Leontief Price Model
Also known as: I-O Price Model, Dual Input-Output Model, Cost-Push Price Model, Input-Output Price Equation
The Leontief price model is the cost-side dual of the quantity input-output system: instead of asking how much each sector must produce to meet final demand, it asks what unit price each sector must charge to cover its intermediate-input costs plus its primary-input (value-added) payments. Solving the dual equation p' = p'A + v' gives p' = v'(I − A)^{-1}, so the same Leontief inverse that propagates quantities also propagates costs, making the model the standard tool for tracing how a change in wages, taxes, or imported-input prices pushes through the entire price structure.
Key highlights
- Provides an exact, economy-wide map of how any primary-input cost shock propagates into final prices, including all indirect channels.
- Uses the same Leontief inverse as the quantity model, so a single inversion serves both quantity and price analysis (duality).
- Ideal for tax-incidence and carbon-pricing studies because it shows full cost pass-through sector by sector.
- Transparent and additive: each price decomposes into the value-added contributions of every upstream sector.
Intuition
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How it works
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When to use it
Use the Leontief price model when you want to trace how a change in primary-input costs — wages, profit margins, indirect taxes, tariffs, or imported-input prices — flows through inter-industry linkages to change the prices of all goods and services. It is the standard input-output approach for cost-push inflation analysis, carbon-tax and energy-price pass-through studies, and tax-incidence work. Like the quantity model it assumes fixed technical coefficients and no behavioral substitution, so it captures full pass-through of costs under unchanged technology and is best for short-run incidence questions rather than situations where firms substitute inputs or absorb costs in margins.
Strengths & limitations
- Provides an exact, economy-wide map of how any primary-input cost shock propagates into final prices, including all indirect channels.
- Uses the same Leontief inverse as the quantity model, so a single inversion serves both quantity and price analysis (duality).
- Ideal for tax-incidence and carbon-pricing studies because it shows full cost pass-through sector by sector.
- Transparent and additive: each price decomposes into the value-added contributions of every upstream sector.
- Assumes complete, instantaneous cost pass-through with fixed margins and no input substitution in response to price changes.
- Holds technical coefficients constant, so it cannot capture the very substitution that price changes are meant to induce.
- Treats value-added components as exogenous, ignoring feedback from prices to wages, profits, and demand.
- Inherits the coarse sectoral aggregation and base-year staleness of the underlying input-output table.
Common pitfalls
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Applications
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Frequently asked
How does the price model relate to the quantity input-output model?
They are duals sharing the same coefficient matrix A and the same Leontief inverse (I − A)^{-1}. The quantity model solves x = (I − A)^{-1} f to find outputs that satisfy final demand; the price model solves p' = v'(I − A)^{-1} to find unit prices that cover value-added costs. Quantities are propagated by reading the inverse down its columns, prices by reading it along its rows.
What does the solution p' = v'(I − A)^{-1} mean economically?
It says each sector's equilibrium unit price equals the sum of value-added (primary-input) costs incurred at every stage of its supply chain, weighted by the total-requirements coefficients. In other words, the price of a good fully reflects the wages, profits, and taxes embodied directly and indirectly throughout all the inputs used to make it.
Can the price model predict how much firms will raise prices?
It predicts full, exact cost pass-through under the assumptions of fixed coefficients and constant margins — the maximum, mechanical price response if all cost increases are passed forward. Actual price changes can be smaller if firms absorb costs in margins or substitute inputs, behaviors the linear model does not capture, so its results are best read as a cost-incidence benchmark.
Sources
- 1.Miller, R. E., & Blair, P. D. (2009). Input-Output Analysis: Foundations and Extensions (2nd ed.). Cambridge University Press.ISBN 9780521739023
- 2.Leontief, W. W. (1936). Quantitative input and output relations in the economic system of the United States. The Review of Economics and Statistics, 18(3), 105–125.
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ScholarGate. (2026, June 22). Leontief Price Model. ScholarGate. https://scholargate.app/economics/leontief-price-model