Chapter 1 · Preparation
Economic Models & Algebra for Economics
Parameters, endogenous variables, equilibrium, and stock-flow equations.
What you will learn
- Solve a market equilibrium
- Evaluate a parameterized model
- Relate a stock to its flows
Worked example
Solve a market equilibrium: follow the method step by step
Try the problem first, then compare your approach with the complete solution and the reasoning behind it.
Solve a market equilibrium
Demand is qᴰ = 40 − 2p and supply is qˢ = 4 + p. Find the equilibrium price p*.
qᴰ = 40 − 2p qˢ = 4 + p Equilibrium: qᴰ = qˢ
In plain English: Quantity demanded falls as price rises, quantity supplied rises as price rises, and the market is in equilibrium when the two quantities are equal.
Bold blue terms in this symbol guide include definitions. Hover, focus, or tap a term. Each row also shows how to read the notation aloud.
- p
- Price: The amount a buyer pays per unit of a good, service, or asset. When buyers and sellers face different prices, specify the price paid or the price received.How to read: price
- qᴰ
- Quantity demanded: The amount of a good or service buyers are willing and able to purchase at a particular price, holding other relevant factors fixed.How to read: quantity demanded
- qˢ
- Quantity supplied: The amount of a good or service sellers are willing and able to offer at a particular price, holding other relevant factors fixed.How to read: quantity supplied
- p*
- Equilibrium price: The price at which quantity demanded equals quantity supplied., where qᴰ = qˢHow to read: equilibrium price
Solution
- Set demand equal to supply: 40 − 2p = 4 + p.
- Subtract 4 from both sides: 36 − 2p = p.
- Add 2p to both sides: 36 = 3p.
- Divide both sides by 3: p* = 12.
- Check: qᴰ = 40 − 2(12) = 16 and qˢ = 4 + 12 = 16.
Visual check
Demand and supply
Quantity is on the horizontal axis and price is on the vertical axis, following the usual economics convention.
- Demand
- Supply
How to read this graph
- Horizontal axis: Quantity, q
- Vertical axis: Price, p
- Curves and lines: Demand, Supply
- Marked points: Open the exact-coordinate table below to read their values.
What the graph shows: Demand slopes downward and supply slopes upward. They intersect at quantity 16 and price 12.
Read the marked points as data
| Feature | Quantity, q | Price, p |
|---|---|---|
| Equilibrium (16, 12) | 16 | 12 |
Logic and solving tips
- An equilibrium value is found by imposing the equilibrium condition, then solving the resulting equation.
- Translate the word equilibrium into an equation first: the two quantity expressions must be equal at the same price.
Where economists use it
A market analyst sets estimated demand equal to supply to forecast the clearing price and sales volume after a crop shortage, tax, or change in production capacity.
Key idea: An equilibrium value is found by imposing the equilibrium condition, then solving the resulting equation.
Applied case study
The egg market after an avian-flu supply shock
Scenario
Economists solve demand and supply together to estimate how market-clearing price and quantity change after a shock. USDA reports that avian influenza left U.S. shell-egg inventories 29 percent lower at the end of 2022 than at the start and pushed wholesale prices higher.
Problem
Model the shock with qᴰ = 90 − 10p and post-shock qˢ = 15 + 5p. Find the new equilibrium price and quantity. Pre-shock supply was qˢ = 30 + 5p.
qᴰ = qˢ 90 − 10p = 15 + 5p
In plain English: Demand and supply are set equal to find the price and quantity at which buyers' planned purchases match sellers' planned sales.
Worked solution
- Set demand equal to post-shock supply: 90 − 10p = 15 + 5p.
- Collect terms: 75 = 15p, so p* = 5.
- Substitute into demand: q* = 90 − 10(5) = 40.
- For comparison, 90 − 10p = 30 + 5p gives the pre-shock values p = 4 and q = 50.
Economic interpretation: The modeled supply loss raises price from 4 to 5 and lowers quantity from 50 to 40, matching the direction observed in the egg market.
Source: USDA ERS: Avian influenza outbreaks reduced egg production The inventory decline and observed price direction are sourced. The linear curves are a teaching model, not USDA estimates.
Reasoning habits that help
- An equilibrium value is found by imposing the equilibrium condition, then solving the resulting equation.
- Variables can change within the model; parameters are treated as fixed for a particular calculation.
- A stock is measured at a point in time; its change is determined by flows measured per unit of time.
Where economists use these methods
- A market analyst sets estimated demand equal to supply to forecast the clearing price and sales volume after a crop shortage, tax, or change in production capacity.
- A policy analyst can insert a household's income into a linear consumption model to forecast spending and estimate how much an income change affects demand.
