Consumer Choice with Three Goods

Why Three-Good Choice Requires Three Marginal Comparisons

When a consumer chooses between two goods, an interior optimum balances the marginal utility gained from spending on one good against the marginal utility gained from spending on the other. With three goods, the same idea applies, but the consumer must compare all three uses of the budget. The relevant comparison is not marginal utility alone: each marginal utility must be measured per dollar spent.

This tutorial develops that three-good condition and shows how to combine it with the budget constraint. The method is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. It assumes the consumer spends a fixed income on three goods and chooses a positive quantity of each.

Definition: Let \(x,y,z\) be quantities of three goods, with positive prices \(p_x,p_y,p_z\), and let income be \(I>0\). The consumer chooses a feasible bundle satisfying \(p_xx+p_yy+p_zz\leq I\) to maximize utility \(U(x,y,z)\). At an interior choice, \(x,y,z>0\). If utility is differentiable, write the marginal utilities as \(MU_x=\partial U/\partial x\), \(MU_y=\partial U/\partial y\), and \(MU_z=\partial U/\partial z\).

The Interior Conditions for Three Goods

Assume prices are fixed and positive, utility is differentiable, and the consumer has locally nonsatiated preferences, so more utility is preferred to spending income without using it. At an interior optimum, the budget is exhausted and each good has the same marginal utility per dollar:

p_xx+p_yy+p_zz=I, \qquad \frac{MU_x}{p_x} = \frac{MU_y}{p_y} = \frac{MU_z}{p_z} =\lambda.

The common value \(\lambda\) is the marginal utility associated with one additional dollar of budget in the constrained optimization problem. The key operational point is that there are two independent equalities among the three marginal-utility-per-dollar ratios, together with the budget equation. These three equations can determine the three quantities.

Why Marginal Utility per Dollar Must Be Equal

Consider a small change that keeps total spending fixed. If the consumer spends a small amount \(\varepsilon\) more on good \(x\), that amount must come from another good, say \(y\). The quantity of \(x\) rises by \(\varepsilon/p_x\), while the quantity of \(y\) falls by \(\varepsilon/p_y\). The resulting first-order change in utility is

\Delta U \approx MU_x\frac{\varepsilon}{p_x} - MU_y\frac{\varepsilon}{p_y} = \varepsilon\left(\frac{MU_x}{p_x}-\frac{MU_y}{p_y}\right).

If \(MU_x/p_x\) were greater than \(MU_y/p_y\), this budget-neutral transfer would increase utility, contradicting optimality. If it were smaller, transferring the spending in the opposite direction would increase utility. Thus the two ratios must be equal. Applying the same argument to the pairs \(x,z\) and \(y,z\) gives equality across all three goods.

The Lagrange multiplier method gives the same result. For an exhausted budget, form

\mathcal{L} = U(x,y,z)+\lambda\left(I-p_xx-p_yy-p_zz\right).

At an interior stationary point, the derivatives with respect to each quantity are zero:

MU_x-\lambda p_x=0,\qquad MU_y-\lambda p_y=0,\qquad MU_z-\lambda p_z=0.

Dividing each equation by its positive price gives \(MU_x/p_x=MU_y/p_y=MU_z/p_z=\lambda\). These first-order conditions identify an interior stationary candidate. If utility is concave, such a candidate is a global maximum; if utility is strictly concave, it is the unique maximum. Without suitable concavity, the first-order conditions alone do not guarantee a global optimum.

Worked Example: A Three-Good Cobb-Douglas Choice

A consumer has utility \(U(x,y,z)=x^{1/2}y^{1/3}z^{1/6}\), income \(I=120\), and prices \(p_x=2\), \(p_y=3\), and \(p_z=1\). Find the interior optimum and verify it. Use the first-order conditions and budget constraint; the common mistake avoided is using the utility exponents as quantities rather than as spending shares.

