Price-Consumption Curves and Demand Curves

From Changing Prices to a Demand Curve

A consumer’s chosen bundle depends on prices, income, and preferences. If income and preferences are held fixed while the price of one good changes, the consumer may move among different optimal bundles. Following those bundles produces a price-consumption curve. Recording just the changing price and the chosen quantity of that good produces an individual demand curve.

These are related but different objects. A price-consumption curve is drawn in a goods-space graph, with quantities of two goods on the axes. An individual demand curve is drawn in a price-quantity graph, with the price of one good and the consumer’s chosen quantity of it on the axes. The conversion from one graph to the other is an organized way to read the consumer’s choices, not a claim that the two curves have the same shape.

The method is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. It requires careful attention to what is held fixed and to whether each chosen bundle is a unique optimum or one of several optimal bundles.

Definition: For two goods \(x\) and \(y\), a price-consumption curve is the set of utility-maximizing bundles \((x^*,y^*)\) as \(p_x\) varies, with income \(m\), the other price \(p_y\), and preferences held fixed. An individual demand curve for good \(x\) records the corresponding chosen quantities \(x^*\) against their prices \(p_x\). If multiple bundles are optimal at a price, demand at that price may be a set of quantities rather than a single quantity.

What Must Be Held Fixed

To trace a price-consumption curve, specify the consumer’s preferences and income, fix the price of good \(y\), and vary only the price of good \(x\). For each value of \(p_x\), solve the consumer’s utility-maximization problem subject to the budget constraint

p_xx+p_yy\leq m,\qquad x\geq 0,\quad y\geq 0.

Here \(x\) and \(y\) are quantities, \(p_x\) and \(p_y\) are prices per unit, and \(m\) is income available for these goods. Under the usual assumption that preferences are locally nonsatiated, the budget constraint binds at an optimum: if some income were unspent, the consumer could afford a preferred bundle. That assumption is not universal, so a problem that allows satiation or states otherwise requires checking whether the budget binds.

The optimal bundle at a given price is found by comparing feasible choices, including boundary choices. For a differentiable interior optimum, the familiar tangency condition can help identify the bundle. But a tangency alone is not a global-optimality check, and it does not apply directly at a corner or kink. The earlier material on individual demand and consumer choice provides the optimization tools; here the emphasis is on repeating the choice as price changes and organizing the results.

Why the Mapping Works

For each price \(p_x\), the consumer’s optimization problem produces an optimal bundle. The price-consumption curve collects those bundles in \((x,y)\)-space. To construct demand for \(x\), take each optimal bundle, keep its \(x\)-coordinate, and pair that quantity with the price that generated the bundle. Thus, if price \(p_x^{(1)}\) produces \((x_1^*,y_1^*)\), the demand graph includes \((p_x^{(1)},x_1^*)\).

For an interior solution, the link between price and the chosen bundle can also be seen from the tangency condition. Let \(U(x,y)\) be a differentiable utility representation, with marginal utilities \(MU_x\) and \(MU_y\). At an interior optimum on a binding budget, a small feasible movement along the budget line satisfies \(p_x\,dx+p_y\,dy=0\), so \(dy/dx=-p_x/p_y\). Along that line, the change in utility is

dU=MU_x\,dx+MU_y\,dy =\left(MU_x-MU_y\frac{p_x}{p_y}\right)dx.

At a differentiable interior maximum, the coefficient on \(dx\) must be zero: otherwise, a small move in one direction would increase utility. Therefore

\frac{MU_x}{MU_y}=\frac{p_x}{p_y}.

This condition explains why a price change can alter the optimal mix: it changes the relative price that the consumer balances against the marginal rate of substitution. It is a local condition, not a substitute for checking feasibility, boundaries, or global optimality.

Worked Example: Tracing an Interior Price-Consumption Curve

A consumer has utility \(U(x,y)=\sqrt{x}+\sqrt{y}\), income \(m=24\), and faces \(p_y=2\). Find the optimal bundles when \(p_x=1,2,\) and \(4\), then construct the corresponding demand observations for \(x\).

Method. The utility is strictly concave, so an interior solution satisfying the first-order condition is the unique maximum when it is feasible. The marginal utilities are \(MU_x=1/(2\sqrt{x})\) and \(MU_y=1/(2\sqrt{y})\). The tangency condition gives

\frac{MU_x}{MU_y}=\sqrt{\frac{y}{x}}=\frac{p_x}{2}, \qquad\text{so}\qquad y=\frac{p_x^2}{4}x.