- A firm subtracts daily shipments from deliveries to predict whether inventory will accumulate or run out; the same accounting tracks debt, wealth, and physical capital.
Chapter vocabulary
Study 28 key terms
Say what the term means before opening it. Then compare your explanation with the definition, notation, and example.
Model
A deliberately simplified representation that preserves the features needed to answer a particular question.
Example: A supply-and-demand diagram leaves out many market details so it can isolate how price coordinates buyers and sellers.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 3Economic model
A set of assumptions and mathematical relationships used to explain or predict economic outcomes.
Example: A market model can combine demand, supply, and a market-clearing condition to determine price and quantity.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 3Cardinality
Numerical information about how much or how many, expressed on a scale where the size of differences is meaningful.
Example: An annual growth rate of 3 percent is cardinal information.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 4Ordinality
Information that ranks outcomes without claiming that the numerical gaps between ranks measure meaningful amounts.
Example: Ranking bundle A above B and B above C describes preferences without saying how much more A is preferred.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 4Variable
A quantity whose value may differ across observations, cases, or solutions of a model.
Example: Market price p is a variable when the model is designed to solve for it.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 4Exogenous variable
A variable whose value is taken as given from outside the model being solved.
Example: Consumer income may be exogenous in a demand model even though another model could explain income.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 4Endogenous variable
A variable whose value is determined within the model by its equations and assumptions.
Example: Equilibrium price and quantity are endogenous in a standard market model.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed pages 4-5Parameter
A value held fixed while a model is solved that describes its environment, technology, or behavioral relationship.
Example: In q^D = a - bp + cy, b measures how strongly quantity demanded responds to price.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed pages 4-5Demand functionq^D = D(p, y, ...)
A rule relating quantity demanded to its determinants, such as own price, income, and prices of related goods.
How to read: quantity demanded equals demand as a function of price, income, and other factors
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 5Supply functionq^S = S(p, ...)
A rule relating quantity supplied to its determinants, such as own price, input prices, and productive capacity.
How to read: quantity supplied equals supply as a function of price and other factors
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 5Behavioral equation
An equation expressing a hypothesis about how an economic agent, market, or economy responds to its determinants.
Example: A demand equation states how buyers' planned purchases respond to price and income.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed pages 5-6Equilibrium condition
A restriction that identifies a state in which the model's relevant plans or forces are mutually consistent.
Example: In a competitive goods market, q^D = q^S is the market-clearing condition.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 5Equilibrium
A solution in which all specified behavioral relationships and equilibrium conditions hold at the same time.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 5Homogeneous good
A good whose units are treated as identical by buyers, so sellers' units are perfect substitutes in the model.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 5Single-equation model
A model in which one equation is sufficient to express the relationship needed to answer the question.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 6Multiple-equation model
A model whose interacting unknowns must satisfy two or more equations simultaneously.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 6System of equations
A collection of equations considered together, with a solution required to satisfy every equation.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 6Interdependence
A relationship in which the outcome in one market, sector, or choice problem depends on outcomes elsewhere in the model.
Example: When demand for each of two goods depends on both prices, the two markets must be solved together.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 6Substitute goods
Goods for which a rise in the price of one increases demand for the other, holding other demand factors fixed.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 6Reduced-form solution
An expression for each endogenous variable written only in terms of exogenous variables and parameters.
Example: Solving simultaneous demand and supply equations for price produces a reduced form for equilibrium price.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 7Static model
A model that analyzes a state or period without explicitly tracing how past states lead to future states.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 7Dynamic model
A model that represents how economic variables evolve through time and links outcomes across dates.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed pages 7-8Flow
A quantity measured over a time interval, such as annual income, quarterly investment, or weekly production.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed pages 7-8Stock
A quantity measured at a particular instant, such as wealth, debt, inventory, or capital at the start of a period.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 8Stock-flow relationshipI_t = K_{t+1} - K_t
An accounting equation linking the change in a stock between dates to the flows that add to or subtract from it.
How to read: investment in period t equals next period's capital stock minus this period's capital stock
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 8Time subscriptK_t
A lower index attached to a variable to identify the date or period at which it is measured.
How to read: capital stock in period t
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 8Discrete time
A representation of time as separate periods, with variables indexed by dates such as t = 0, 1, 2, ... .
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 8Continuous time
A representation in which time can take every value in an interval and variables may change at each instant.
Textbook reference: Chapter 1, Section 1.1 What Is an Economic Model?, printed page 8Chapter foundations
Practice the chapter foundations
The textbook does not include a Chapter Review for Chapter 1, so MFE provides its own foundation practice for this preparation chapter.
Practice all 3 foundation problems