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PROBABILIDAD DE FRACTURA DE CADERA AISLADA A 10 AÑOS
In this section, the setting satisfies all criteria of the neoclassical framework. The firm seeks to maximize its (expected) profits and has full knowledge of the
5Note that this allocation, namely the proportionate distribution of fixed expenses to individ-
ual units, is apparent for the single-good case, but much less so when the firm produces multiple products; here, an allocation base such as labour hours is necessary to achieve a sensible fixed cost allocation. We discuss the influence of internal costing systems necessary for a correct allocation of overhead costs in Chapter 2.
demand and cost functions, as well as on the distribution of the error term that will be introduced later.
3.2.1.1 Certainty
As a benchmark case, we will analyse profits of the monopolist under certainty, where the firm has to decide on which costing system to implement. Profits under certainty will be denoted asπcert andωcert.
Variable Costing
Under a variable costing regime, profits under certainty are6 πcert = pvcq(pvc)−cq(pvc)−F
= −c(m−ac(1+α)) +c(1+α)(m−ac(1+α))−F
= cα(m−ac(1+α))−F (3.1)
Differentiating with respect toαand setting equal to zero yields αcert∗ = m−ac
2ac (3.2)
The second order condition is also satisfied. Maximum profits are
πcert∗ = a
2c2−4aF−2acm+m2
4a (3.3)
Full Costing
In the case of full costing, profits are ωcert = pf cq pf c −cq pf c −F = −c(m−a(c+ f)(1+λ)) + (c+ f)(1+λ) ·(m−a(c+ f)(1+λ))−F = −(f + (c+ f)λ)(−m+a(c+ f)(1+λ))−F (3.4)
6The model was analysed with Mathematica 7.0.0. The source code is available upon request
from the author. Most numbered expressions are labeled accordingly in the Mathematica source code for the readers convenience.
Differentiation with respect toλ, setting equal to zero and plugging in f = qF
e gives the optimal mark-up
λcert∗ = mqe−2aF−acqe
2aF+2acqe (3.5)
which also fulfills the second order condition. Inserting intoωcertgives max- imum profits under full costing of:
ωcert∗ = a
2c2−4aF−2acm+m2
4a (3.6)
Comparison
Comparing results under the optimal mark-ups in both cases, we see that
pcertvc ∗ =pcertf c ∗ = ac+m
2a (3.7)
and accordingly
πcert∗ =ωcert∗
This shows that the firm is indifferent between variable and full costing and can achieve maximum profits under both costing systems. Note that λcert∗ becomes negative if c+ f > p∗ ≥ c, while α never turns negative since this would result in p<c. Comparison of the mark-up gives the intuitive result
α∗ >λ∗ (3.8)
since under variable costing, the mark-upαcert∗is applied to a comparatively small cost basec, while under full costing the smaller mark-upλ∗is multiplied with the larger cost basec+ f.
The optimal profit mark-up under full costingλcert∗increases with expected quantityqe: ∂λcert∗ ∂qe = F(ac+m) 2a(F+cqe)2 >0 (3.9)
As the full costing firm expects to sell more units, the allocated fixed cost per unit f decreases and thus lowers the cost base, which in turn makes a higher
mark-up necessary to set the optimal price. Note that an error-ridden calcu- lation of the expected sold quantity might prevent the firm from deriving the optimal mark-up.
3.2.1.2 Additive Demand Uncertainty
We now introduce a demand shock that the firm cannot observe before it fixes the selling price. We assume that the firm knows the distribution of the er- ror parameter. Concerning the dichotomization by Knight (1933), who defines “uncertainty” as a situation where agents have no knowledge about the proba- bility function over the random variable, and “risk” as the situation where they do, we will follow Aiginger (1987) by referring to the latter.
We defineuas a shock which is characterized by a symmetric Beta distribu- tion within the domain delimited by the minimumvand the maximumw. For reasons of simplicity, we assume the distribution ofuto be symmetrical around its mean. The according probability density function ofuis then given as7
f(u;β) = (u−v)
β−1
(w−u)β−1(w−v)1−2β
B(β,β) (3.11)
where β is a shape parameter and B(β,β) is the Euler Beta function. We delimit the probability density function by v = −1 and w = 1. The expected value and the variance ofuare then given as:
E(u) = 0 (3.12)
Var(u) = 1
1+2β (3.13)
Figure 3.1 depicts the probability density function of u for different values of the shape parameter β. It can be seen that this distribution incorporates a variety of possible characteristics regarding the dispersion ofu.
7The general form of the Beta probability density function is given as
f(z;γ,β,v,w) = 1
B(γ,β)
(z−v)γ−1(w−z)β−1
(w−v)γ+β−1 (3.10)
where γand βare the shape parameters of the distribution. The symmetric version is
0 0 Β =1 Β =2 Β =5 Β =10
Figure 3.1: The Probability Density Function ofufor Different Values of β We will, again following Aiginger (1987), distinguish between additive and multiplicative demand shocks. To keep the main mechanism of the model in focus, we will assume that profits are evaluated through a linear utility func- tionU(x) = x. Firms are hence risk neutral and, in the neoclassical approach, maximize expected utility. For the sake of clarity, we will omit the notation of the utility function.8 In the following, variables for additive uncertainty in the neoclassical model will be denoted with the superscript auc, while in the case of multiplicative uncertainty (introduced below), the superscript muc will be used.