Substitute this relationship into the binding budget. For any listed \(p_x\),

p_xx+2\left(\frac{p_x^2}{4}x\right)=24, \qquad x=\frac{48}{p_x(p_x+2)}, \qquad y=\frac{p_x^2}{4}x.

At \(p_x=1\), \(x=48/(1\cdot3)=16\), and \(y=(1/4)(16)=4\). Expenditure is \(1(16)+2(4)=24\).

At \(p_x=2\), \(x=48/(2\cdot4)=6\), and \(y=(4/4)(6)=6\). Expenditure is \(2(6)+2(6)=24\).

At \(p_x=4\), \(x=48/(4\cdot6)=2\), and \(y=(16/4)(2)=8\). Expenditure is \(4(2)+2(8)=24\).

The price-consumption curve therefore passes through \((16,4)\), \((6,6)\), and \((2,8)\) for these prices. To construct demand, pair each price with the bundle’s \(x\)-quantity: the observations are \((p_x,x^*)=(1,16),(2,6),(4,2)\). The budget checks confirm feasibility; strict concavity and the interior first-order condition confirm optimality. The common mistake avoided here is plotting \((x,y)\) as if it were the demand curve: those are price-consumption coordinates, not price-quantity coordinates.

Corner Choices and Ties

A price-consumption curve need not be smooth. It may contain boundary optima, and at some prices the consumer may have more than one optimal bundle. The individual demand curve must reflect the actual optimum at each price rather than an assumed smooth path. In particular, a single price does not imply a range of optimal quantities unless the consumer is truly indifferent among those quantities.

Worked Example: A Price Tie Is Not the Same as a Worse Choice

A consumer has utility \(U(x,y)=2x+y\), income \(m=10\), and \(p_y=1\). Find the optimal choice when \(p_x=1,2,\) and \(3\), and state the demand for \(x\) at each price.

Method. With linear utility, compare utility gained per dollar: \(x\) yields \(2/p_x\) utility units per dollar, while \(y\) yields \(1\) per dollar. If one good gives strictly more utility per dollar, spending on the other good cannot be optimal. If the ratios are equal, every affordable mix is optimal.

At \(p_x=1\), \(x\) gives \(2\) units per dollar and \(y\) gives \(1\). Spending all \(10\) on \(x\) gives \(x=10\), \(y=0\), and utility \(20\). Spending on \(y\) instead would give utility \(10\), so the optimum is \((10,0)\) and demand is \(x^*=10\).

At \(p_x=2\), both goods give \(1\) utility unit per dollar. Every point on the budget line \(2x+y=10\) is optimal: utility is \(2x+y=10\) at every such point. The possible optimal quantities of \(x\) range from \(0\) to \(5\), with \(y=10-2x\). Demand at \(p_x=2\) is therefore the interval \(0\leq x^*\leq5\), not a single quantity.

At \(p_x=3\), \(x\) gives \(2/3\) utility units per dollar, while \(y\) still gives \(1\). The unique optimum is to spend all \(10\) on \(y\), giving \((x,y)=(0,10)\), utility \(10\), and demand \(x^*=0\).

The price-consumption choices are \((10,0)\) at price \(1\), every bundle \((x,10-2x)\) with \(0\leq x\leq5\) at price \(2\), and \((0,10)\) at price \(3\). In the demand graph, the observations are \(x^*=10\) at \(p_x=1\), the interval \(0\leq x^*\leq5\) at \(p_x=2\), and \(x^*=0\) at \(p_x=3\). The trap is to treat a lower utility per dollar as equality: at price \(3\), the consumer is not indifferent, so mixtures are not optimal.

Using a Numerical Schedule

A schedule is a compact way to present the demand observations generated by a price experiment. Each row should state the changed price and the corresponding optimal bundle, so the reader can verify both the price-consumption curve and the demand quantity. The following example uses fixed-proportion preferences, for which optimal bundles lie along a kink. The purpose is to trace and map the choices, not to assume an interior tangency.

Worked Example: Building a Schedule from Fixed-Proportion Choices

A consumer’s utility is \(U(x,y)=\min(x,2y)\), income is \(m=30\), and \(p_y=2\). Find the optimal bundle and demand for \(x\) at \(p_x=1,2,\) and \(4\).

Method. Utility is limited by whichever of \(x\) and \(2y\) is smaller. Any bundle with excess of one good relative to the ratio \(x=2y\) spends income without increasing utility as much as a bundle that restores that ratio. Thus an optimum can be chosen with \(x=2y\). Substituting \(y=x/2\) into the binding budget gives

p_xx+2\left(\frac{x}{2}\right)=30, \qquad x=\frac{30}{p_x+1}, \qquad y=\frac{15}{p_x+1}.

