue available at time t

I~t~ = The amount of revenue allocated to new investment in fixed inputs
at time t.

W~t~ = price of variable input

P~t~ = price of the fixed input

At any time, e.g., t =1, the government must decide how much of its
revenue should be allocated to purchasing variable inputs, and how much
should be allocated to purchasing new capital. Let us assume the
following:

\(1\) The price of government services is the numeraire good, equal to
one. Thus Q, the quantity of government output is also the value of
government output.

\(2\) Output is determined by the following generalized production
function; which we will assume is differentiable.

\(1\) Q~t~ = f(K~t~, V~t~)

where

R~t~ - P~t~I~t~

\(2\) V~t~ =

W~t~

Now assume that the objective function of the government is to maximize
the present value of the stream of government output subject to the
production function and total revenue constraints.

Let *~Q~* = the discount rate of the government

Then

T f(K~t~,V~t~) I~t~ R~t~-P~t~I~t~

\(3\) Q = £ \-\-\-\-\-\-\-- where K~t~ = K~t-1~ = \-\-- and V~t~ =
\-\-\-\-\-\--

i=1 (1+ *~Q~*)^t^ P~t~ W~t~

or

T f(K~t-1~ + I~t~ R~t~-P~t~I~t~

\(4\)

Q = £

t=1 (1 = *~Q~*)^t^

or

T t-1 I~i~ R~t~-P~t~ /

\(5\) £ f(£ K~0~ + \-\-\-- , \-\-\-\-\-\-\-\--) /(1+*~Q~*)^t^

t=1 1=0 P~i~ W~t~

Taking the partial derivative of Q with respect to I~j~, and setting the
result equal to zero, we get

-P~j~^f^~2j~ T ^f^~1t~

\(6\) \-\-\-\-\-\-\-\-\-- + £ \-\-\-\-\-\-\-\-\-\-- = 0

W~j~(1+*~Q~*)^j^ t=j +1 P~j~(1+*~Q~*)^t^

P~j~ ^f^~2j~ T ^f^~1t~

\(7\) \-\-\-\-\-- \-\-\-\-\-- = £ \-\-\-\-\-\-\--

W~j~ (1+*~Q~*)^j^ t=j+1 P~j~(1+*~Q~*)^t^

where

*6*Q~t~ *6*Q~j~

^f^~1t~ = \-\-\-\-\-\-- ; ^f^~2j~ = \-\-\-\-\-\--

*6*K~t~ *6*V~j~

The interpretation of the result seems straightforward. The government
should allocate resources until the present value of the marginal value
product of an additional unit of the variable input is exactly equal to
the present value of an additional unit of fixed inputs.

When this decision rule is not followed, either because of LDC
government policy which underallocates resources to the variable inputs,
or because donors limit the fungibility of their assistance making it
impossible for LDC\'s to allocate resources efficiently, then there is a
recurrent cost problem.

Note that there are really two allocation decisions. The first concerns
the amount of resources which is to be allocated to public production as
opposed to private production. Up to this point we have not investigated
that question in this Appendix. The second allocation decision is the
choice between fixed and variable inputs, and in terms of our model, a
recurrent cost problem exists when

P~j~^f^~2j~ T ^f^~it~

\(8\) \-\-\-\-\-\-\-\-\-- \> £ \-\-\-\-\-\-\--

W~j~(1+*~Q~*)^j^ t=j+1 P~j~(1+*~Q~*)^t^

Let us now consider the allocation of resources between public and
private production. It is clear that by allowing revenues to vary, so
that the government can compete for scarce resources, the optimal
allocation rule is:

\(9\) PVMPV^g^~t~ = PVMPV^p^~t~; PVMPK^g^~t~=

PVMPK^p^~t~

Where

PVMPV^g^~t~= the present value of the marginal product of an extra unit
of variable input into government production

PVMPK^g^~t~= the present value of the marginal product of an extra unit
of fixed input into government production

PVMPV^p^~t~ , PVMPK^p^~t~ have similar meanings for private production
thus the present value of the marginal product of any input should be
equal in the public and private sectors.

If government cannot claim enough resources so that its marginal
production is as valuable as that of the private sector, then resource
allocation is not optimal. The same result holds if the government
claims too many resources.

From the point of view of project analysis we are interested in both
allocation questions. If either

PVMPV^g^~t~ \> PVMPV^g^~t~

\(10\)

[PVMPV^g^~t~]{.underline} \> [PVMPK^g^~t~]{.underline}

W~t~ P~t~

then the shadow price of government revenue is greater than one.

In calculating rates of return for new projects, the use of such a
shadow price recurrent expenditures and revenues will enable one to make
investment decisions so as to economize on recurrent resources (cf.
Appendix II for a detailed example).

The model is easily expanded to include three inputs: a fixed input, a
locally produced variable input such as labor, and an imported variable
input such as petrol.