Theoretical Economics Letters, 2011, 1, 41-45
doi:10.4236/tel.[phone] Published Online August 2011 (http://www.scirp.org/journal/tel)
Inventories, Interest Rates, and Markups
David Glenn Bivin
Department of Economics, Indiana University Purdue University Indianapolis, Indianapolis, USA
Received June 7, 2011; revised July 13, 2011; accepted July 25, 2011
Abstract
This note explains why inventories might rise with interest rates. Higher real interest rates not only increase
the carrying cost of inventories they also reduce the present value of the markup on delayed sales. When the
markup is large enough, it is profitable to increase stocks in order to avoid sales delays. Another possibility
is that the firm has an incentive to smooth its total stocks so that an increase in the real interest rate causes
finished goods to fall but the reduction is partially offset by an increase in raw materials.
Keywords: Inventories, Interest Rates
1. Introduction
An enduring puzzle in the inventory literature is that in-
ventories do not appear to decline when the real interest
rate rises. Inventories provide a bridge between the time
that stocks are delivered, the time they are processed, and
the time they are sold. During this time span, the pur-
chase and/or processing cost must be financed. When the
real interest rate rises, the firm has an apparent incentive
to speed up the process in order to receive the revenue to
pay off these financing costs more rapidly. The idea is
intuitive. Indeed, Blinder and Maccini [1] capture the
view of the profession when they state that “the idea is so
simple that it is hard to imagine how it can be wrong” (p.
82).1
The empirical evidence provides only sparse support for
the hypothesis.2 Recently, Maccini, Moore, and Schaller
[2] developed an innovative test for a long-run inverse
relationship between the interest rate and finished goods
inventories in the context of the linear quadratic model.
Their estimates suggest a significant inverse relationship
in a number of non-durables finished goods for the time
period examined. Bivin [4] extends the model to account
for durables producers and work-in-process and raw ma-
terials inventories, as well as the more recently available
NAICS data. Among this larger set of cases, the results
are far less robust.
This paper adopts a different strategy from that of the
empirical literature. The goal here is to explain why in-
ventories might actually rise in response to an increase in
the real interest rate. The key ingredient is backorders that
provide the firm with a second opportunity to make a sale
when the first opportunity is lost due to inadequate stocks.
The reward to filling a backorder is smaller than the re-
ward for filling an order when it first arrives due to dis-
counting. While a higher real interest rates raises invent-
tory-carrying cost, it also raises the penalty for failing to
fill a new order.
The model is similar in spirit to that of Kahn [5]. Kahn’s
model features explicit stock outs of finished goods and
random demand. One of the versions of his model allows
backorders and in that model finished goods are inde-
pendent of the real interest rate. It is shown here that add-
ing a carrying cost of finished goods to the model causes
the firm's safety stock to rise as the real interest rate rises.
A model that is similar in spirit is then applied to the case
of a firm that holds only raw materials.
A third model demonstrates why raw materials might
rise in the context of a linear-quadratic model with invent-
tories of both finished goods and raw materials. In this
case, the increase is due to an effort to smooth total stocks.
1In fairness, they also point that interest rates influence inventories throug
h
a number of channels including the discounting of revenue. That is the crux
of the models developed in sections 2 and 3.
2Maccini, Moore, and Schaller [2] provide an excellent review of the
literature and point out that interest in the topic seems to have waned.
Ramey and West [3] survey some of the more prominent work in the field.
Their results suggest that significant inventory responses typically occur in
less than 25% of the models that allow for an interest rate effect (Table 11,
p
. 907).
2. Backorders, Finished Goods and Carrying
Cost
Output available for sale is equal to finished goods on

D. G. BIVIN 42

hand at the beginning of the period plus output produced
during the period. Sales are the minimum of demand and
output available for sale:

1
min ,SQXF

 (1)
where S is sales, Q is demand, X is output, and F is the
end-of-period stock of finished goods. F–1 is finished
good inventory inherited from the prior period. Unfilled
orders are backlogged and added to total demand in the
following period. Total demand is

