Abstract
The situation we have looked at so far, where only one good is produced, plays a role in reality which is not to be underestimated because, like every economic theory, it too only really applies where the conditions are broadly met. For example, if a by-product is produced in addition to the main product and this provides only a small fraction of the revenue, we can then safely apply the theory of “single” supply. We therefore deduct the revenue of the by-product from total costs to perhaps achieve greater accuracy and regard the difference to be the principal product’s total costs.
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Notes
- 1.
See Fanno, M. (1914). It would be too complicated to compare and contrast this paper because our explanations have completely different basic ideas and structures.
- 2.
This vectorial relationship is clearly visible. From Fig. 3.2 we can see that these equations apply:
$${x_1}=r.\cos \ \varphi \ {x_2}=r. \ \sin \ \varphi.$$It is hence
$${{\mathfrak{r}}_1}=({x_1},{x_2})=(r.\cos \varphi,r.\sin\varphi )=r.(\cos\varphi,\sin\varphi ).$$On the one hand, since the equation e 1 2 + e 2 2 = 1 applies to vector \(\mathfrak{e}\), whose components we describe with e 1, e 2; on the other hand, if we describe the length of \(\mathfrak{e}\) using \(|\mathfrak{e}|\), we obtain:
$${e_1}=|\mathfrak{e}|.\cos \ \varphi \ {e_2}=|\mathfrak{e}|. \ \sin \ \varphi$$and we hence derive:
$$|\mathfrak{e}{|^2}=|\mathfrak{e}{|^2}.{\cos^2}\varphi +|\mathfrak{e}{|^2}.{\sin^2}\varphi =1.$$Since we only know positive lengths, \(|\mathfrak{e}|=1\) and as a result \(\mathfrak{e}\) = (cos ϕ, sin ϕ), then ultimately
$${\mathfrak{r}}=r. \ \mathfrak{e} \ [\varphi ].$$ - 3.
Hence ϕ = constant.
- 4.
For an arbitrary function Φ (\(\mathfrak{x}\)) of product vector \(\mathfrak{x}\), we hence derive this relationship:
$$\varPhi (\mathfrak{x})=\varPhi (\mathfrak{e}.r)=\varPhi \left[ r \right].$$ - 5.
We therefore derive:
$$G\left[ r \right]=E\left[ r \right]-K\left[ r \right].$$ - 6.
We derive:
$$b\left[ \varphi \right];\ q\left[ \varphi \right];\ p\left[ \varphi \right];\ s\left[ \varphi \right].$$ - 7.
Then the following applies:
$$P\left[ \varphi \right]={P_1}.\cos \varphi +{P_2}.\sin \varphi .$$ - 8.
It is indeed very clear here that an allocation of total costs to individual products, or even only variable costs, is completely pointless. The various product combinations appear to be various quantities of one and the same product. Each individual velocity of production costs just as much as the entire product vector if we ask the question, “What must be sacrificed to obtain the relevant velocity of production?” And conversely, each velocity of production costs nothing if we ask the question, “What can we save if we give up producing a good in order to produce other goods in unchanged quantities?”
- 9.
Translator's note: ‘Betriebsoptimumkurve’ and ‘Betriebsminimumkurve’ respectively in German, literally ‘firm's optimum/minimum curve’, but von Stackelberg seems to use this to apply to a firm as well as an enterprise. For clarity, we have opted to use ‘optimum/minimum position curve’.
- 10.
s can only follow a path inside q [ϕ] in the appropriate case of the monopoly.
- 11.
Proposition (XXVI).
- 12.
In terms of the principle of the satisfaction of needs and wants, we can now say that an additional subsidiary principle also becomes necessary in the monopoly economy and that of all the highest possible velocities of production (each direction has such a velocity of production), the velocity which should be obtained is the one which has the lowest average costs. Apart from the length, the direction must now also be determined somehow.
- 13.
Each of these families of curves must fulfil the requirement that its curves do not intersect each other. Otherwise the requirement for the invertible definite covariance matrix of points and their coordinates would not be fulfilled.
- 14.
Each of these families of curves must fulfil the requirement that its curves do not intersect each other. Otherwise the requirement for an invertible definite covariance matrix of points and their coordinates would not be fulfilled.
- 15.
See Chapter 1, footnote 22.