Step 1: Find the marginal utilities. Differentiating gives \(MU_x=\frac12 U/x\), \(MU_y=\frac13 U/y\), and \(MU_z=\frac16 U/z\). At an interior optimum, set each marginal utility per dollar equal to the same \(\lambda\):

\frac{U}{4x} = \frac{U}{9y} = \frac{U}{6z} =\lambda.

Since \(U>0\), equating the first ratio with the second gives \(9y=4x\), or \(y=4x/9\). Equating the first with the third gives \(6z=4x\), or \(z=2x/3\).

Step 2: Use the budget. Substitute these quantity relationships into \(2x+3y+z=120\):

2x+3\left(\frac{4x}{9}\right)+\frac{2x}{3} = 2x+\frac{4x}{3}+\frac{2x}{3} =4x=120.

Therefore \(x=30\), \(y=4(30)/9=40/3\), and \(z=2(30)/3=20\). The optimal bundle is \((30,40/3,20)\).

Step 3: Check the conditions. Spending is \(2(30)+3(40/3)+1(20)=60+40+20=120\), so the full budget is used. Because the utility exponents sum to one, the marginal-utility-per-dollar ratios are \(U/(4x)\), \(U/(9y)\), and \(U/(6z)\). At the solution, their denominators are \(120,120,120\), respectively. All three ratios are equal. The utility function is concave on the positive quantities, so this interior candidate is a global optimum.

Solving by Equalizing Marginal Utility per Dollar

Not every utility function needs to be handled by writing three equations in full. In some cases, each first-order condition directly expresses a quantity in terms of the common value \(\lambda\). The budget then determines \(\lambda\), after which the quantities follow.

Check the setup.
Identify the three prices, income, utility function, and the assumption that the optimum has positive quantities and an exhausted budget.

Differentiate.
Find \(MU_x\), \(MU_y\), and \(MU_z\), then divide each by its own good’s price.

Set the ratios equal.
Use two independent equalities to relate the quantities, or express all three quantities in terms of \(\lambda\).

Apply the budget and verify.
Solve for the quantities, check total expenditure, and confirm that all three marginal utilities per dollar match.

Worked Example: Log Utility with Unequal Weights

Suppose \(U(x,y,z)=\ln x+2\ln y+\ln z\), income is \(I=96\), and prices are \(p_x=4\), \(p_y=2\), and \(p_z=6\). Find and check the interior optimum. The method is to solve the equal-ratio conditions for quantities in terms of \(\lambda\); this avoids the mistake of treating unequal utility weights as equal.

Step 1: Differentiate and divide by prices. The marginal utilities are \(1/x\), \(2/y\), and \(1/z\). Thus:

\frac{1}{4x}=\frac{2}{2y}=\frac{1}{6z}=\lambda.

Solving each equation gives \(x=1/(4\lambda)\), \(y=1/\lambda\), and \(z=1/(6\lambda)\).

Step 2: Substitute into the budget. The expenditure on each good is \(p_iq_i\), so:

4\left(\frac{1}{4\lambda}\right) +2\left(\frac{1}{\lambda}\right) +6\left(\frac{1}{6\lambda}\right) = \frac{1}{\lambda}+\frac{2}{\lambda}+\frac{1}{\lambda} =\frac{4}{\lambda} =96.

Therefore \(\lambda=1/24\). Substitution gives \(x=6\), \(y=24\), and \(z=4\).

Step 3: Check the result. The expenditure is \(4(6)+2(24)+6(4)=24+48+24=96\). The marginal utilities per dollar are \(1/(6\cdot4)=1/24\), \(2/(24\cdot2)=1/24\), and \(1/(4\cdot6)=1/24\). The log utility is strictly concave for positive quantities, so the verified interior solution is the unique global optimum.

Worked Example: Square-Root Utility and a Direct Lambda Solution

A consumer’s utility is \(U(x,y,z)=2\sqrt{x}+3\sqrt{y}+\sqrt{z}\), with income \(I=32\) and prices \(p_x=1\), \(p_y=3\), and \(p_z=1\). Solve for the interior optimum. The method expresses each quantity in terms of \(\lambda\); the common error avoided is dividing marginal utility by a quantity rather than by its price.