11
QQ QS



 (2)
where Q and
 are positive constants and  is a stan-
dard normal variate. The demand shock is observed after
the output decision has been made. Backorders are added
to new orders to yield total demand. Finished goods
evolve as
1
F
FXS

 (3)
The cost function is linear. The decision variable is
output and the solution satisfies the Bellman equation:
 
1
X
max F VFEpScXbFVFXS 

  

1


(4)
where p, c, and bF are the sales price, per unit produc-
tion cost, and the per-unit carrying cost for finished
goods, respectively. The parameter
 is the single-period
discount factor defined as
 = 1/(1 + r) where r is the
real interest rate.
Let
 denote the probability that demand is satisfied:


 
111
11
Pr FXQQS
FXQQS
1





 

  









 
11 0EVp cbEV    
(5)
where  (•) is the cumulative standard normal density.
Taking the first derivative of (4) with respect to X yields
XFF
Since output is immediately available for sale, finished
goods on hand and current output are perfect substitutes
and the shadow value of finished goods is simply E(VF)
= c. Substituting this result into (6) and solving for
(6)

yields.





1
11
1
1
F
F
F
r
pb pb
p
r





 

pc pcpc
b
  
(7)
If bF = 0 this expression is identical to that in Kahn [5].
As bF rises,

declines implying a greater willingness
to delay a sale. The derivative of (7) with respect to r is



2
1F
rrp rb




–
F
bpc
  (8)
This expression is clearly positive as long as bF > 0
and it increases as p-c rises. This supports the contention
that the counterintuitive behavior is driven by the mar-
kup. Solving for X + F–1 from (5) yields


1
1X
X
FEQEQ k



  

min ,XQM
 (9)
where kX =

–1(
 ). In the steady-state, X = E(Q) and  kX
is target finished goods (when kX > 0) or backorders
(when kX < 0). Since kX is an increasing function of
 , it
follows that target finished goods are an increasing func-
tion of r. The reason is that the implicit cost of backor-
ders rises when the real interest rate rises because the
markup is discounted more heavily. Thus the firm has an
incentive to raise its safety stock as a means of econo-
mizing on this penalty.
3. Production to Order3
Under production to order, goods are customized in some
sense and production cannot commence until the order
arrives. Thus the current demand shock is known prior to
the production decision. Here, the firm accumulates ho-
mogeneous raw materials in anticipation of orders. It has
an incentive to do so because of a delivery lag that pre-
vents newly ordered materials from arriving early
enough in the period to be processed and sold by the end
of the period. As a result of the delivery lag, the firm
targets it deliveries on fulfilling unmet demand in the
current period along with new orders that are expected to
arrive in the following period. The buffer against demand
shocks is now fulfilled by raw materials rather than fin-
ished goods. When the buffer is exhausted, output is
starved and the firm accumulates backorders until the
following period. Since the firm observes the demand
shock prior to placing its materials orders it knows at that
time the level of backorders it will have on hand at the
end of the period.
Inputs are defined such that one unit of raw material is
required for each unit of output. The firm has an incen-
tive to fill its orders as soon as possible and so output is
defined as4
1
where M–1 is raw materials on hand at the end of the
prior period. Demand in the following period is now de-
fined as
(10)


1
max, 0QQ QM
11

  
 (11)
3The model in this section is similar to that of Kahn [6] and Bivin [7].
Both assume the presence of unfilled orders, raw materials, and
binding stock out constraints.
4It will not be profitable to produce more than total demand when the
cost of finished goods is sufficiently high and/or the final products are
so individualized that the demand for the specific product produced is
unlikely to eventually appear. For a model of the production-to- stock/
p
roduction-to-order decision see Krane [8].