- 16.
x 1 is defined by the curve to be a function of x 2 and vice versa, and in point of fact, the following always applies:
$$\frac { {d{x_1}}} {{d{x_2}}}< 0 ; \ \frac{ {d{x_2}}} {{d{x_1}}} < 0.$$For example, where the indifference cost curve has the point k 1 on axis 1 and point k 2 on axis 2, it transforms entirely into a rectangle which has these corner points:
$$(0,0),({k_1},0),({k_1},{k_2}),(0,{k_2}).$$Within this rectangle they must monotonically decrease, but they can follow an arbitrary path.
- 17.
For example, they can be closed curves which are located around a point, the maximum point (Translator’s note: see Cassel, G. (1923): Book 2, Chapter VII, § 29: 267) of the revenue function.
- 18.
Here the tangential indifference revenue line would lie between the indifference cost curve and the origin, that is, in exact opposition to the claim put forward.
- 19.
The method for indifference curves was first used in theoretical economics by Francis Ysidro Edgeworth and, with groundbreaking success, by Vilfredo Pareto (1927): 540, footnote 1).
- 20.
See German ed. pp. 63.
- 21.
Variable costs for x 1 and x 2 are: K II (x 1, x 2).
Where x 1 is not obtained, the following variable costs occur: K II (0, x 2). Hence the following saving is made:
$${K_{\rm II }}({x_1},{x_2})-{K_{\rm II }}(0,{x_2}).$$Where x 2 is not obtained, a saving of
$${K_{\rm II }}({x_1},{x_2})-{K_{\rm II }}({x_1},0)$$is hence made. If neither x 1 nor x 2 is obtained, then K II (x 1, x 2) is saved. However, it is now usually
$${K_{\rm II }}({x_1},{x_2})\ne {K_{\rm II }}({x_1},{x_2})-{K_{\rm II }}(0,{x_2})+{K_{\rm II }}({x_1},{x_2})-{K_{\rm II }}({x_1},0),$$that is, it is usually:
$${K_{\rm II }}({x_1},{x_2})\ne {K_{\rm II }}({x_1},0)+{K_{\rm II }}(0,{x_2}).$$The equation exists only if K II (x 1, x 2) can be shown to be the sum of the functions of only one of the variables, when we can then write:
$${K_{\rm II }}({x_1},{x_2})={K_{\rm II},_{1}} ({x_1})+ {K_{\rm II},_{2}} ({x_2}).$$Therefore the allocation of variable costs to individual goods is only possible in this case.
- 22.
We differentiate:
$$G ({x_1}, {x_2}) = {x_1} . {P_1} + {x_2} . {P_2}-K ({x_1}, {x_2})$$using x 1 and x 2 and obtain
$$\frac {{\partial G}} {{\partial {x_1}}} = {P_1}-{{K^{\prime}}_1} ; \ \frac {{\partial G}} {{\partial {x_2}}} = {P_2}-{{K^{\prime}}_2}.$$Both derivatives of G must vanish for x 1 = s 1 and x 2 = s 2.
- 23.
It must follow that:
$$\frac{{{\partial^2}G}}{{\partial {x^2}_1}}=-{{K^{\prime\prime}}_{11 }}<0$$and
$$\frac{{{\partial^2}G}}{{\partial {x^2}_2}}=-{{K^{\prime\prime}}_{22 }}<0$$\({{K^{\prime\prime}}_{11 }}>0\mathrm{ and}{{K^{\prime\prime}}_{22 }}<0.\) hence:
- 24.
See Chapter 3, footnote 12.
- 25.
Hence:
$$\frac{{K({x_1},{x_2})}}{{\sqrt{{{x^2}_1+{x^2}_2}}}}$$ - 26.
The rather complex issues in this section are necessary to provide an accurate description of these mathematical notions and their origins.
References
Fanno, M. (1914). “Contributo alla teoria dell’offerta a costi congiunti” (Published in English as: A contribution to the theory of supply at joint cost, Forewords by Punzo, L. and Morishima, M., London, Macmillan, 1999) in Supplemento al Giornale degli Economisti e Rivista di Statistica, October, Athenaeum, Rome, via Calamatta, 16, p. 142.
Pareto, V. (1927). Manual of political economy, Paris (2nd ed.). (p. 540).
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von Stackelberg, H. (2014). Costs in Joint Production. In: Foundations of a Pure Cost Theory. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34537-1_3
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DOI: https://doi.org/10.1007/978-3-642-34537-1_3
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