Taylor error control and optimal value sensitivity

A marginal approximation is useful only if its error is understood over the contemplated change. Likewise, an optimal decision at one price need not remain optimal at a new price. Taylor bounds quantify finite-change errors, and differentiating the optimality equation describes how a regular interior optimum moves. Feasible boundaries create a separate regime.

1 An error bound from curvature

For $f\in C^2$ on the interval between $q$ and $q+h$, twice applying the fundamental theorem gives

$$f(q+h)=f(q)+f'(q)h+h^2\int_0^1(1-t)f''(q+th)dt.$$

To derive it, write the function along $q+th$ and integrate its derivative twice. Thus if $|f''|\le M$ throughout that interval,

$$|f(q+h)-f(q)-f'(q)h|\le\frac M2h^2.$$

If $f''\ge0$, the linear tangent approximation is a lower bound, not merely a close guess. The sign follows from the nonnegative integral remainder. Higher-order versions follow by repeated integration or the one-variable Taylor theorem with explicit derivative bounds.

For the cost model $C(q)=100+4q+0.02q^2$, $C''=0.04$, so the exact error of the marginal estimate over increment $h$ is $0.02h^2$. A ten-unit change gives error 2, not the one-unit error 0.02. The derivative is a local rate; the planned step size matters quadratically in this model.

Exponential example. For $f(q)=e^{0.1q}$, increasing $q$ from 0 by $h\in[0,1]$ gives linear estimate $1+0.1h$. Since $f''(q)=0.01e^{0.1q}$, the error is nonnegative and at most $0.005e^{0.1}h^2$. This is a bound over a specified interval, not a bound inferred from $f''(0)$ alone.

2 Elasticity and exact logarithmic changes

For positive $C^1$ functions $f(q)$ on a positive interval, elasticity is $E(q)=qf'(q)/f(q)$. The chain rule gives $d\log f/d\log q=E(q)$. Hence the exact finite-change identity is

$$\log\frac{f(q_1)}{f(q_0)}=\int_{q_0}^{q_1}\frac{E(q)}q\,dq.$$

If elasticity is constant $E_0$, integration gives $f(q)=Kq^{E_0}$ on a positive connected interval. Conversely this power law has constant elasticity. A first-order percentage approximation replaces the integral by $E(q_0)(q_1-q_0)/q_0$; it is not generally exact for large changes.

For $f(q)=q^2$, a 10% increase in $q$ yields a 21% increase in $f$, not exactly 20%. The log identity gives $\log(f(q_1)/f(q_0))=2\log1.1$, equivalent to the exact factor $1.1^2$.

3 Sensitivity of an interior optimum

Consider profit $\Pi(q,p)=pq-C(q)$ with price parameter $p$. An interior stationary choice satisfies $C'(q_*(p))=p$. If $C\in C^2$ and $C''(q_*)>0$, the local implicit-function theorem supplies a differentiable stationary branch and

$$\frac{dq_*}{dp}=\frac1{C''(q_*)}.$$

The formula follows by differentiating the stationary equation. Positive curvature makes profit locally concave and the choice increase with price. If $C$ is strictly convex throughout the feasible interval, an interior stationary point is its unique profit maximizer there. The local derivative formula alone does not prove global optimality or continued feasibility.

Let $V(p)=\Pi(q_*(p),p)$ on a smooth interior maximizing branch. The chain rule gives

$$V'(p)=\Pi_q(q_*,p)q_*'(p)+\Pi_p(q_*,p)=q_*(p),$$

because the stationary first term vanishes. This is the envelope calculation: the first-order value change uses the parameter effect at the optimum, even though the optimizer itself moves.

4 Boundaries change the regime

For $C(q)=F+aq+bq^2$ with $b>0$, capacity $Q>0$, and feasible $0\le q\le Q$, price-independent fixed cost $F$ does not affect the maximizing quantity in this mathematical model. Completing the square or examining the derivative gives

$$q_*(p)=\begin{cases}0,&p\le a,\\(p-a)/(2b),&a<p<a+2bQ,\\Q,&p\ge a+2bQ.\end{cases}$$

In the interior regime, $q_*'=1/(2b)$. In either saturated regime, $q_*'=0$. At the transition prices the choice function is continuous but not differentiable. The optimal value remains differentiable here, with derivative $q_*$, as direct piecewise differentiation confirms. The interior sensitivity cannot be extrapolated through a capacity boundary.

For $a=4$, $b=0.1$, $Q=50$, price 10 gives $q_*=30$. A small price increase of 0.2 moves the optimum by 1 while it remains interior. At price 15, the capacity constraint already binds, so the same increase does not change the chosen quantity. Fixed-cost interpretations such as shutdown decisions require a model specifying whether the fixed cost is avoidable; that business rule is not encoded by this objective alone.

5 Graded practice

E1 Foundation. If $|C''|\le0.6$ over a five-unit output increase, bound the marginal-approximation error. What extra sign information follows if $C''\ge0$?

E2 Exact elasticity. For $f(q)=3q^{-2}$, compute elasticity and the exact relative change when $q$ rises by 20%.

E3 Interior sensitivity. Let $C(q)=20+2q+q^2/2$. For feasible $q\ge0$ and price $p>2$, find the optimizer, the optimal value and its derivative.

E4 Capacity. Add $q\le4$ to E3. Give the full optimal quantity and identify where the unconstrained sensitivity formula stops applying.

6 Full solutions

S1. The magnitude is at most $(0.6/2)5^2=7.5$ cost units. If curvature is nonnegative, actual cost increase is at least the initial marginal estimate, so the error has a known nonnegative sign as well as an upper bound.

S2. $E=-2$. The exact factor is $1.2^{-2}=25/36$, giving relative change $-11/36$, approximately $-30.56\%$. The first-order estimate $-40\%$ is not exact for this finite step.

S3. $C'=2+q$, so $q_*=p-2$, feasible for $p>2$. Strict convexity gives the unique global maximum. $V(p)=(p-2)^2/2-20$, so $V'=p-2=q_*$ and $q_*'=1$.

S4. The optimizer is 0 for $p\le2$, $p-2$ for $2<p<6$, and 4 for $p\ge6$. Sensitivity is 1 only in the middle regime and 0 outside. At 2 and 6 the optimizer's derivative does not exist, so there is no single interior derivative formula valid across the thresholds.

Summary

Curvature bounds turn local marginal rates into controlled finite-change estimates. Elasticity integrates to an exact logarithmic-change formula. Optimality equations give local parameter sensitivity, while the envelope calculation describes value changes. Feasible boundaries create regime changes that must be checked before extrapolating any interior formula.