We investigate the effect of a learning curve on the production of durable goods by examining a durable-goods monopolist in a two-period model. If the monopolist faces a learning curve, the model shows that the equilibrium quantity of the first- (second-) period products will be smaller (larger) than if there were no learning curve. Consequently, in cases where the original production cost is sufficiently large, the presence of a learning curve drives down total profits.

Atsuo Utaka

English

Research Papers in Economics

The Learning Curve and Durable−Goods Production 
Atsuo Utaka
Osaka University
Abstract
We investigate the effect of a learning curve on the production of durable goods by
examining a durable−goods monopolist in a two−period model. If the monopolist faces a
learning curve, the model shows that the equilibrium quantity of the first− (second−) period
products will be smaller (larger) than if there were no learning curve. Consequently, in cases
where the original production cost is sufficiently large, the presence of a learning curve
drives down total profits.
Citation: Utaka, Atsuo, (2001) "The Learning Curve and Durable−Goods Production." Economics Bulletin, Vol. 12, No. 5 pp.
1−8
Submitted: October 27, 2001.  Accepted: November 7, 2001.
URL: http://www.economicsbulletin.com/2001/volume12/EB−01L10004A.pdf1.  Introduction 
 
     The purpose of this paper is to investigate the effect of a learning curve on the 
production of durable goods.   
     There is said to be a learning effect if experienced producers have detailed 
knowledge allowing them to decrease production cost.
1  Economies of scale are 
realized if mass production in a “certain” period reduces that period’s production cost.  
In contrast, when there is a learning effect, it is the “cumulative” quantity of the 
products that reduces the production cost, which is denoted by a learning curve.     
     We analyze the effect of a learning curve on the production of durable goods by 
using a two-period model of durable-goods monopolist.
2  If a learning curve is 
introduced in a two-period model, the quantity of the first-period products affects the 
second-period production cost.  In the case of non-durable goods, the existence of a 
learning curve increases the quantity of products in not only the first period but also the 
second period.  Namely, the producer will intentionally increase the quantity of the 
first-period products so as to reduce the second-period production cost, which in turn 
stimulates production activities in the second period.   
          On the other hand, there is a “commitment problem” in the production of durable 
goods.
3  Therefore, we would expect that the effect of a learning curve on the 
production of durable goods to be different from that of a learning curve on the 
production of non-durable goods.
4  This paper examines this difference by using a 
durable-goods monopolist’s model.   
 
 
2.  The  Model      
 
     Here, we investigate the effect of a learning curve on the durable-goods 
producer’s behavior.   
     Rental demand functions of durable goods at the first and second period are 
                                                  
1 For an analysis of duopoly competition with a learning curve, see Spence (1981), and Fudenberg 
and Tirole (1983). Majd and Pindyck (1989) show the implication of a learning curve under 
uncertainty. See Cabral and Riordan (1994) for the effect of a learning curve on predatory pricing. 
2  See Bulow (1982) concerning the durable-goods monopolist’s model. 
3  On the commitment problem, see Bulow (1982), Waldman (1996), and Utaka (2000). 
4 Olsen (1992) investigates the validity of the Coase conjecture in the production of durable goods 
with a learning curve. In this paper, we examine the effect of the learning curve on the equilibrium 
results.   2
denoted as 
 
ra b q 11 =− ,                                                      ( 1 )  
 
() ra b q q 21 2 =− + ,                                                 ( 2 )         
 
where  qi represents the quantity of products, and  ri  ( i=1 or 2) the rental price of 
period i. 
     We assume that durable goods do not depreciate physically, and the discount 
factor is 1.  The marginal cost of producing durable goods in the first period is c.  
Moreover, we assume that the larger the quantity of first-period products, the smaller 
the second-period production cost, namely, 
 
() ( ) cc q c 2 2 12 0 =<    '.  
 
