Exchange Curves: Reading a Firm’s Own Costs
Agenda
- One item first: what a chosen operating point implies
- The aggregate curve, and the two quantities that cannot be chosen separately
- Reading a firm’s practice backwards
- The variety index
- When ordering costs differ across items
A Different Question
Last session: given costs and a constraint, find the quantities.
Today: given the quantities a firm is actually using, what does that say about the costs it is implicitly asserting?
This is the more useful direction in practice, because most organizations cannot defend their cost parameters but can report what they currently do.
One Item First
A planner decides to order an item twice a year and, doing so, carries an average of $400 of it in stock.
\[ \frac{\bar{I}}{\overline{\mathit{OF}}} = \frac{400}{2} = 200 = \frac{k}{\gamma} \]
So the planner’s choice implies \(k = 200\gamma\), whether or not the planner knows it.
If the firm’s carrying charge is \(\gamma = 0.20\) per year, the implied ordering cost is
\[ k = \$40 \]
That Number Can Be Checked
Suppose purchasing reports that placing an order genuinely costs $120.
Then the ratio has to be \(120/0.20 = 600\), and the planner is operating at 200.
The order quantity in use is too small, and the item is being ordered far too often for the stated carrying charge.
Or Read It the Other Way
Take the practice as correct instead, and solve for the carrying charge:
\[ \gamma = \frac{k}{200} = \frac{120}{200} = 0.60 \]
A carrying charge of 60% per year.
So one of two things is true. Either the item is ordered too often, or the firm’s carrying charge is three times what it says it is.
Both are defensible. The calculation does not settle which, but it forces somebody to make one of the two claims.
What We Just Did
We obtained a second reading of the ordering cost, from behavior rather than from accounting.
Neither reading is authoritative.
Their disagreement is the information.
Now Take Many Items
Define two aggregates over the whole portfolio:
\[ \bar{I}^{a} = \sum_{j} c_j\frac{Q_j}{2} \qquad N^{a} = \sum_{j}\frac{\lambda_j}{Q_j} \]
The first is dollars, and appears on a balance sheet.
The second is orders per year, and is what the receiving department feels.
They trade against one another. The question is exactly how.
Substitute the Economic Quantities
Suppose every item is ordered in its economic quantity, with a common \(k\) and a common \(\gamma\), so \(Q_j = \sqrt{2k\lambda_j/(\gamma c_j)}\).
\[ \bar{I}^{a} = \sqrt{\frac{k}{\gamma}}\cdot\frac{1}{\sqrt{2}}\sum_{j}\sqrt{\lambda_j c_j} \]
\[ N^{a} = \sqrt{\frac{\gamma}{k}}\cdot\frac{1}{\sqrt{2}}\sum_{j}\sqrt{\lambda_j c_j} \]
Two things follow at once.
The Product, and the Ratio
\[ \bar{I}^{a}\cdot N^{a} = \frac{1}{2}\left(\sum_{j}\sqrt{\lambda_j c_j}\right)^{2} \]
\(k\) and \(\gamma\) have both cancelled.
\[ \frac{\bar{I}^{a}}{N^{a}} = \frac{k}{\gamma} \]
The product is fixed by the item data alone, and defines a single hyperbola: the exchange curve.
The ratio \(k/\gamma\) is a ray from the origin.
Where the ray crosses the curve is where the portfolio operates.
The Curve
Reading It Forwards
Follow the curve left to right. Aggregate investment falls as the number of replenishments rises. That is the trade the product forces.
The curve flattens toward the right, so the stock given up for each additional order gets smaller as the portfolio orders more often.
Two rays, two operating points: \(k/\gamma = 12\) meets the curve at 100 orders and $1,200; \(k/\gamma = 3\) meets it at 200 orders and $600.
A firm does not choose a point on the curve directly.
It chooses a cost ratio, and the ratio chooses the point.
The Point Above the Curve
The triangle sits at 200 replenishments and $1,200 of stock.
\[ 200 \times 1200 = 240{,}000 \ne 120{,}000 \]
No set of economic order quantities produces it. A portfolio found there is ordering in quantities that are not economic.
That portfolio can move onto the curve and improve both measures at once, with no trade-off to argue about and no new money.
This is the rare case in which an inventory recommendation costs nothing.
Reading It Backwards
A distributor places 100 orders a year and carries $1,200 of cycle stock, on the curve.
\[ \frac{\bar{I}^{a}}{N^{a}} = \frac{1200}{100} = 12 = \frac{k}{\gamma} \]
Case 1, the ordering cost is known. Estimation puts \(k = \$4\):
\[ \gamma = \frac{4}{12} = 0.333 \text{ per year} \]
A charge of 33% is what would make current practice optimal. If the true figure is 20%, the ratio that should govern is \(4/0.20 = 20\), not 12, so the firm is ordering too often for the stock it holds.
Two More Readings
Case 2, neither cost is known. The ratio is still informative, because it is what the firm’s own behavior asserts.
A manager who cannot produce \(k\) or \(\gamma\), but who rejects \(k/\gamma = 12\) as absurd, has learned something about current practice.
Case 3, the operating point is constrained. Suppose transportation fixes the portfolio at 200 orders a year. The curve gives \(\bar{I}^{a} = \$600\) and a required ratio of 3.
Reaching that ratio cheaply means reducing \(k\), since \(\gamma\) is a property of the cost of capital and is not easily moved.
