Private and Social Marginal Curves

Why Private and Social Marginal Curves Differ

A buyer or seller typically makes decisions by considering the benefits or costs that fall directly on them. Other people may also be affected. When those effects are not included in the decision maker’s own calculation, the private marginal curve can differ from the social marginal curve. Comparing the two makes the external effect visible in a common unit: the additional cost or benefit associated with one more unit of activity.

For a consistent comparison, use the same measure of activity, time period, and currency for all four curves. For example, if $q$ is units produced per month and costs are measured in dollars per month, each marginal cost at $q$ is measured in dollars per additional unit at that activity level. The same units apply to marginal benefits.

Definition: Marginal private cost (MPC) is the additional cost borne by the decision maker when activity increases by one unit. Marginal external cost (MEC) is the additional cost that the activity imposes on other people. Marginal social cost (MSC) is the additional cost to everyone affected: $MSC=MPC+MEC$.

Definition: Marginal private benefit (MPB) is the additional benefit received by the decision maker when activity increases by one unit. Marginal external benefit (MEB) is the additional benefit received by other people. Marginal social benefit (MSB) is the additional benefit to everyone affected: $MSB=MPB+MEB$.

“Private” identifies who directly bears a cost or receives a benefit; it does not mean that the effect is confidential or belongs to a privately owned firm. “Social” means the total for the affected parties, not a claim that every person receives an equal share. If an external effect is adverse, its marginal external cost is positive. If an external effect benefits others, its marginal external benefit is positive. The relevant external effect may vary with the activity level.

How to Read the Gaps Between Curves

At a fixed $q$, the vertical gap between MSC and MPC is MEC: $MSC(q)-MPC(q)=MEC(q)$. Similarly, the vertical gap between MSB and MPB is MEB: $MSB(q)-MPB(q)=MEB(q)$. The comparison holds at the same activity level. A horizontal distance between curves does not measure the external effect in dollars per additional unit.

The formulas describe the marginal curves whether the external effects arise on the production side or the consumption side. The source of the effect determines who is affected, but the accounting is the same: include the decision maker’s marginal cost or benefit and add the marginal effect on others. For an external benefit, the social benefit curve lies above the private benefit curve when MEB is positive. For an external cost, the social cost curve lies above the private cost curve when MEC is positive.

These relationships do not require the external effect to be constant. A fixed MEC creates a constant vertical gap; a MEC that changes with $q$ creates a gap that changes with $q$. The same is true of MEB. The curves should be interpreted using marginal values at the stated activity level, rather than one average external effect applied everywhere.

Why the Addition Rule Holds

The addition rule follows from adding the total effects first and then asking how those totals change when activity increases. Suppose $C_p(q)$ is total private cost and $D(q)$ is total external damage imposed on others. Under a differentiable model, total social cost is $C_s(q)=C_p(q)+D(q)$. Differentiating both sides with respect to $q$ gives the marginal relationship:

MSC(q)=\frac{dC_s(q)}{dq} =\frac{dC_p(q)}{dq}+\frac{dD(q)}{dq} =MPC(q)+MEC(q).

The same reasoning applies to benefits. If $B_p(q)$ is total private benefit and $E(q)$ is total external benefit to others, then $B_s(q)=B_p(q)+E(q)$, so differentiating gives $MSB(q)=MPB(q)+MEB(q)$. This justification assumes the total effects are differentiable; in a discrete setting, the corresponding statement uses the change in each total from one activity level to the next. The formula does not mean that a marginal value is the total effect: it is the change in the total for an additional unit.

Worked Example: Finding Marginal Social Cost from Total Cost

A hypothetical workshop produces $q$ batches per month. Its total private cost is $C_p(q)=40q+q^2$ dollars. Its production also causes total external damage $D(q)=2q^2$ dollars per month. Find MPC, MEC, and MSC at $q=6$, and check the result using total costs.

Method. Differentiate the private cost and external damage separately to obtain their marginal values, then add them. This avoids treating the total damage at six batches as the marginal damage of one more batch.

Step 1: Find MPC. Since $C_p(q)=40q+q^2$, $MPC(q)=dC_p/dq=40+2q$. At $q=6$, $MPC=40+2(6)=40+12=\$52$ per additional batch.

Step 2: Find MEC. Since $D(q)=2q^2$, $MEC(q)=dD/dq=4q$. At $q=6$, $MEC=4(6)=\$24$ per additional batch.

Step 3: Add the marginal effects. $MSC(6)=MPC(6)+MEC(6)=52+24=\$76$ per additional batch.

Check using total social cost. $C_s(q)=C_p(q)+D(q)=40q+q^2+2q^2=40q+3q^2$, so $MSC(q)=dC_s/dq=40+6q$. At $q=6$, $MSC=40+36=\$76$, matching the separate calculation.

Marginal Benefits Use the Same Accounting

For benefits, the private curve records the marginal benefit to the direct decision maker, while the social curve also includes benefits to others. A positive external benefit therefore raises MSB above MPB at the same activity level. The calculation is not a discount or a percentage adjustment: it is addition of marginal benefits expressed in the same units.

Worked Example: Adding an External Benefit to Private Benefit

A hypothetical community program has total private benefit $B_p(q)=120q-3q^2$, measured in dollars per month, where $q$ is the number of service units per month. Each unit also creates a total external benefit $E(q)=12q$ dollars per month for people who do not use the service. Find MPB, MEB, and MSB at $q=10$, then verify MSB from total social benefit.

Method. Differentiate each total benefit with respect to activity and add the two marginal benefits. This avoids using the total external benefit as though it were the benefit of just one additional unit.