At \(p_x=1\), \(x=30/2=15\) and \(y=15/2=7.5\). Expenditure is \(1(15)+2(7.5)=30\).

At \(p_x=2\), \(x=30/3=10\) and \(y=15/3=5\). Expenditure is \(2(10)+2(5)=30\).

At \(p_x=4\), \(x=30/5=6\) and \(y=15/5=3\). Expenditure is \(4(6)+2(3)=30\).

The price-consumption schedule consists of bundles \((15,7.5)\), \((10,5)\), and \((6,3)\). The corresponding demand observations are \((1,15),(2,10),(4,6)\). In each row, \(x=2y\) and the full budget is spent, which checks the fixed-proportion choice and its feasibility. A common mistake is to use the kink condition alone without checking the budget: the ratio identifies the efficient mix, while the budget determines its scale.

Reading and Checking the Two Curves

When drawing the price-consumption curve, put \(x\) and \(y\) quantities on the axes and mark the optimal bundles as \(p_x\) changes. When drawing demand for \(x\), put \(x^*\) on the horizontal axis and \(p_x\) on the vertical axis, following the conventional economics orientation. Transfer each bundle’s \(x\)-coordinate to the demand graph and label it with the price that generated the bundle. If the exercise instead specifies a different axis convention, follow that convention and label the axes explicitly.

A demand curve summarizes the consumer’s chosen quantity at alternative prices with other determinants fixed. The chosen quantity need not decline whenever price rises for every possible preference and income combination; the construction records the choices rather than assuming a direction in advance. Likewise, a discrete schedule supplies points, not automatically a complete smooth curve between them. Connecting points is a graphical convention unless the model gives a reason to specify choices at intermediate prices.

Keep income, preferences, and the other good’s price fixed throughout the price comparison.
For each price, find a global optimum, including corners and kinks where relevant.
Verify that each reported bundle satisfies the budget constraint.
Map the \(x\)-quantity together with its generating price; do not confuse bundle coordinates with price-quantity coordinates.
At a tie, report all optimal quantities or the demand correspondence, rather than inventing a unique choice.

Key takeaway: A price-consumption curve follows optimal bundles in goods space as one price varies. An individual demand curve is obtained by pairing each varied price with the chosen quantity of the good whose price changed. Correctly handling corners and ties is essential to both graphs.

Check Your Understanding

For each problem, hold all stated fixed conditions constant and map the resulting choice into demand for \(x\).

A consumer has \(U(x,y)=\sqrt{x}+\sqrt{y}\), income \(m=18\), and \(p_y=1\). Find the optimal bundle and \(x\)-demand at \(p_x=1\) and \(p_x=3\). (Answer: At \(p_x=1\), \((x,y)=(9,9)\) and \(x^*=9\); at \(p_x=3\), \((x,y)=(1.5,13.5)\) and \(x^*=1.5\).)
A consumer has \(U(x,y)=3x+y\), income \(m=12\), and \(p_y=2\). State demand for \(x\) at \(p_x=4,6,\) and \(8\), including any tie. (Answer: \(x^*=3\) at \(p_x=4\); \(0\leq x^*\leq2\) at \(p_x=6\); and \(x^*=0\) at \(p_x=8\).)
A consumer has \(U(x,y)=\min(2x,y)\), income \(m=20\), and prices \(p_x=2,p_y=1\). Find the optimal bundle, then find the optimal \(x\)-quantity if \(p_x\) rises to \(4\) with other conditions unchanged. (Answer: Initially \((x,y)=(5,10)\); after the price rise, \((x,y)=(10/3,20/3)\), so demand changes from \(5\) to \(10/3\).)
A consumer has \(U(x,y)=\sqrt{x}+\sqrt{y}\), income \(m=30\), \(p_y=3\), and \(p_x=2\). Find the optimal bundle and verify its expenditure. Then state the \(x\)-quantity when \(p_x=3\). (Answer: At \(p_x=2\), \((x,y)=(9,4)\), with expenditure \(2(9)+3(4)=30\); at \(p_x=3\), \(x^*=5\).)
A consumer has \(U(x,y)=x+4y\), income \(m=24\), and \(p_y=3\). Find demand for \(x\) at \(p_x=0.50,0.75,\) and \(1\), taking care to identify the tie price. (Answer: \(x^*=48\) at \(p_x=0.50\); \(0\leq x^*\leq32\) at \(p_x=0.75\); and \(x^*=0\) at \(p_x=1\).)

What You'll Learn