Ratios produce averages and per-dollar comparisons

The denominator supplies the unit.

Name that denominator aloud before comparing the resulting values.

AFC=FC/Q, AVC=VC/Q, and ATC=TC/Q=AFC+AVC. AP_L=Q/L. Utility maximization compares MU_X/P_X across goods. Cost minimization compares MP_L/P_L across inputs. Equal numerators alone do not optimize when prices differ.

Derived demand links a change and a value: MRP_L=MP_L× MR. In a competitive product market, MR=P, so VMP_L=MP_L× P. A monopsonist compares MRP with MRC, not directly with the wage.

A four-step formula chain

Output rises from 150 with six workers to 168 with seven. The seventh worker’s MP is 18. If product price and MR are $5, MRP is $90. At an $82 marginal hiring cost, the worker adds $8 to profit. Each number has a distinct unit and role.

A ratio converts a total into an amount per unit. The denominator tells you what “per” means. Average fixed cost is fixed dollars per unit of output. Average product is output per worker. Marginal utility per dollar is additional satisfaction per dollar spent. Writing the unit beside a ratio prevents comparisons that look algebraically tidy but make no economic sense.

Cost identities are especially useful. Since TC=FC+VC, dividing every term by output gives ATC=AFC+AVC. If fixed cost is $120, variable cost is $280, and output is 40, then AFC is $3, AVC is $7, and ATC is $10. A distractor of $400 is total cost, not average cost. As output rises, AFC falls because the same fixed amount is spread over more units.

Per-dollar comparisons solve constrained allocation problems. A consumer with MU_X/P_X=6 and MU_Y/P_Y=4 should shift a dollar toward X and away from Y, assuming both goods can be adjusted. Diminishing marginal utility lowers X’s ratio as more X is consumed and raises Y’s ratio as less Y is consumed, moving toward equality. Equal marginal utilities alone are optimal only when prices are equal.

The analogous producer rule compares marginal product per input dollar. Suppose the last labor dollar produces 8 units while the last machine dollar produces 5. Reallocating spending toward labor can produce more output at the same total cost. Cost minimization requires equal marginal products per dollar for adjustable inputs, subject to corner solutions and technological constraints.

Keep marginal and average units separate

Seven workers produce 210 units, and the eighth raises output to 232. Average product with eight workers is 232/8=29 units per worker. The eighth worker’s marginal product is 22 units. If output sells competitively for $4, that worker’s MRP is $88 per worker. None of these is total revenue.

Ratios also expose profitability. (P-ATC) is profit per unit, so multiplying by quantity gives total economic profit. The formula fails if ATC and price refer to different output levels. Similarly, dividing fixed cost by zero output is undefined. The firm still incurs total fixed cost even though AFC cannot be computed at zero.

Before choosing a ratio, translate the stem: “per unit” usually calls for division, “the next unit” calls for a change, and “per dollar” calls for dividing a marginal benefit or product by price. This language-first method prevents visual formula matching.