Step 1: Write the conditions. The marginal utilities are \(1/\sqrt{x}\), \(3/(2\sqrt{y})\), and \(1/(2\sqrt{z})\). Dividing by the corresponding prices gives:

\frac{1}{\sqrt{x}} = \frac{1}{2\sqrt{y}} = \frac{1}{2\sqrt{z}} =\lambda.

Step 2: Express quantities in terms of \(\lambda\). Squaring the positive solutions gives \(x=1/\lambda^2\), \(y=1/(4\lambda^2)\), and \(z=1/(4\lambda^2)\).

Step 3: Use the budget.

\frac{1}{\lambda^2} +3\left(\frac{1}{4\lambda^2}\right) +\frac{1}{4\lambda^2} = \frac{2}{\lambda^2} =32.

Thus \(\lambda^2=1/16\), and since \(\lambda>0\), \(\lambda=1/4\). The quantities are \(x=16\), \(y=4\), and \(z=4\).

Step 4: Verify. Total spending is \(1(16)+3(4)+1(4)=16+12+4=32\). The three marginal utilities per dollar are \(1/4\), \((3/(2\cdot2))/3=1/4\), and \((1/(2\cdot2))/1=1/4\). Square-root utility is strictly concave for positive quantities, confirming that this is the unique global optimum.

Common Mistake: Equalizing Marginal Utilities Instead of Their Dollar Values

Equal marginal utilities are not generally the correct condition when prices differ. For the log-utility example, incorrectly setting \(1/x=2/y=1/z\) gives \(x=t\), \(y=2t\), and \(z=t\). Applying the budget would produce \(4t+2(2t)+6t=14t=96\), so \(t=48/7\). This bundle is feasible, but it is not optimal.

At that bundle, \(MU_x/p_x=7/192\), \(MU_y/p_y=7/96\), and \(MU_z/p_z=7/288\), which are unequal. The correct comparison accounts for the number of dollars needed to buy each unit: use \(MU_x/p_x=MU_y/p_y=MU_z/p_z\), not \(MU_x=MU_y=MU_z\). A good with a higher price may have a different marginal utility at the optimum while still delivering the same marginal utility per dollar.

Key takeaway: For a differentiable, interior three-good optimum with positive prices and an exhausted budget, equate all three marginal utilities per dollar, then use the budget equation. Verify both the expenditure total and each ratio; satisfying the budget alone does not establish optimality.

Check Your Understanding

For each problem, find or assess the three-good interior choice using the budget and marginal-utility-per-dollar conditions.

A consumer has \(U=x^{1/2}y^{1/4}z^{1/4}\), income \(80\), and prices \((2,1,4)\). Find the interior bundle. (Answer: \(x=20,\ y=20,\ z=5\).)
For \(U=\ln x+\ln y+2\ln z\), income is \(132\), and prices are \((3,6,2)\). Find the optimal quantities. (Answer: \(x=11,\ y=5.5,\ z=33\).)
A consumer has \(U=3\sqrt{x}+\sqrt{y}+2\sqrt{z}\), prices \((1,2,1)\), and income \(54\). Find the interior optimum. (Answer: \(x=36,\ y=1,\ z=16\).)
For \(U=\ln x+2\ln y+\ln z\), prices are \((2,1,4)\). At the bundle \((10,20,5)\), calculate each marginal utility per dollar and state the direction of a budget-neutral shift that would improve utility. (Answer: the ratios are \(0.05,\ 0.10,\ 0.05\); shift spending from \(x\) or \(z\) toward \(y\).)
For \(U=xyz\), prices are \((2,3,1)\), and income is \(60\). A proposed bundle is \((10,10,10)\). Check feasibility and the marginal-utility-per-dollar condition. (Answer: it is feasible since spending is \(60\), but it is not optimal; the ratios are \(50,\ 100/3,\ 100\), so they are unequal.)

What You'll Learn