Under additive demand uncertainty, the demand function is given as
qauc(p) = m−ap+u
Variable costing
Knowing the distribution of the error, the firm faces the following expected utility of profits in the case of variable costing:
E(πauc) = w ˆ v (c(1+α)(m+u−ac(1+α)) −c(m+u−ac(1+α))−F) f(u)du
= w ˆ v (cα(m+u−ac(1+α))−F) f (u)du = w ˆ v u f(u)du+ (cα(m−ac(1+α))−F) = cα(m−ac(1+α))−F (3.14) Maximizing and solving forα yields again
αauc∗ = m−ac
2ac (3.15)
which is equal to the optimal mark-up under certainty,αcert∗. Facing additive demand uncertainty, the variable costing firm does not alter its strategy com- pared to the certainty case. Hence, expected utility of profits under additive uncertainty are the same as profits under certainty:
E(πauc∗) = a 2c2−4aF−2acm+m2 4a =π cert∗ (3.16) Full costing
Expected profits under full costing are
E(ωauc) = w ˆ v ((c+ f)(1+λ)(m+u−a(c+ f)(1+λ))−c(m+u −a(c+ f)(1+λ))−F) f(u)du = w ˆ v (−(f + (c+f)λ)(a(c+ f)(1+λ)−m−u)−F) f(u)du = w ˆ v u f (u)du+ (−(f + (c+ f)λ)(a(c+ f)(1+λ)−m)−F) = −(f + (c+ f)λ)(a(c+ f)(1+λ)−m)−F (3.17)
Differentiating with respect to λ, setting equal to zero, using f = qF
e and solving forλgives
λauc∗ = mqe−2aF−acqe 2aF+2acqe
=λcert∗ (3.18)
Of course, since mark-ups and price are equal to the values in the case of certainty, expected profits under uncertainty are identical:
E(ωauc∗) = a
2c2−4aF−2acm+m2
4a =ω
cert∗
(3.19) As we can see, the presence of an additive demand shock with a zero mean has no effect on the firm’s behaviour under either costing system.
3.2.1.3 Multiplicative Demand Uncertainty
Under multiplicative demand uncertainty, the demand function is given as
qmuc(p) = u(m−ap)
where we assume E(u) = 1. This is obtained by changing the limits of the distribution given in (3.11) tov=0 andw =2.
Again, we will consider the effects of this form of demand shock under both costing regimes.
Variable costing
Expected profits under variable costing and multiplicative uncertainty are
E(πmuc) = w ˆ v (cu(1+α)(m−ac(1+α)) −cu(m−ac(1+α))−F)f (u)du = w ˆ v (cuα(m−ac(1+α))−F) f (u)du = cα(m−ac(1+α))−F (3.20)
which is again identical to the profit function under certainty,πcert. Thus, the optimal mark-up is again given as
αmuc∗ = m−ac 2ac =α
cert∗
(3.21) and expected profits are equal to the profits under certainty, E(πmuc∗) = πcert∗.
Full costing
If the firm employs a full costing method and demand is subject to a multi- plicative shock, expected profits are
E(ωmuc) = w ˆ v ((c+ f)u(1+λ)(m−a(c+ f)(1+λ))−cu(m −a(c+f)(1+λ))−F) f (u)du = w ˆ v (−u(f + (c+f)λ)(−m+a(c+f)(1+λ))−F) f (u)du = −(f + (c+ f)λ)(−m+a(c+ f)(1+λ))−F (3.22) Again, maximization gives
λmuc∗ = mqe−2aF−acqe 2a(F+cqe)
=λcert∗ (3.23)
The full costing firm thus also faces the same expected profits as under cer- tainty,E(ωmuc∗) = ωcert∗.
3.2.1.4 Summary of the Results of the Neoclassical Approach in Monopoly
As we can see, the optimal mark-ups do not change under additive or multi- plicative demand uncertainty as long as the firm is risk neutral.9 As a conse- quence, the firm suffers no disadvantages by using the - from a neoclassical
9Analysis of the case of a risk seeking/averse firm is omitted here for the sake of brevity, but
there is no reason to suspect that the optimal solution could not be attainable for full-cost pricing under these circumstances.
standpoint - economically counter-intuitive full-cost approach, which includes sunk costs in the pricing decision. It anticipates the higher cost base in the mark-up calculation when maximizing the expected profit function and arrives at the optimal price just like under variable costing.
If profit mark-ups are assumed to be non-negative, the optimal price will not be attainable under full-cost pricing if it lies between variable and aver- age unit costs. However, there are several empirical studies that suggest that prices exceed marginal cost and - under common assumptions regarding the cost structure - also average costs in almost all sectors (Baba (1997), Hall (1988), Martins et al. (1996)). Therefore, and because of the obvious possibility of neg- ative profit mark-ups, we have reason to suspect that this limitation of the full- costing approach does not play a large role for the pricing behaviour of firms. It can thus be seen that the neoclassical model fails to explain the wide use of full-cost pricing. In the same vein, this example shows that full-cost pricing does not stand in contrast to neoclassical ideas, but can be easily incorporated into its general framework without leading to different implications or making marginalist methods in theoretical research on pricing obsolete. It is thus not very surprising that the recognition of the widespread use of full-cost pricing failed to induce a persistent dent in the popularity of neoclassical economics.