μMicroeconomics LabGRAPHICAL EXPLORATIONSOffer Curve Lab ↗

EXPLORE · ADJUST · EXPLAIN

From firm costs to market entry

Change the number of price-taking firms and follow the market price back to each firm’s profit.

Costs & industry

Technology stays fixed at T = √2. Input prices determine the quadratic cost coefficient: a = √(wr) ≈ 5.
Firms take the market price as given. Fixed cost d is committed in the short run and avoidable through exit in the long run.
Shared price scale: 0–150 · P = 70

1 · A representative firm

Cost curves, market price, and profit or loss

Marginal average variable and average total costs with shutdown and exit thresholds003306609901212015150min ATCq*Firm output · qPrice / cost per unit
MCAVCATCMarket price

■ p ≤ 20: short-run shutdown.
■ 20 < p < 44.49: produce now; exit in the long run.

Output per firm5
Profit per firm95

2 · Market clearing

P(Q) = 120 − 1Q

Market demand and industry supply determine equilibrium price00303060609090120120150150EquilibriumIndustry output · QPrice · P
Market demandSupply · 10 firmsMarket price
Available firms10
Market price70
Industry quantity50
Positive economic profit

Short-run equilibrium calculations

Cost & supply formulas

Cost curves and short-run supply
C(q)=wr q2+20q+30AVC(q)=aq+bATC(q)=aq+b+dqMC(q)=2aq+bq(P)=max⁡ ⁣{0,P−b2a}\begin{aligned}C(q)&=\sqrt{wr}\,q^2+20q+30\\ \mathrm{AVC}(q)&=aq+b\\ \mathrm{ATC}(q)&=aq+b+\frac d q\\ \mathrm{MC}(q)&=2aq+b\\ q(P)&=\max\!\left\{0,\frac{P-b}{2a}\right\}\end{aligned}

AVC approaches b as q approaches zero; it is not U-shaped here. At P = b, the firm chooses zero output. ATC reaches its minimum at q = √(d/a) = 2.45.

Market clearing

Market equilibrium
qn=max⁡ ⁣{0,A−b2a+Bn},n≥1Qn=nqn,Pn=A−BQnπn=aqn2−d\begin{aligned}q_n&=\max\!\left\{0,\frac{A-b}{2a+Bn}\right\},\quad n\ge1\\ Q_n&=nq_n,\qquad P_n=A-BQ_n\\ \pi_n&=aq_n^2-d\end{aligned}
With your parameters
qn≈5Pn≈70πn=5⋅5 qn2−30≈95\begin{aligned}q_n&\approx5\\ P_n&\approx70\\ \pi_n&=\sqrt{5\cdot5}\,q_n^2-30\\&\approx95\end{aligned}

For q > 0, the profit rectangle has width q and height P−ATC(q). At shutdown, the fixed loss remains even though the rectangle disappears.

Where the quadratic cost can come from

One explicit production technology—not a unique inference from the cost curve

Cost minimization

Hold target output q̄ fixed
min⁡L,H≥0wL+rH+bqˉ+dsubject toT(LH)1/4≥qˉa=2wrT2\begin{aligned}\min_{L,H\ge0}\quad&wL+rH+b\bar q+d\\\text{subject to}\quad&T(LH)^{1/4}\ge\bar q\\ a&=\frac{2\sqrt{wr}}{T^2}\end{aligned}

L is labor and H is machine-hours. Materials cost b per unit; overhead is d. This technology has decreasing returns to scale: doubling both inputs multiplies output by √2.

T = √2 is held fixed, so a = √(wr) ≈ 5. Raising either input price raises variable costs and changes the market equilibrium. Relative input prices also change the cost-minimizing input mix.

1 · Bang per buck

Equalize marginal output per dollar
MPLw=MPHrqˉ4Lw=qˉ4Hr⟹wL=rHL∗=(qˉT)2rwH∗=(qˉT)2wr\begin{aligned}\frac{\mathrm{MP}_L}{w}&=\frac{\mathrm{MP}_H}{r}\\ \frac{\bar q}{4Lw}&=\frac{\bar q}{4Hr}\quad\Longrightarrow\quad wL=rH\\ L^*&=\left(\frac{\bar q}{T}\right)^2\sqrt{\frac rw}\\ H^*&=\left(\frac{\bar q}{T}\right)^2\sqrt{\frac wr}\end{aligned}

2 · Lagrangian

First-order conditions for positive output
L=wL+rH+bqˉ+d+λ[qˉ−T(LH)1/4]w=λqˉ4L,r=λqˉ4HT(LH)1/4=qˉ\begin{aligned}\mathcal L={}&wL+rH+b\bar q+d\\&+\lambda\left[\bar q-T(LH)^{1/4}\right]\\ w&=\lambda\frac{\bar q}{4L},\qquad r=\lambda\frac{\bar q}{4H}\\ T(LH)^{1/4}&=\bar q\end{aligned}

Divide the first two conditions, then use the binding output constraint. Substituting the resulting inputs yields C(q) = [2√(wr)/T²]q²+bq+d.

Optimal input use
qˉ=5L∗=12.5,H∗=12.5wL∗+rH∗=aqˉ2≈125\begin{aligned}\bar q&=5\\ L^*&=12.5,\qquad H^*=12.5\\ wL^*+rH^*&=a\bar q^2\\&\approx125\end{aligned}