3 Technical efficiency and finance constraints
3.2 Technical efficiency and finance constraints: a partial equilibrium approach
Consider an industry with I = 1, . . ., N firms. I assume that all firms have limited access to the credit market (which means that they do not have suffi- cient financial capital to rent the desired amount of physical capital) and are internally inefficient. The firms in this industry are exposed to different credit constraints; also I assume that some firms are more efficiently organized than others. In the Introduction to the chapter I have outlined the many reasons why a firm is inefficient; however, here I assume that the inefficiency derives from the hold-up problem due to the fact that the manager of the firm has to make costly investments in the organization of the firm for which he can only hope to recoup a fraction of the return through its salary. However, some firms will be more efficient than others as they may be more successful at inducing the manager to align his interests to those of the firm. The model shows that a
tightening of the credit constraint in a particular firm, which is not currently on the frontier, can help to catch up with firms on the frontier because the manager is induced to invest more effort in the firm; in other words, the tight- ening of credit conditions for a firm can be followed by an increase of tech- nical efficiency for this same firm at a given point in time. Each period the firm produces with the following (convex) production technology:
, or
a > 0 and >< 0.
Output is being produced by two inputs (managerial effort and capital) and is being sold at an exogenously fixed price, normalized to 1. γ(which can be either positive or negative) indicates whether marginal product of effort vested in organization is more or less productive in a firm with access to more capital or not. First, ki,t is the amount of physical capital equipment rented from a competitive market at the price of 1. The expenditure has to be paid up front and thus has to be financed through credit. As mentioned above, I assume that the firm is credit rationed and that the maximum amount of financial capital avail- able for firm i in period t is k−
i,tand that this constraint is always binding. The debt is paid back in full at the end of each period and, without loss of generality, I assume that the interest rate is nil. Second, the manager provides effort: he has to decide on his effort one period before production takes place. Once the decision has been made, it cannot be undone immediately. The manager makes the decision in anticipation of how much capital can be rented in the future which in turn is determined by the credit constraint. To capture differences in the internal organization of different firms in a simple way, I assume that the manager in firm i is rewarded for organizing the production process by a share si of the net profit yi,t−ki,t. This can be thought as representing an incomplete con- tracting situation where, after production has taken place, the manager and the owners of the firm negotiate about the share of the surplus that the manager should retain. In this situation a standard hold-up problem arises and so the para- meter s can then be interpreted as a measure of the inefficiency in the internal organization of the firm.
The per period utility function of the manager is:
and his budget constraint is Life-time utility is then:
,
)
( , ,
,
,t it i it it
i w s y k
c = = −
) , ( , 1 ,
,t it it
i F e k
y = −
1 , , , 2
1 , 1 ,
,t = it− −0.5 it− + it + it it−
i ae e bk k e
y γ
∑
= − −= T
t
t t i t i i t
i s y k e
U
0
2 ,
, ) 0.5 ]
( δ [
2 ,
,t it 0.5 t
i c e
u = −
where δis the discount rate ande−1= 0. The b in the production function must be sufficiently large to insure that the firm will actually borrow money up to the limit 1r.
The model can be solved by backwards induction. In period 2, production takes place:
and the manager’s income is:
.
In period 1, the manager faces the problem:
. The first-order condition is:
and his optimal effort choice is:
.
The manager’s payoff in period 2 is denoted by V2*. In period 0, the manager faces a similar problem:
and he chooses similarly. In both cases, the optimal effort is:1
It can be easily checked that optimal effort increases if the finance constraints gets more binding as long as γ < 0. Indeed, the manager makes his effort decision based on his expectations about the availability of credit in the future. If he anticipates that more credit is going to be available in the future and thus more rented capital equipment is going to be available he may decide to spend more effort in the firm. In particular, he spends less effort if the marginal pro- ductivity of his effort is lower in an environment with more capital equipment available (γ< 0).
Consider now technical efficiency. yˆtis the output produced in period t by the firm with the best practice technique supported by a combination of efficient internal organization (high siand lax external credit constraints (high k−). I can
i t i i i
t s
k a e s
δ γ δ
+
= + +
1
)
( , 1
*
, for t = 0,1.
