Why Relative Prices Matter
A consumer’s budget does not change only when income changes. A change in the price of one good also changes the trade-off the budget permits: buying more of that good requires giving up more of the other. Whether the best bundle shifts smoothly or jumps to a boundary depends on the shape of preferences. With perfect substitutes, a consumer generally chooses one good alone; with smooth, strictly concave preferences, the consumer may choose both goods until a boundary condition is reached.
Let \(x,y\geq 0\) denote quantities, let \(p_x,p_y>0\) be their prices, and let income be \(m>0\). A feasible bundle satisfies \(p_xx+p_yy\leq m\). The relative price of \(x\) in units of \(y\) is \(p_x/p_y\): it is the number of units of \(y\) whose expenditure equals the cost of one unit of \(x\). The comparisons below hold income and the price of \(y\) fixed while varying \(p_x\).
Perfect Substitutes: Compare Utility per Dollar
Suppose preferences are represented by \(u(x,y)=ax+by\), with \(a,b>0\). One dollar spent on \(x\) produces \(a/p_x\) units of utility, while one dollar spent on \(y\) produces \(b/p_y\). Thus the optimum depends on which good gives more utility per dollar. If the two amounts are equal, every fully spent bundle is optimal.
Worked Example: A Price Change Reverses the Preferred Endpoint
A consumer has utility \(u(x,y)=6x+3y\), income \(m=12\), and price \(p_y=2\). Compare the optimal bundles when \(p_x=3\), \(p_x=4\), and \(p_x=5\).
Method. Compare utility per dollar for each good. With \(p_y=2\), good \(y\) gives \(3/2=1.5\) utility units per dollar.
When \(p_x=3\), good \(x\) gives \(6/3=2\) utility units per dollar, which exceeds \(1.5\). Spend all income on \(x\): \(x=12/3=4\), \(y=0\). The resulting utility is \(6(4)+3(0)=24\).
When \(p_x=4\), good \(x\) gives \(6/4=1.5\) utility units per dollar, equal to good \(y\). The budget equation is \(4x+2y=12\), or \(y=6-2x\), with \(0\leq x\leq3\). For every bundle on this segment, \(u=6x+3(6-2x)=18\). The consumer is indifferent among all these bundles, including both endpoints.
When \(p_x=5\), good \(x\) gives \(6/5=1.2\) utility units per dollar, less than \(1.5\). Spend all income on \(y\): \(y=12/2=6\), \(x=0\). Utility is \(6(0)+3(6)=18\).
Check. The selected bundles spend \(3(4)=12\), \(4x+2y=12\), and \(2(6)=12\), respectively. The middle case is a tie, not a unique bundle. The common mistake is to compare the coefficients \(6\) and \(3\) without dividing by prices; the relevant comparison is utility per dollar.
Smooth Preferences: Interior Choices and Boundary Tests
With smooth, strictly concave preferences, the consumer generally balances marginal utility per dollar rather than choosing the good with the larger average utility per dollar. If both quantities are positive, the tangency condition \(MU_x/p_x=MU_y/p_y\) applies. As a price changes, that equation predicts a candidate interior bundle. The candidate is an interior optimum only if both quantities are positive and the budget is satisfied.
A useful boundary test follows by considering a consumer who currently consumes only \(y\). A small increase in \(x\) costs \(p_x\); to stay on the budget, the consumer must reduce \(y\) by \(p_x/p_y\) units per unit of \(x\). The utility change per small unit of \(x\), evaluated at \(x=0\), is therefore \(MU_x-(p_x/p_y)MU_y\). If this is positive, moving away from the boundary improves utility, so the \(y\)-only bundle cannot be optimal. If it is nonpositive and utility is concave, the boundary may be optimal; the complete first-order conditions determine the result.
This condition is the boundary counterpart of tangency. In the interior, marginal utility per dollar is equal across goods. At a zero-consumption boundary, the marginal utility per dollar of the excluded good must not be greater than that of the good being consumed. For differentiable, concave utility, satisfying the full optimality conditions is sufficient for a global maximum. Strict concavity makes that maximum unique. Strict concavity alone is not enough to make an arbitrary bundle satisfying tangency and the budget optimal: monotonicity and the relevant boundary and multiplier conditions matter.
Worked Example: A Smooth Optimum Reaches a Boundary
A consumer has utility \(u(x,y)=\ln(1+x)+\ln(1+y)\), income \(m=8\), and \(p_y=1\). Find the optimum at \(p_x=2\), and determine when the consumer instead chooses \(x=0\) as \(p_x\) rises.
Method. Utility is increasing and strictly concave on \(x,y\geq0\), so the budget binds. For an interior bundle, \(MU_x=1/(1+x)\) and \(MU_y=1/(1+y)\), and the tangency condition is \(MU_x/MU_y=p_x/p_y\). Here it becomes \((1+y)/(1+x)=p_x\).
At \(p_x=2\), tangency gives \(1+y=2(1+x)\), hence \(y=1+2x\). The binding budget is \(2x+y=8\). Substitution gives \(2x+1+2x=8\), so \(4x=7\) and \(x=7/4\). Then \(y=1+2(7/4)=9/2\). Both quantities are positive, and spending is \(2(7/4)+9/2=7/2+9/2=8\), as required. Strict concavity and the full optimality conditions make this the unique optimum.