The curve gives guidance on changing the environment, not merely operating within it. It says which cost to attack, and by how much.
A Consequence Worth Memorizing
Since \(\bar{I}^{a} \times N^{a}\) is constant along the curve, cutting investment by a factor scales the workload by its reciprocal.
Cut inventory investment by 30%:
\[ \frac{1}{0.70} = 1.4286 \]
Workload rises 42.86%.
And that factor is the same wherever on the curve you start, because the curve is a rectangular hyperbola and the individual costs cancelled out of the product entirely.
When Ordering Costs Differ
The simple curve assumed a common \(k\). Real portfolios do not have one.
A municipal utility storeroom carries nine items. Order handling costs \(\kappa = \$55\) per hour, and item \(j\) takes \(w_j\) hours per order regardless of size, so \(k_j = \kappa w_j\).
The curve survives. Only one aggregation is required:
\[ \sum_{j}\sqrt{k_j c_j \lambda_j} = 19{,}755.87 \]
\[ \bar{I}^{a}\cdot\overline{\mathit{OC}}^{a} = \frac{1}{2}(19{,}755.87)^{2} = 1.951\times 10^{8} \]
Tracing It With the Multiplier
\[ \bar{I}^{a} = \frac{\sum_j\sqrt{k_jc_j\lambda_j}}{\sqrt{2\theta}}, \qquad \overline{\mathit{OC}}^{a} = \sqrt{\frac{\theta}{2}}\sum_j\sqrt{k_jc_j\lambda_j} \]
| \(\theta\) | \(\bar{I}^{a}\) | \(\overline{\mathit{OC}}^{a}\) | Order handling | Product |
|---|---|---|---|---|
| 0.05 | $62,474 | $3,124 | 57 hours | \(1.951\times 10^{8}\) |
| 0.10 | $44,175 | $4,418 | 80 hours | \(1.951\times 10^{8}\) |
| 0.25 | $27,939 | $6,985 | 127 hours | \(1.951\times 10^{8}\) |
| 0.50 | $19,756 | $9,878 | 180 hours | \(1.951\times 10^{8}\) |
| 1.00 | $13,970 | $13,970 | 254 hours | \(1.951\times 10^{8}\) |
The last column is constant by construction, and checking it is the arithmetic check for this section.
Where This Storeroom Sits
The firm’s own carrying charge is the multiplier, so \(\theta = \gamma = 0.25\) puts it on the third row.
$27,939 of cycle stock against $6,985 a year of order handling, which is 127 hours of a storekeeper’s time.
Since \(\bar{I}^{a}/\overline{\mathit{OC}}^{a} = 1/\theta\), the storeroom holds exactly four dollars of stock for every dollar a year it spends ordering.
Halving the carrying charge to 0.125 moves it between rows two and three: more stock, less workload, at an exchange rate the curve fixes exactly.
The Variety Index
\[ J^{*} = \frac{\left(\sum_j \sqrt{k_j c_j\lambda_j}\right)^{2}}{K^{w}\lambda^{a}c^{w}} \]
where \(\lambda^{a}\) is aggregate demand and \(K^{w}\), \(c^{w}\) are dollar-usage weighted averages.
It is bounded: \(1 \le J^{*} \le J\).
For the storeroom, with denominator \(6.870\times 10^{7}\):
\[ J^{*} = \frac{(19{,}755.87)^{2}}{6.870\times 10^{7}} = \mathbf{5.68} \]
Nine items behave in aggregate like fewer than six identical ones.
What Drives It
The transformer alone contributes 6,133 of the 19,756 in that sum.
\(J^{*}\) reaches its upper bound \(J\) when the items are identical in the relevant product, and falls toward 1 when a single item dominates.
That concentration is the same thing ABC classification measures, arrived at a different way.
And the curve decomposes:
\[ \frac{1}{2}J^{*}K^{w}\lambda^{a}c^{w} = \frac{1}{2}(5.6813)(123.67)(14{,}445)(38.46) = 1.951\times 10^{8} \]
Four Things Set the Frontier
\[ \bar{I}^{a}\cdot\overline{\mathit{OC}}^{a} = \frac{1}{2}J^{*}K^{w}\lambda^{a}c^{w} \]
- \(J^{*}\): how varied the items are
- \(K^{w}\): what an order costs
- \(\lambda^{a}\): how much is demanded
- \(c^{w}\): what the goods are worth
Nothing else. No policy, no multiplier, no service target.
Which is why the curve can be drawn before anything is optimized.
What We Did Today
The product is fixed by the item data; the ratio is \(k/\gamma\). A firm chooses a ratio, and the ratio chooses the point.
A point off the curve is free money. Both measures improve at once.
A point on the curve asserts a cost ratio, which can be compared against the costs the firm claims. A discrepancy says practice is optimal for costs the firm does not have.
Cutting investment 30% raises workload 42.86%, from anywhere on the curve.
The variety index says how alike the items are. 5.68 for nine items.
For Next Time
Read before next session: Joint Replenishment.
We have done the shared resource. Next is the shared setup, which couples the cost functions themselves and leaves the items unconstrained.
Next session:
- Why a shared order cost is a different problem, not a variant of this one
- Power-of-two policies, and why a schedule that nests is one a dock can run
- The 6.07% bound, and where it comes from
- Why coordination beats cleverness by a wide margin
The flatness of the EOQ cost curve returns next session, applied to cycle lengths instead of order quantities. It does the same work.