* 2 2 2
0 , ,
1 , 1
* ,
0 argmax s(y k ) 0.5e V
e i = δ i i − i − i +δ s
k a e i si i
δ γ δ
+
= + 1
)
( ,2
* , 1
0 )
( − 1,i+ 1,i − 1,i =
i a e k e
s γ
δ
2 1 , ,
2 , 2
* ,
1i argmax si(y i k i) 0.5ei
e = δ − −
) ( 2,i 2,i
i y k
s −
e b+γ i,t
i i i t i i
i ae e bk k e
y2, = 1, +0.5 2, + 2, +γ 2, 1,
measure technical efficiency in firm i in period t by comparing it to this benchmark:
, where
My main interest is to find out how technical efficiency periods 1 and 2 in. firm i thus defined is affected by a lasting, but unexpected change in the credit constraint in period 1. The fact that it is unexpected implies that it could not be taken into account when effort was decided in period 0. The fact that it is lasting implies that the manager, once he has observed the change in period 1, would wish to adjust his effort choice made in that period recognizing that credit con- ditions in period 2 could also have changed. To simplify the analysis, I assume that initially. It is also important to note that the change in credit conditions is local to firm i, that is, it does not apply to all the other firms in the industry. Therefore, I can take the benchmark as given.
A change in the credit constraint can affect the firm’s output through three channels. First, the direct effect is obvious: if more credit becomes available in a given period, more capital can be rented and more can be produced. The second effect is an interaction effect and its direction depends on the sign of γ. If γ< 0, more credit would reduce output through this channel because it crowds out the productive value of effort; if γ > 0, more credit would enhance the productive value of the effort. The third effect works through a change in effort in anticipa- tion of changing credit conditions in the future. Thus, this effect cannot affect technical efficiency in period 1, but will have an impact on efficiency in period 2.
Again the effect depends critically on the sign of γ. For γ< 0, the manager would want to spend less effort on organizing production because the extra rented capital makes such effort less productive at the margin. As a consequence, output would fall in period 2. The opposite happens, of course, if γ > 0. The overall impact on output is then determined by the interaction between these effects.
With this in mind, consider first what happens to technical efficiency in period 1. Since the decision on effort has already been made in period 0 based on the expected credit conditions, I get:
.
Next, consider period 2. After the change has been observed in period 1, the manager can adjust his effort choice to accommodate the new environ- ment in period 2. The change in technical efficiency in period 2 is therefore given by:
. 1 )
) ( (
ˆ 1 ˆ
1
1 1
* 0 , ,
i i i i
i t i
s k a b s
y y
e k
TE
δγ γδ
γ
+ + + + =
∂ =
∂
i i
i k k
k1, = 2, =
* 1 , , , 2
* 1 ,
* 1 ,
*
,t = it− −0.5( it−) + it + it it−
i ae e bk k e
y γ for t = 1,2.
t t i t
i y
TE y ˆ
* ,
, =
] )
ˆ [(
1 *
1 ,
* 1
* , 1 , 1 2 ,
i i
i i i i
i b e
k k e e y a k
TE γ + +γ
∂ + ∂
−
∂ =
∂
Using the FOC from the manager’s effort decision problem I get:
by substitution of the optimal effort decision at time 1. It is clear that technical efficiency increases in both periods if γ >−0. However, this is not the case when γ <0. Indeed in this case it is possible to identify four values of γsuch that:
1 Technical efficiency in period 1 is decreasing in the credit constraint for and non-decreasing otherwise.
2 Technical efficiency in period 2 is decreasing in the credit constraint for and non-decreasing otherwise.
For small values of , technical efficiency increases in both periods because the interaction effect and the effort effect are small. For values of γ smaller than γ1, the effort effect is sufficiently large to ensure that technical effi- ciency in period 2 falls, while technical efficiency in period 1 increases unless γ< γ−1because the interaction effect is not strong enough to dominate the direct effect. For γ yet lower, technical efficiency in both periods falls as the inter- action effect is now sufficient to overcome the direct effect. Interesting, for very low values of γ, technical efficiency increases in both periods. The reason is that effort is very low and the interaction effect is dampened for that reason. Thus, there is a non-monotonic relationship between γand the impact of credit con- straints on technical efficiency. The most realistic case is the one with γ >γ−2. This is the case where the interaction effect in itself is not sufficient to overcome the direct effect (and thus technical efficiency in period 1 is enhanced by the availability of more credit) while the combination of the effort effect and the interaction effect is sufficient to reduce technical efficiency in period 2. So in this case a restriction in the available credit reduces output immediately, but the manager in anticipation of the lack of machinery in the next period puts more effort into organizing production. This compensates for the lack of capital and may in some cases more than compensate. If so, technical efficiency improves in the next period.