D. G. BIVIN43
Raw materials evolve as
1
M
MDX

 (12)
where D is the delivery of raw materials in the current
period.
Based on results from previous models of this type, we
posit the following solution:
 
*
01 1
max, 0
D
D
M
EQkQQMk



 (13)
where the inventory buffer is now captured by  kD. It
then follows that Pr(M* ≥ Q
1) = Pr(kD ≥
 ). As before,
this probability will be denoted as

. The inventory rule
implies a constant probability of unmet orders. This is
consistent with the linear cost function described below.
From (13) it follows that
 
**
1
1
1
max,0min,
D
D
DMXM
QQM kQMM
QQM k






 
 
1 1

(14)
Note that a one unit increase in M–1 yields a one unit
decrease in D* regardless of whether the extra unit of
materials is used in current production. Intuitively, D* +
M–1 are total materials available for production dur- ing
the current and following periods. These stocks are tar-
geted on current actual demand plus next period’s ex-
pected demand and a safety margin. Demand not met in
the current period is added to next period’s demand so
that total demand over the next two periods is independ-
ent of M–1.
Output in the following period may now be defined as


*
11

min ,
max,0min,
XQM
QQM k
11D



 
1



The significance of (15) is that next period's output
depends upon last period’s inventories.
(15)
Costs are still linear and the objective function is
 
1
max M
DVMp cXvDbMEVMDX


 


(16)
where v is the purchase price for one unit of raw materials
and bM is the per-unit carrying cost of raw materials. The
first-order condition requires

M
V
EvbE
D



 




M
V (17)
The marginal shadow value of raw materials is that it
enables the firm to sell its output sooner. By the envelope
theorem:




1
111
11
11
11
11
M
M
VM XD
EpcEvEbE
MMM
XD
pcE vE
MM
VM M
bE E
MM






 
 
 
  

  
 

 
 

 

  


  





According to the model, the firm fulfills all of its ord-
ers in the period in which they arrive with probability  .
Those orders not filled in the period in which they arrive
are fulfilled in the following period. Thereafter, M–1 is
irrelevant for the firm’s performance. The decision for
D1 is independent of M–1 and so is M1. Therefore, based
on (13) through (15), it follows that
1
–1 1
11
–11 1
1π, 1,
1
–, 0
X XD
EE E
MM M
DM
M
EEE
MMM




 


 
 
 


  

 

 

and it follows that (18) may be rewritten as






11
MM
EVp cbv 

 

(19)
Substituting (19) into (17) and rearranging yields
 
11
M
M
vb pc bv




 

(20)
Solving for  :


1
1
1
M
M
pc vb
pc b




 
 (21)
The expression is a legitimate probability as long as
p–c––1(v+bM)>0. This expression is the per-unit markup
when materials are purchased this period and processed
and sold in the following period. Therefore it is
non-negative. The influence of a change in the real in-
terest rate on  is not apparent from the expression but it
can be shown that ∂

∂r > 0 if

2
2
1
–––( )
M
MM
M
vb
pcvb pcb
b



 (22)
The expression on the right is a “quasi-markup” with
raw materials used up in production valued at their re-
placement cost. It is larger than the markup defined ear-
lier and thus must be positive. As long as this markup is
sufficiently high, kD increases when the real interest rate
rises. For  near one, the expression on the right-hand
side of (22) is likely to be small. It is worth noting that
this result holds even when bM = 0.
Since kD = 
–1(
 ), it follows from (13) that M* is an
increasing function of r when  is an increasing function
of r.
1
M
M




18
4. Multi-Stage Stocks in the
Linear- Quadratic Model5
Finally, it is straightforward to show that steady-state
raw materials may rise in the linear quadratic model in
which increasing costs are assigned to finished goods,
5 The model in this section is based upon Humphreys, et al. [9] and
Bivin [4].

D. G. BIVIN 44

t
raw materials, and total stocks. The last cost, which to
my knowledge has not been introduced before, carries
some intuition. It suggests that costs are lower for a firm
with an excess unit of finished goods and a one-unit
shortage of raw materials than for a firm with an excess
unit of both finished goods and raw materials. This could
be the case if finished goods and raw materials share the
same warehouse space. Also, in a more general model
than that presented here, the price response to inventory
disequilibria may be smaller the closer that total stocks
are to their equilibrium.
Consider a production-to-stock firm in which sales are
equal to news orders. Sales are typically random and
exogenous but here they are treated as a known constant
along with the remaining exogenous variables.6 The ob-
jective function is
2
0
22
11
2
11
{
()()
[()]}
k
ttt
t
FtFM tM
It tI
CostcXwXvD
bFSb MS
bM FS








 