We specify this function as follows: 
 
() cq ce q
2 11 =− .                                                   ( 3 )             
 
In other words, the second-period production cost is a linear decreasing function of the 
quantity of first-period products. 
     We investigate the subgame-perfect Nash equilibrium.  At first, let us analyze 
the monopolist’s behavior in the second period.    The monopolist’s problem becomes as 
follows: 
 
() {} ()
q
ab qq ce qq
2
2 2 12 1 2 max   =  q π− + − − .                           ( 4 )        
 
From the first order condition, we obtain 
 
()
q
ac q be
b
2
1
2
=
−− −
.                                               ( 5 )      
 
     Therefore, the monopolist’s problem in the first period can be expressed as  3
follows: 
 
     
()
()
() ()
() ()
q
rr q r c q c e q q
ab q
ac q be
b
ab qb
ac q be
b
cq c eq
ac q be
b
1
11 12 22 1 1 2
1 1
1
1
1
1
11
1
22
2
max
.
   =  q q
                =  q
                      q +  
                     
π+ + − − −
−
+
−− − 




 −− ⋅
−− − 





−− − ⋅
−− −
        ( 6 )    
 
From the first order condition of this problem, the equilibrium quantity of the 
first-period products becomes 
 
() ~ q
ab e a c
bb e e
1 22
2
54
=
+−
+−
.                                                ( 7 )  
 
By substituting (7) into (5), we obtain 
 
()
()
~ q
ab bc e a c
bb e e
2 22
35 5 3
25 4
=
−+ −
+−
.                                          ( 8 )  
  
From (7) and (8), we obtain the following theorem. 
 
Theorem 1 
     If the monopolist faces a learning curve, the equilibrium quantity of the first 
(second) period products becomes smaller (larger) than if there were no learning curve.   
 
Proof 
     See  Appendix  1. 
 
     Since  the  existence  of  a  learning  curve reduces the second-period production cost, 
the quantity of the second-period products becomes larger, which inevitably decreases  4
the value of the first-period products.  Therefore, the monopolist determines to lower 
the production level in the first period.   
     Next, by considering (4), (6), (7), (8), and theorem 1, we obtain the following 
theorem. 
 
Theorem 2 
 
     If the monopolist faces a learning curve, the second-period profits become larger 
than if there were no learning curve.    On the other hand, the total profits become larger 
(smaller) in cases where  () ca <>
1
5
. 
. 
Proof 
     See  Appendix  2. 
 
     Especially, it is worth noting that the existence of a learning curve decreases the 
total profits as long as the original production cost is large (namely, in cases where 
a c
5
1
> ).  The intuition behind this result is as follows.  The existence of a learning 
curve decreases not only the second-period production cost but also the quantity of the 
first-period products, which leads to the excessive production in the second period (see 
theorem 1).  On the other hand, the decrease in the quantity and price of the 
first-period products seriously lowers the first-period profits.  Therefore, the decrease 
in the first-period profits exceeds the increase in the second-period profits.     
          On the contrary, in cases where the original production cost is small, the increase 
in the second-period profits outweighs the decrease in the first-period profits.  This 
outcome is resulted mainly from the greatly lower cost in the second-period production.   
 
 
3.  Conclusion 
 
     We have investigated the effect of a learning curve on the production of durable 
goods.  In the case of non-durable goods, it is known that the existence of a learning 
curve increases the quantity of products in not only the first period but also second 
period, which leads to the larger total profits.  On the contrary, our paper has shown 
that the equilibrium quantity of the first-period products becomes smaller.    5
Concequently, especially in cases where the original production cost is large, the 
presence of a learning curve drives down the total profits. 
     The  existing  empirical  studies  find  out  the presence of a learning curve in various 
sorts of industry (for the case of chemical processing industry, see Lieberman (1984)).  
However, our paper has shown that in the production of durable goods, the existence of 
a learning curve can decrease the total profits.    Therefore, it is necessary to investigate 
empirically what effect a learning curve has on the durable-goods producer’s profits.   
          It is noted that our analysis is local one.    In other words, we investigate only the 
case where the learning effect is sufficiently small.  Therefore, there is room for 
investigating the case where the larger learning effect exists.       
 
