Measuring Output Gains from Reallocating Resources

Measuring a Reallocation in Physical Output

Comparative advantage identifies which producer can make a good at lower opportunity cost. Measuring the consequence of reallocating production requires one further step: translate the change in output into the labor hours released or required, then calculate what that labor produces in the other sector. The result is a change in combined physical output, not automatically a change in income or welfare.

This tutorial considers two countries, two goods, and labor as the resource being shifted between sectors. A reallocation can hold combined output of one good constant while increasing the other. That comparison makes the output gain measurable without adding unlike goods together.

Definition: A physical output gain from reallocation occurs when resources are shifted between sectors so that the combined quantity of one good is unchanged and the combined quantity of the other good increases. The calculation aggregates each good across producers separately; it does not add quantities of different goods.

Assume each country has a fixed amount of labor, can move labor between its two sectors, and has constant labor productivity in each sector. Constant productivity means that one labor hour produces the same number of units regardless of how many hours are already assigned to that sector. Each country’s production possibilities frontier is therefore linear. Labor may be reallocated within each country; it need not cross a national border.

The opportunity cost of a good is the quantity of the other good forgone to produce it. For example, if a country produces \(x\) units of X or \(y\) units of Y per labor hour, producing one additional unit of X requires \(1/x\) labor hours and forgoes \(y/x\) units of Y. Thus its opportunity cost of X is \(y/x\) units of Y per unit of X.

The Net-Gain Calculation

Suppose country R has the lower opportunity cost of X, while country S has the higher opportunity cost of X. Increase R’s X production by \(q\) units and reduce S’s X production by the same \(q\). Combined X output is unchanged. R must move labor from Y to X, while S can move labor released from X into Y.

\begin{aligned} \text{Y forgone in R} &= q\frac{y_R}{x_R},\\ \text{Y gained in S} &= q\frac{y_S}{x_S},\\ \Delta Y_{\text{combined}} &= q\left(\frac{y_S}{x_S}-\frac{y_R}{x_R}\right). \end{aligned}

Here \(x_R\) and \(y_R\) denote R’s units of X and Y produced per labor hour; \(x_S\) and \(y_S\) have the same meanings for S. The expression is positive precisely when S has the higher opportunity cost of X. That is why shifting X production toward R, the lower-opportunity-cost producer, increases combined Y while holding combined X fixed.

Result: If \(q\) units of X move from a producer with higher opportunity cost of X to one with lower opportunity cost, combined X stays constant and combined Y rises by \(q\) times the difference between those opportunity costs. The calculation is valid only if the required labor shifts are feasible and the stated productivity assumptions hold.

Find each producer’s opportunity cost.
Divide its Y output per labor hour by its X output per labor hour to obtain the Y forgone per X produced.

Choose the direction.
To hold combined X fixed and gain Y, move X production toward the country with lower opportunity cost of X.

Convert output changes into labor hours.
Divide the required change in X by the relevant X productivity. Use the released or reassigned hours to calculate the change in Y.

Aggregate by good and check the sign.
Confirm that the increase and decrease in X offset, and that the net change in Y agrees with the opportunity-cost comparison.

Worked Calculations

Worked Example: Moving X Production Toward Its Lower-Cost Producer

Country A produces 5 units of X or 2 units of Y per labor hour. Country B produces 2 units of X or 3 units of Y per hour. Reallocate production so that A makes 10 more units of X and B makes 10 fewer. Find the change in combined Y, and verify that combined X is unchanged.

Method. Compare opportunity costs, convert each X change into labor hours, and compute the associated Y changes. This avoids treating the labor hours shifted in the two countries as if their productivities were identical.

A’s opportunity cost of X is \(2/5=0.4\) units of Y per X. B’s is \(3/2=1.5\) units of Y per X. Therefore, X is being shifted toward the lower-opportunity-cost producer, A.

A needs \(10/5=2\) labor hours to produce 10 additional units of X. Those hours move from Y production, so A gives up \(2\times2=4\) units of Y. B reduces X production by 10 units and releases \(10/2=5\) labor hours. Moving those hours to Y produces \(5\times3=15\) additional units of Y.

The net change in Y is \(15-4=11\) units. The changes in X are \(+10\) for A and \(-10\) for B, so the net change in X is \(10-10=0\). The formula gives the same result: \(10(1.5-0.4)=11\) units of Y gained. Check: combined X is unchanged and combined Y rises by 11 units, as expected when X shifts toward its lower-cost producer.

Worked Example: Checking the Aggregate Totals

Country C produces 4 units of X and 6 units of Y per labor hour. Country D produces 3 units of X and 2 units of Y per hour. Initially, C assigns 8 labor hours to X and 4 to Y; D assigns 4 hours to X and 8 to Y. Shift 12 units of X production from C to D. Calculate both countries’ new outputs and the combined output change.

Method. First compute the labor adjustments, then compare the initial and final aggregate totals. This also checks that the change in X is exactly zero.

C reduces X by 12 units. At 4 units of X per hour, that releases \(12/4=3\) hours. C moves those hours to Y and gains \(3\times6=18\) units of Y. Its X output falls from \(8\times4=32\) to \(32-12=20\); its Y output rises from \(4\times6=24\) to \(24+18=42\).

D increases X by 12 units. At 3 units of X per hour, that requires \(12/3=4\) hours, moved from Y production. D’s X output rises from \(4\times3=12\) to \(12+12=24\); its Y output falls from \(8\times2=16\) to \(16-(4\times2)=8\).

