Construct a continuous entry equilibrium

Construct a continuous entry equilibrium

A differentiated seller reports zero economic profit. Does that force price to equal marginal cost? Build a graph that can answer the question rather than importing a rule from perfect competition.

Constructed functions, with units

This is an original fictional market. Q measures continuous output units, with 0 < Q ≤ 5. P is credits per unit. Total economic cost includes opportunity costs:

TC = 8 + 2Q + Q².

The 8-credit fixed cost remains when output changes in this comparison. Average total cost divides cost by output: ATC = 8/Q + 2 + Q. It is undefined at Q=0. Marginal cost, the instantaneous extra cost per unit, is MC = 2 + 2Q.

After an illustrative entry adjustment, the firm’s demand is P = 10 − Q. Revenue is TR = 10Q − Q², giving marginal revenue MR = 10 − 2Q. A single price applies to all output, so selling more lowers the price on earlier units too. That is why MR lies below demand. This continuous MR is a slope, not the finite revenue change from adding a whole unit.

Solve quantity, price and profit in that order

Set MR=MC: 10−2Q=2+2Q, so Q=2. Read demand at that quantity: P=8. Total cost is 8+4+4=16; revenue is 2×8=16. Economic profit is zero, ATC=8, and MC=6. Thus P=ATC while P>MC.

Check that the crossing is a maximum rather than merely assuming it. Profit at any positive allowed Q is:

TR−TC = 8Q−2Q²−8 = −2(Q−2)².

A square cannot be negative, so this expression never exceeds zero and equals zero only at Q=2. This algebra proves the optimum across the specified continuous domain. It is stronger evidence than testing only integer outputs.

Plot before reading the interpretation

Q	Demand P	MR	MC	ATC
1	9	8	4	11
1.5	8.5	7	5	8.8333
2	8	6	6	8
2.5	7.5	5	7	7.7
3	7	4	8	7.6667
4	6	2	10	8
5	5	0	12	8.6

Draw Q horizontally and credits per unit vertically. The three linear curves can be connected exactly. ATC is curved: do not join a few sampled points with straight lines and claim that proves tangency. Use the formula for intermediate points, leave ATC undefined at zero, and distinguish the curves with labels as well as colors.

At Q=1.9, ATC≈8.1105 and demand=8.1. At Q=2.1, ATC≈7.9095 and demand=7.9. Demand lies below ATC on either side and touches it at Q=2. Indeed, ATC−P = 2(Q−2)²/Q, which is positive except at Q=2 for Q>0. This is an algebraic check of the touch, not a claim inferred from pixel placement.

Optional calculus gives the same result: the ATC slope is 1−8/Q², which is −1 at Q=2, equal to the demand slope. These slope formulas are extensions; the square identities establish the essential result without differentiation.

Minimum cost per unit is a different output

ATC is smallest at Q=√8≈2.828, with minimum ATC≈7.657. One non-calculus check is that Q+8/Q is at least 2√8 for positive Q, with equality at Q=√8. Alternatively, compare a fine table around that quantity; that suggests, but does not prove, the minimum.

The difference √8−2≈0.828 output unit is excess capacity in this standard comparison. It is not a measured factory utilization rate or a physical maximum. The firm would not voluntarily choose the minimum-ATC output at the stated demand: profit is maximized at Q=2. Differentiation may give consumers variety, so the existence of a markup alone is not a complete welfare evaluation.

Reconstruct the entry comparison

Before entry, suppose demand was P=12−Q, with unchanged costs. MR is then 12−2Q. Equating MR and MC yields Q=2.5, P=9.5. Revenue is 23.75; total cost is 19.25; profit is 4.5 credits. Positive economic profit can attract entry where barriers permit it, dividing demand among more differentiated sellers. The supplied after-entry curve is an assumed equilibrium endpoint, not a model estimating the number of entrants, timing, product quality or demand elasticities.

For your assessment, draw both demand/MR pairs with the same cost curves. Mark the short-run profit rectangle at Q=2.5, with ATC=7.7 and height 9.5−7.7=1.8. Its area is 2.5×1.8=4.5. Then identify the long-run tangency, markup and excess capacity separately. Explain why a rectangle’s zero profit area does not imply a zero markup.

A follow-up practice case uses demand P=11−Q and the same costs. Calculate output, price, revenue, cost and profit before checking the Apply feedback. Decide whether this demand curve has reached the zero-profit endpoint. State one real institutional condition, such as entry barriers, that could prevent the assumed adjustment. No result here is an empirical claim about a named industry.

Inspect the worked continuous graph

The table above is the numerical alternative. Linear curves are exact; the smooth ATC is drawn from closely spaced formula values over the visible range, with values above 14 clipped by omission. Its minimum marker uses the analytic value. Before viewing this worked graph, construct your own; then explain the separate intersections and the dotted Q=2 guide. The graph illustrates the fictional functions and is not a plot of observed firms.