"supply curves" wrong ?

[Widdekind]

Q: Do "Supply curves" make sense ?

Simplistically, Total Costs ($/yr) for a production run, may be modeled mathematically, as a function of the production Quantity (#/yr), as the sum, of Fixed Costs (e.g. "rent"), plus Total Unit Costs (e.g. "a minute of labor per shirt"):

TC(Q) = FC + Q x UC​

Then, Average Unit Cost is

<C> = TC / Q = FC/Q + UC ----> UC​

The only requirement, for profitable production, is that unit Prices exceed Average Unit Costs:

P > <C>​

Thus, since Fixed Costs amortize, or "cost out", over all units produced, Average Costs tend to decrease, with increasing produced Quantity, towards the Unit Cost. So, and low Prices, only large Quantities are profitable to produce; whereas at high Prices, small Quantities are also profitable to produce. Seemingly, there is no "Supply Curve"; only a "Supply Region", defined by P > <C>, i.e. "above & right of the Average Cost curve". Conversely, "Demand Curves" seem logical, i.e. consumers will demand high volumes at low prices, et vice versa. If so, then any production volume rate Q (widgets/year), profitable to produce, i.e. P > <C>; and agreeable to consumers, i.e. on the demand curve P(QD); will satisfy both producers & consumers. Producers would want to manage production, to maximize Profits:

Profits = Revenues - Expenses
..........= P x Q - (Q x UC + FC)
..........= Q x (P - UC) - FC
..........= Q x dP - FC​

contours of constant Profit satisfy

Profit = constant
Q x dP = constant​

thus are "hyperbolas" in the "region of Profitable production"; Profits form a mathematical "mountain", or "Profit peak", increasing "up & right". Producers would seek to "scale the Profit peak", as much as they may, according to customer Demand curves. The following figure depicts hypothetical Average-Cost & Demand curves:

• elastic Demand-curve (De)
• inelastic Demand-curve (Di)
• typical Demand-curve (D)

which intersect the "zone of Profitability" at different places of P,Q plane, implying different Profit-maximizing production quantities (+). Naively, the conventional economics concept, of "Supply-curves" rising "up & right", represent "Producers' fantasies", whereby they would wish to "climb the Profit peak", i.e. sell more-and-more, at higher-and-higher prices. But, limited by consumer Demand, Producers can produce any level of production, on the Demand-curve whilst within the "zone of Profitability", and still break-even or better.