Optimal Consumption Calculator

Enter each good's marginal utility and price, and the calculator reports both ratios, their relative gap, whether they match within your chosen tolerance, and which good the consumer should buy more of when they do not match. The result is a directional economic diagnosis, not a complete demand solution, because income, quantities, and the full utility function are not supplied.

What the MRS and price-ratio condition means

For two goods X and Y, the marginal rate of substitution used here is MUx divided by MUy. It describes how much marginal utility the consumer gains from X relative to Y at the current bundle. The market price ratio is Px divided by Py. It describes how many units of Y's price must be given up to pay for one unit of X. At an interior optimum, these two trade-offs align: MUx/MUy = Px/Py. The same condition can also be written as equal marginal utility per currency unit, MUx/Px = MUy/Py. When the ratios match, moving a small amount of spending from one good to the other does not improve utility at the margin. This calculator evaluates that equality with positive marginal utilities and positive prices. It does not assume that the submitted values are quantities, and it does not calculate a budget line. Instead, it focuses precisely on the local optimality test economists use after marginal utilities at a candidate bundle are known.

How to read the buy-more recommendation

If MRS is greater than Px/Py, good X provides more marginal utility relative to good Y than the market requires for X's relative cost. Equivalently, marginal utility per currency unit is higher for X. The direction of improvement is therefore to buy more X and less Y while keeping spending constrained. Under the usual assumption of diminishing marginal utility, that reallocation lowers MRS toward the price ratio. If MRS is below Px/Py, the reverse applies: Y currently offers the higher marginal return per currency unit, so the consumer should buy more Y and less X. The boolean output fields make this direction explicit, while the recommendation states it in a sentence. “Buy more” describes a marginal reallocation, not an unlimited purchase. As the bundle changes, marginal utilities generally change too, so the ratios should be recomputed. The endpoint never claims an exact new quantity because that would require the utility function, current quantities, income, and the budget constraint.

Tolerance, assumptions, and boundary solutions

Exact equality is uncommon when marginal utilities are estimated, surveyed, or rounded. The tolerance input therefore defines a relative band around the price ratio. Its default value of 0.01 treats a gap of up to one percent in either direction as a match. Set tolerance to zero when exact arithmetic equality is required, or increase it when the underlying marginal-utility estimates are noisy. The returned relative_gap is positive when MRS exceeds the price ratio and negative when it falls below; gap_percent expresses the same comparison in percentage points for easier reporting. This is an interior-optimum test and relies on differentiable preferences, positive prices, positive marginal utilities, and the ability to adjust both goods. A corner solution may be correct when one good is not consumed, preferences are not smooth, goods are indivisible, or constraints prevent marginal reallocation. The calculator also does not test whether the budget is exhausted or whether the candidate bundle is globally optimal. Use its result as a focused first-order condition check within a broader consumer-choice analysis.