Editing Natural monopoly (section)

=== Formal definition ===
[[William Baumol]] (1977)<ref>Baumol, William J., 1977. "On the Proper Cost Tests for Natural Monopoly in a Multiproduct Industry", ''American Economic Review'' 67, 809–22.</ref> provides the current formal definition of a natural monopoly. He defines a natural monopoly as "[a]n industry in which multi-firm production is more costly than production by a monopoly" (p.&nbsp;810). Baumol linked the definition to the mathematical concept of [[subadditivity]]; specifically, subadditivity of the [[cost curve|cost function]]. Baumol also noted that for a firm producing a single product, scale economies were a sufficient condition but not a necessary condition to prove subadditivity, the argument can be illustrated as follows:

Proposition: Strict economies of scale are sufficient but not necessary for ray average cost to be strictly declining.<ref>W. J. Baumol, 1976. "Scale Economies, Average Cost and the Profitability of Marginal-Cost Pricing"</ref>

Proposition: Strictly declining ray average cost implies strict ray subadditivity.

:{| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
!Proof
|-
|Consider n output vectors <math>v_1y,...,v_ny</math>, when ray average costs are strictly declining:

<math>C(\sum v_jy)/\sum v_j < C(\sum v_iy)/\sum v_i</math>

Therefore:

<math>\frac{C(\sum v_jy)}{\sum v_j}<\sum \frac {v_i} {\sum v_j} \frac{C(v_iy)}{v_i}=\frac{\sum C(v_iy)}{\sum v_j}</math>

Which gives:

<math>C(\sum v_jy)<\sum C(v_iy)</math>

Therefore the cost function is strictly subadditivite.
|}

Proposition: Neither ray concavity nor ray average costs that decline everywhere are necessary for strict subadditivity.

:{| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
!Proof
|-
|Let <math>c_1 < c_2</math> in the piecewise-linear cost function:

<math>C(y) = \begin{cases} c_1, & y < y_B \\ c_2, & y > y_B \end{cases}</math>

Let <math>y_r < y_b < y_s</math>. The cost function is not concave, average cost increases after <math>y_b</math> and ray average cost is greater at <math>y_s</math> than <math>y_r</math>. Also:

total cost of any output y by a single firm <math> \le c_2 < 2 c_1 \le</math> total cost of production by more than one firm

Therefore the cost function is strictly subadditivite.
|}

Combining all propositions gives:

Proposition: Global scale economies are sufficient but not necessary for (strict) ray subadditivity, the condition for natural monopoly in the production of a single product or in any bundle of outputs produced in fixed proportions.

==== Multiproduct case====

On the other hand if firms produce many products scale economies are neither sufficient nor necessary for subadditivity:

Proposition: Strict concavity of a cost function is not sufficient to guarantee subadditivity.

:{| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
!Proof
|-
|Let <math>0 <a <1</math> and <math>0 < k <1/2 </math> in the cost function for two outputs:

<math>C=y_1^a+y_1^k y_2^k+y_2^a</math>

C is strictly concave and not subbaditive:
<math>C(1,1)=3>C(1,0)+C(0,1)=2</math>
|}

Therefore:

Proposition: Scale economies are neither necessary nor sufficient for subadditivity.