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Derivation and Determination of the Long-Run Industry Supply Curve

The long-run supply curve of an industry under perfect competition represents the relationship between the market price and the total quantity supplied by all firms after all possible long-run adjustments have taken place. Specifically, the long-run supply of the industry is derived by multiplying the long-run optimum production of an individual firm by the total number of firms operating within that industry:

$S = Q \times N$

Where:

• $S$ represents the long-run supply of the industry.
• $Q$ represents the optimum output of an individual firm.
• $N$ represents the number of firms operating in the industry.

While the short-run supply curve of an industry invariably slopes upward due to the operation of the law of diminishing returns across fixed plant capacities, the geometric shape and slope of the long-run industry supply curve depend entirely on which law of production applies to the industry as it expands or contracts. The three prevailing production conditions are:

1. If production obeys the law of constant cost, the long-run supply curve of the industry is a horizontal straight line parallel to the horizontal axis ($X$-axis).
2. If production obeys the law of decreasing cost, the long-run supply curve of the industry slopes downward from left to right.
3. If production obeys the law of increasing cost, the long-run supply curve of the industry slopes upward from left to right.

Constant Cost Industry and Supply Curve

A constant cost industry refers to an industry wherein an increase or decrease in aggregate production causes no change in the average cost of production or the equilibrium price; both remain strictly constant over the long run. Consequently, the long-run supply curve ($LS$) of a constant cost industry is a horizontal straight line.

Individual Firm Mechanics

Within a constant cost framework, an individual firm can adjust its production capacity by altering plant sizes without experiencing changes in unit costs:

• Short-run marginal cost and short-run average cost curves for a small-scale plant are designated as $SMC$ and $SAC$.
• Short-run marginal cost and short-run average cost curves for a large-scale plant are designated as $SMC_1$ and $SAC_1$.
• The minimum average cost of production for both plant configurations is identical at price level $OP$.
• If the firm utilizes the small plant, it produces output quantity $OM$ at minimum average cost $OP$.
• If the firm expands to the larger plant, it produces a larger output quantity $ON$ while maintaining the exact same minimum average cost $OP$.
• At price $OP$, the firm satisfies the full equilibrium equality:

$SAC = SMC = SAC_1 = SMC_1 = MR = AR = P$

Long-run marginal cost ($LMC$) and long-run average cost ($LAC$) curves coincide with this level. All firms operating within this industry produce under constant cost conditions.

Industry Equilibrium and Supply Derivation

The industry long-run supply schedule is derived through shifts in market demand:

• The initial market demand curve $DD$ intersects the initial market supply curve $SS$ at equilibrium point $A$.
• At point $A$, the industry produces a total output of $OQ$ at equilibrium price $OP$.
• When market demand increases from $DD$ to $D_1D_1$, a new equilibrium is established at point $B$ where $D_1D_1$ intersects the expanded supply curve $S_1S_1$.
• At point $B$, the industry supplies an expanded quantity $OQ_1$ at the original price $OP$.
• The industry is capable of supplying any required aggregate quantity at price level $OP$.
• Connecting initial equilibrium point $A$ with subsequent equilibrium point $B$ traces the industry's long-run supply curve, designated as line $AB$ (or $LS$), which is a horizontal line parallel to the $X$-axis.

Decreasing Cost Industry and Supply Curve

A decreasing cost industry refers to an industry in which per-unit production costs progressively decrease as the industry expands. When total market output increases in response to an increase in demand, long-run average costs decline continuously.

As the industry expands, individual firms reap external economies of scale. The emergence of these external economies causes both marginal cost and average cost curves of existing and entering firms to shift bodily downward. The full long-run equilibrium condition for the firm requires that production occurs at the minimum point of the long-run average cost curve, satisfying:

$LAC = LMC = AR = MR = SAC = SMC$

Individual Firm Adjustments

• A representative firm is characterized by short-run marginal cost $SMC$, short-run average cost $SAC$, and long-run average cost $LAC$.
• The firm is initially in equilibrium at point $E$, producing output $OM$ at price $OP$.
• As the industry expands, the firm experiences external economies, shifting its cost schedules downward to $SMC_1$, $SAC_1$, and $LAC_1$.
• A new firm equilibrium is established at point $E_1$, expanding individual firm output from $OM$ to $ON$.
• The trajectory connecting equilibrium points $E$ and $E_1$ forms the long-run supply curve of the firm ($EE_1$).

Industry Adjustments and Downward-Sloping Supply Curve

• Initial market demand curve $DD$ intersects industry supply curve $SS$ at point $A$, determining initial price $OP$ and equilibrium output $OQ$, where quantity demanded equals quantity supplied.
• An outward shift in market demand to $D_1D_1$ causes industry supply to expand to $OQ_1$.
• Owing to external economies and falling unit costs across all firms, the equilibrium price falls from $OP$ to $OP_1$.
• The new industry equilibrium settles at point $B$.
• Connecting equilibrium point $A$ to point $B$ yields the long-run industry supply curve $AB$ (or $LS$), which slopes downward from left to right.

Increasing Cost Industry and Supply Curve

An increasing cost industry refers to an industry wherein an expansion in aggregate output leads to an increase in production costs.