(23)
where c, bF, bM, bI, w > 0 and 
F,

M,

I ≥ 0, and w is
unit labor cost. As before it is assumed that each unit of
out-put requires one unit of raw materials. The firm
minimizes (23) with respect to Xt and Dt subject to the
inventory investment identities:
1ttt
F
FXS

 (24)
1tt tt
M
MDX

 (25)
The Lagrangian form of the cost minimization problem
is




 
2
2
1
0
2
1111
,t 1
,1
k
tttFtF
t
tt
Ftt t
Mt tttt
CostcXwXvDb FS
bM FS
FFX S
MM DX


















(26)
where

F,t and

M,t are Lagrangian multipliers. In addition
to the inventory investment identities, the first-order
condition requires
,,
0.5 0
tFtMt
t
Cost cX w
X



(27a)
,0 Mt
t
Cost v
D

 
 (27b)


,,
0.5
0.5
0
FtF
t
1
I
ttIFt Ft
Cost bF S
F
bFM S







(27c)


,,1
0.5
0.50
MtM
t
Itt IMtMt
Cost bM S
M
bFMS






 
(27d)
Our interest is in the steady-state in which D* =X* = S,

F,t =

F, and

M,t =

M. From (27b) it is apparent that

M
= v and it follows from (27a) that
0.5
FcSv w


 (28)
This is the marginal replacement cost of finished
goods in the steady state.
Note that (1–  ) 
F/
 = r 
F and (1–
 ) 
M/
 = r 
M.
Equations (27c) and (27d) may be written as




**
0.50.5
0.5 0.5
FI I
FF II
bbF bM
bb crSrvw



(29a)



**
0.50.5
0.5
MI I
MM II
bbMbF
bbSrv 


(29b)
Solving:

 
*0.5 0.5
0.5
MI
F
bcSwvbcSw
FSr T 
 
 (30)


*
0.5
FI
M
MS
rT 
 0.5bvbcSw
(31)
where T = (bM + bI)(bF + bI) – bI
2 = bMbF + bI(bM + bF) >
0. Neither 
F nor

M depend upon the real interest rate.
When bI = 0 both stocks fall when the real interest rate
arise and, from (30), finished goods fall regardless of the
value of bI. Moreover, adding (30) and (31) together:



**
0.5
0.5
FM
MF
FM S
bcSwvb
rT
 

v
(32)
and it is clear that total stocks decline when the real in-
terest rate rises.
However, according to (31), M* rises if bI is suffi-
ciently large relatively to bF. Specifically, an increase in
r causes M* to rise if
0.5
I
F
bv
bcS
w
(33)
The numerator of the expression on the right is the
marginal cost of a delivery in the steady state and the
denominator is the marginal cost of a unit of output in
the steady state. If the latter is sufficiently large relative
to the former, the firm will respond to an increase in the
6This is merely for convenience. Even if sales are random, certainty
equivalence applies and the optimal solution can be found by replacing
the random variables with their expected values. The influence of un-
certainty on inventories is built into the model through the specification
of the inventory target functions. The conclusions do not depend upon
this specification of the targets.
real interest rate by reducing its finished goods and raising
its raw materials to maintain overall stock equilibrium.

D. G. BIVIN 45
Given the choice, the firm disinvests in finished goods
rather than raw materials because finished goods incor-
porate processing costs (which must be financed) while
raw materials do not. Thus, an increase in the real inter-
est rate effectively delays production leading the firm to
accumulate pre-production stocks at the expense of
post-production stocks.
5. Conclusions
This paper presents three scenarios in which finish goods
or raw materials may rises in response to an increase in
the real interest rate. In the first two, production lags,
backorders, and stockout constraints are responsible for
the counter-intuitive result: the firm accumulates addi-
tional stocks despite the increased carrying cost in order
to avoid delaying its revenue which, of course, is also
discounted. In the third scenario, the firm is subject to a
cost assigned to its total stocks. When the real interest
rate rise, the firm reduces finished goods but, under certain
combinations of costs, raises raw materials as a means of
smoothing total stocks. The interesting feature of these
conclusions is that they are generated within the context
of standard models and require only straight forward as-
sumptions.
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