 
Appendix 1 
 
          Here, we conduct theorem 1.    First, from (7), we obtain 
 
()
() () () ( ) {} []
()
() ( ) {}
dq
de bb e e
a c b be e b e ab e a c
bb e e
a c e abe c a b
~
.
1
22 2
22
22 2
22
1
54
54 42 2
1
54
45 3
=
+−
−+ − − − + −
=
+−
−+ − +      
 
 
Therefore, it is shown that 
 
()
() {}
dq
de b
ca b
e
~
1
0
22
2 1
5
53 0
=
=− + < , 
 
in other words, if the monopolist has a learning curve, the equilibrium quantity of the 
first-period products becomes smaller than if there were no learning curve.         
     Next,  from  (8),  we  obtain  
  6
()
() () () () {} []
()
() {} []
dq
de bb e e
a c b be e b e ab bc e a c
bb e e
ab b c e ab bc e a c
~
.
2
22 2
22
22 2
22
1
25 4
53 54 42 3 5 53
1
25 4
13 5 6 10 5 3
=
+−
⋅− + − −− −+ −
=
+−
++ − +−
               
     
 
 
Therefore, it follows that 
 
() ()
dq
de bb e e
ab b c
e
~
2
0
22 2
22 1
25 4
13 5 0
=
=
+−
+> . 
 
 
Appendix 2 
 
     First, we investigate the effect of a learning curve on the second-period profits.  
From (4), (7), and (8), it is shown that 
 
() () ()
()
()
() ()
()
()
~ ~~ ~ ~ ~ π 2 2 12 1 2
22
22 22 22
35 5 3
25 4
2
54
35 5 3
25 4
2
54
3
=− + − −
=
−+ −
+−
⋅−
+−
+−
−
−+ −
+−
−+
+−
+−



 



 
=
qab qq ce qq
ab bc e a c
bb e e
ab
ab e a c
bb e e
b
ab bc e a c
bb e e
ce
ab e a c
bb e e
b
   
       
   ()
()
()
ab bc e a c
bb e e
bq
−+ −
+−



 



 
=⋅
55 3
25 4
22
2
2
2 . ~ .  
 
 
Therefore, by considering theorem 1, we obtain 
  7
d
de e
~
.
π 2
0
0
=
>  
 
     Next,  let  us  investigate  the  effect of a learning curve on the total profits.     
     From  (6),  we  obtain 
 
()
d
de
d
dq
dq
de
q
b
ace q b q
ππ
=⋅+ − + −
1
11
11 2
2.  
 
Since 
d
dq
~ π
1
0 = , it follows that 
 
()
d
de
q
b
ace q b q
~ ~
~~ π
=− + −
1
11 2
2.  
 
By considering  ~ q
a
b
1
2
5
=   in the case of  e = 0, we obtain 
 
()
d
de
a
b
ac
e
~ π
=
=−
0
2 25
5 . 
 
Therefore,  
 
()
d
de e
~ π
=
><
0
00   , in cases where  () ca <>
1
5
.                 
 
 
References 
 
Bulow J. I. (1982) “Durable-Goods Monopolists” Journal of Political Economy  90, 
314-332. 
Cabral. L. M. B. and M. H. Riordan (1994) “The Learning Curve, Market Dominance 
and Predatory Pricing” Econometrica 62, 1115-1140. 
Fudenberg, D. and J. Tirole. (1983) “Learning-By-Doing and Market Performance” Bell 
Journal of Economics 14, 522-530. 
Lieberman, M. B. (1984) “The Learning Curve and Pricing in the Chemical Processing  8
Industry” RAND Journal of Economics 15, 213-228. 
Majd, S., and R. S. Pindyck (1989) “The learning Curve and Optimal Production under 
Uncertainty” RAND Journal of Economics 20, 331-343. 
Olsen, T. (1992) “Durable Goods Monopoly, Learning by Doing and the Coase 
Conjecture” European Economic Review 36, 157-177. 
Spence, M. A. (1981) “The Learning Curve and Competition” Bell Journal of 
Economics 12, 49-70. 
Utaka, A. (2000) “Planned Obsolescence and Marketing Strategy” Managerial and 
Decision Economics 21, 339-344. 
Waldman M. (1996) “Planned Obsolescence and the R&D Decision” RAND Journal of 
Economics 27, 583-595. 
 

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