Initially, combined X is \(32+12=44\) and combined Y is \(24+16=40\). After reallocation, combined X is \(20+24=44\) and combined Y is \(42+8=50\). Thus combined X changes by \(44-44=0\), and combined Y increases by \(50-40=10\) units.

As a check using opportunity costs, C’s cost of X is \(6/4=1.5\) Y per X, while D’s is \(2/3\) Y per X. Moving X from C to D therefore raises Y by \(12(1.5-2/3)=12(5/6)=10\) units. Check: this matches the aggregate totals.

Worked Example: Finding the Reallocation Needed for a Target Gain

Country P produces 6 units of X and 3 units of Y per labor hour. Country Q produces 2 units of X and 5 units of Y per hour. How many units of X must be shifted from Q to P to increase combined Y by 21 units while keeping combined X unchanged? State the labor-hour changes and check the result.

Method. Calculate the net Y gain per unit of X shifted, then divide the target gain by that amount. The key trap is reversing the direction of the opportunity-cost difference.

P’s opportunity cost of X is \(3/6=0.5\) Y per X. Q’s is \(5/2=2.5\) Y per X. Each unit of X shifted from Q to P therefore gains \(2.5-0.5=2\) units of Y in net.

Let \(q\) be the X output shifted. The target equation is \(2q=21\), so \(q=21/2=10.5\) units of X. P needs \(10.5/6=1.75\) labor hours moved from Y to X. P gives up \(1.75\times3=5.25\) units of Y. Q releases \(10.5/2=5.25\) hours from X and uses them in Y, gaining \(5.25\times5=26.25\) units of Y.

The net Y gain is \(26.25-5.25=21\) units. P gains 10.5 units of X and Q loses 10.5, so combined X is unchanged. Check: both the direct labor calculation and the opportunity-cost formula give the required 21-unit increase in combined Y.

Why the Formula Holds—and When It Does Not

The formula follows from the labor requirements for the offsetting X changes. To make \(q\) additional units of X, R moves \(q/x_R\) labor hours out of Y. Those hours would have produced \((q/x_R)y_R\) units of Y, so that is R’s Y loss. S cuts X output by \(q\), releasing \(q/x_S\) hours; moving them to Y adds \((q/x_S)y_S\) units of Y. Subtract the loss from the gain:

\Delta Y_{\text{combined}} = \frac{q}{x_S}y_S-\frac{q}{x_R}y_R = q\left(\frac{y_S}{x_S}-\frac{y_R}{x_R}\right).

This is an output accounting result, not a claim that every reallocation creates a gain. If X shifts in the opposite direction, the sign reverses. If both countries have the same opportunity cost of X, the expression is zero: the reallocation changes who produces X but does not change combined Y when combined X is held fixed.

The result depends on its assumptions. With rising opportunity costs, productivity changes as labor moves between sectors; the amount of Y gained per additional unit of X shifted will not generally stay constant. Then a single linear formula is not enough for a large reallocation. A shift may also be infeasible if a country lacks enough labor in the sector from which workers must move. The calculation presumes sufficient available labor, divisible labor, unchanged technology, and no transition losses.

Finally, units of X and Y should not be added together as if they were interchangeable. Saying that output rises by 11 units of Y while X is unchanged is precise. Saying that the economy gains “11 units of output” without identifying the good is not. A physical output gain also does not, by itself, identify who gains or establish how the result affects well-being.

Common Exam Trap: Reversing the Direction

A frequent error is to subtract opportunity costs in the wrong order, or to move production toward the country with the higher opportunity cost. In the target-gain example, the incorrect calculation would be \(10.5(0.5-2.5)=-21\), and describing that as a 21-unit gain would reverse the sign. The correct direction is from Q, with opportunity cost 2.5, to P, with opportunity cost 0.5; the net change is \(10.5(2.5-0.5)=21\) units of Y. Always name which country gains X and which gives it up before calculating the net effect.

A second check is to recompute the result using labor hours. If the X increase and decrease do not cancel, or if the Y gain does not equal the gain in one country minus the loss in the other, revisit the labor conversions. This method of measuring physical output changes is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Key takeaway: For a fixed increase in one good’s combined output, move production toward the producer with lower opportunity cost of that good. The other good’s net output change equals the output gained from released labor minus the output forgone in the expanding sector.

Check Your Understanding

For each problem, track the output change by good and verify the direction of the reallocation.

Country K produces 3 units of X and 6 units of Y per hour; country L produces 5 units of X and 2 units of Y per hour. Shift 10 units of X from K to L. How does combined Y change if combined X is held constant? (Answer: Combined Y increases by 16 units; combined X is unchanged.)
Country A produces 4 units of X and 3 units of Y per hour; country B produces 2 units of X and 5 units of Y per hour. Shift 8 units of X from B to A. Calculate the net change in combined Y. (Answer: Combined Y increases by 14 units; combined X is unchanged.)
Country M produces 4 units of X and 2 units of Y per hour; country N produces 3 units of X and 6 units of Y per hour. Shift 12 units of X from N to M. Show the Y lost in M and gained in N, then find the net change. (Answer: M loses 6 units of Y, N gains 24 units of Y, and combined Y increases by 18 units; combined X is unchanged.)
Country L produces 5 units of X and 4 units of Y per hour; country H produces 2 units of X and 4.6 units of Y per hour. How many units of X must move from H to L to increase combined Y by 18 units? (Answer: 12 units of X.)
Country J produces 4 units of X and 2 units of Y per hour; country K produces 6 units of X and 3 units of Y per hour. Shift 8 units of X from K to J. Calculate the change in combined Y and explain the result using opportunity costs. (Answer: Combined Y is unchanged; both countries have an opportunity cost of 0.5 units of Y per X.)

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