To find the price at which the consumer reaches \(x=0\), evaluate the boundary test at \(x=0\), \(y=8\). The marginal-utility ratio there is \((1+y)/(1+x)=9\). The boundary condition is therefore \(9\leq p_x\), or \(p_x\geq9\). At \(p_x=9\), the optimum is \(x=0,y=8\), with equality in the boundary condition; for \(p_x>9\), the same \(y\)-only bundle is optimal.
For prices \(0<p_x<9\), the interior equation and budget give \(y=p_x(1+x)-1\) and \(p_xx+y=8\). Combining them yields \(2p_xx+p_x-1=8\), so \(x=(9-p_x)/(2p_x)>0\). The corresponding \(y=(p_x+7)/2>0\). Thus the interior solution is feasible throughout this price range and meets the boundary precisely at \(p_x=9\).
Check. At the stated price \(p_x=2\), the bundle \((1,4)\) would spend \(2(1)+4=6\), not the full income of \(8\), and it fails the tangency equation. Solving both equations gives \((7/4,9/2)\). The common mistake is to use tangency without checking the budget and positivity of both quantities.
A Relative-Price Change Need Not Create a Corner
Not every smooth preference pattern has a price threshold at which a good disappears. For example, consider \(u(x,y)=\ln x+3\ln y\), with \(x,y>0\), positive prices, and positive income. The utility is increasing and strictly concave; its marginal utility becomes unbounded as either good approaches zero. The optimum is interior for any positive prices and income. This contrasts with the shifted logarithms in the previous example, whose marginal utilities at zero are finite and permit a boundary optimum.
Worked Example: Relative Prices Change an Interior Bundle
A consumer has utility \(u(x,y)=\ln x+3\ln y\), income \(m=120\), and \(p_y=2\). Compare the optimal bundles at \(p_x=1\) and \(p_x=4\).
Method. Since both goods are consumed at an interior optimum, \(MU_x=1/x\), \(MU_y=3/y\), and tangency requires \((1/x)/(3/y)=p_x/p_y\). Rearranging gives \(y=3(p_x/p_y)x\). Together with the binding budget, this determines the bundle.
At \(p_x=1\), the tangency equation gives \(y=(3/2)x\). Substituting into \(x+2y=120\) gives \(x+3x=120\), so \(x=30\) and \(y=45\). Spending is \(1(30)+2(45)=120\).
At \(p_x=4\), tangency gives \(y=6x\). The budget \(4x+2y=120\) becomes \(4x+12x=120\), so \(x=7.5\) and \(y=45\). Spending is \(4(7.5)+2(45)=30+90=120\).
The higher relative price reduces the chosen quantity of \(x\) and raises its share of expenditure only through the changed price; the quantity of \(y\) happens to remain \(45\) in these particular calculations. Both bundles are interior and satisfy tangency. The common mistake is to assume every price increase must produce a corner; here the logarithmic preferences keep both goods positive at every positive price.
A Reliable Comparison Across Price Cases
When a problem asks how an optimum changes as a relative price changes, use a consistent sequence. Identify the utility shape first: linear preferences call for a utility-per-dollar comparison, while smooth preferences call for marginal conditions. For a smooth candidate, solve tangency together with the binding budget, then check that the quantities are nonnegative. If a candidate reaches zero, test the relevant boundary condition rather than extending the interior equation into an infeasible region.
Use positive prices and income, nonnegative quantities, and increasing utility when concluding that the budget binds. Check concavity before relying on first-order conditions for a global optimum.
For linear utility, compare \(a/p_x\) with \(b/p_y\). For smooth interior preferences, set \(MU_x/MU_y=p_x/p_y\).
Substitute into the budget, verify signs, and apply the marginal-utility inequality if one good is zero. Treat equality at a threshold as a possible boundary optimum or tie, as appropriate.
A frequent exam trap is to report an interior tangency solution even when it gives a negative quantity, or to claim that strict concavity alone validates a candidate. A candidate must be feasible and satisfy the appropriate optimality conditions. At a price threshold, equality may mean the interior bundle has just reached the boundary; for perfect substitutes, equality instead makes an entire budget segment optimal.
Check Your Understanding
For each problem, identify the relevant price comparison or optimality condition before calculating.
- A consumer has \(u(x,y)=5x+2y\), income \(18\), \(p_x=2\), and \(p_y=3\). Which good is chosen, and what is the optimal bundle? (Answer: \(x=9,\ y=0\).)
- A consumer has \(u(x,y)=6x+3y\), income \(12\), \(p_x=4\), and \(p_y=2\). Describe the set of optimal bundles. (Answer: Every bundle satisfying \(4x+2y=12\), with \(x,y\geq0\), is optimal.)
- A consumer has \(u(x,y)=\ln(1+x)+\ln(1+y)\), income \(8\), \(p_y=1\), and \(p_x=3\). Find the optimal bundle. (Answer: \((x,y)=(1,5)\).)
- A consumer has \(u(x,y)=\ln x+3\ln y\), income \(80\), \(p_x=2\), and \(p_y=1\). Find the optimal bundle and verify the budget. (Answer: \((x,y)=(40/3,40)\); spending is \(2(40/3)+40=200/3\), so this answer would not exhaust income. Recheck the budget and give the corrected bundle.)
- For \(u(x,y)=\ln(1+x)+\ln(1+y)\), income \(10\), and \(p_y=1\), find the price threshold at which the optimum reaches \(x=0\), then give the optimum at \(p_x=12\). (Answer: The threshold is \(p_x=11\); at \(p_x=12\), \((x,y)=(0,10)\).)