Step 1: Find MPB. $MPB(q)=dB_p/dq=120-6q$. At $q=10$, $MPB=120-6(10)=120-60=\$60$ per additional service unit.

Step 2: Find MEB. $MEB(q)=dE/dq=12$. Therefore, at $q=10$, $MEB=\$12$ per additional service unit.

Step 3: Find MSB. $MSB(10)=60+12=\$72$ per additional service unit.

Check using total social benefit. $B_s(q)=B_p(q)+E(q)=120q-3q^2+12q=132q-3q^2$. Thus $dB_s/dq=132-6q$, and at $q=10$, $MSB=132-60=\$72$. This agrees with the sum of the private and external marginal benefits.

Putting Costs and Benefits Together

A complete description may include both an external cost and an external benefit. Calculate each curve separately at the same activity level; do not combine MEC with MPB or MEB with MPC. Cost effects belong in the comparison between MPC and MSC, while benefit effects belong in the comparison between MPB and MSB. Keeping those comparisons separate makes it easier to identify which private curve omits which effect.

Worked Example: Comparing All Four Marginal Curves

A hypothetical service has marginal private cost $MPC(q)=6+4q$, marginal external cost $MEC(q)=4$, marginal private benefit $MPB(q)=30-2q$, and marginal external benefit $MEB(q)=3$. Find all four marginal values at $q=0.5$, state each social curve, and check the gaps.

Method. Evaluate each private and external marginal curve at the stated activity level, then add cost components together and benefit components together. This avoids mixing costs with benefits or evaluating one curve at a different $q$.

Step 1: Evaluate the costs. $MPC(0.5)=6+4(0.5)=6+2=8$. The marginal external cost is constant, so $MEC(0.5)=4$. Thus $MSC(0.5)=8+4=12$.

Step 2: Evaluate the benefits. $MPB(0.5)=30-2(0.5)=30-1=29$. The marginal external benefit is constant, so $MEB(0.5)=3$. Thus $MSB(0.5)=29+3=32$.

Step 3: Check the vertical gaps. $MSC-MPC=12-8=4=MEC$, and $MSB-MPB=32-29=3=MEB$. The values are consistent, with costs and benefits each measured per additional unit at $q=0.5$.

Common Mistakes and Why the Distinction Matters

Using a total effect as a marginal effect. If total external damage is $D(q)$, MEC is the change in $D$ for an additional unit, or $D'(q)$ in a differentiable model. It is not generally $D(q)$.
Using an average effect in place of a marginal one. Dividing total external damage by $q$ gives an average amount per unit, not necessarily the cost of the next unit. These quantities are equal only under special conditions, such as a constant marginal external cost starting from zero damage.
Adding the wrong components. The social cost curve is MPC plus MEC; the social benefit curve is MPB plus MEB. Adding a benefit to a cost does not produce either social curve.
Evaluating at different activity levels. The vertical gap is meaningful only when the private and social values are compared at the same $q$.

For example, suppose MPC is $6+4q$, MEC is $4$, and $q=0.5$. A mistaken calculation might report $MPC=10$, then $MSC=10+4=14$. The error is evaluating $6+4(0.5)$ as 10. The correct arithmetic is $6+4(0.5)=6+2=8$, so $MSC=8+4=12$. Checking the gap gives $12-8=4$, exactly the stated MEC.

This distinction matters because private decisions reflect private costs and benefits, while an assessment of effects on everyone must account for external marginal effects as well. Identifying which curves differ is a necessary first step in analyzing externalities. The calculations here define and compare those curves; they do not by themselves determine a particular policy or establish an efficient activity level.

Key takeaway: At each activity level, add marginal external costs to MPC to obtain MSC, and add marginal external benefits to MPB to obtain MSB. Compare curves at the same quantity, and use marginal—not total or average—external effects.

Check Your Understanding

For each question, keep the activity level and units consistent, and distinguish marginal from total effects.

A hypothetical bakery’s total private cost is $C_p(q)=25q+2q^2$ dollars per day, and total external damage is $D(q)=3q^2$ dollars per day. Find MPC, MEC, and MSC at $q=4$. (Answer: MPC $=\$41$, MEC $=\$24$, MSC $=\$65$ per additional unit.)
A neighborhood program has total private benefit $B_p(q)=80q-2q^2$ and total external benefit $E(q)=5q$, in dollars per week. Find MPB, MEB, and MSB at $q=10$. (Answer: MPB $=\$40$, MEB $=\$5$, MSB $=\$45$ per additional unit.)
At $q=0.5$, a hypothetical producer has $MPC(q)=6+4q$ and $MEC(q)=4$, both in dollars per unit. Calculate the two marginal costs and verify the external-cost gap. (Answer: MPC $=\$8$, MEC $=\$4$, MSC $=\$12$; the gap is $12-8=\$4$ per additional unit.)
A service has $MPB(q)=50-3q$ and $MEB(q)=2q$, in dollars per unit. At $q=6$, calculate MPB and MSB, and state their vertical gap. (Answer: MPB $=\$32$, MEB $=\$12$, MSB $=\$44$; the gap is $\$12$ per additional unit.)
Total private cost is $C_p(q)=10q+q^2$, total external damage is $D(q)=q^2$, total private benefit is $B_p(q)=70q-2q^2$, and total external benefit is $E(q)=6q$, all measured in dollars per period. Find MPC, MEC, MSC, MPB, MEB, and MSB at $q=5$. (Answer: MPC $=\$20$, MEC $=\$10$, MSC $=\$30$, MPB $=\$50$, MEB $=\$6$, and MSB $=\$56$ per additional unit.)

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