Chapter 14 · Test 3
Comparative Statics for Economics
Parameter changes, implicit differentiation, and the envelope theorem.
What you will learn
- Differentiate a reduced-form equilibrium
- Apply implicit differentiation
- Interpret the envelope theorem
Worked example
Differentiate a reduced-form equilibrium: follow the method step by step
Try the problem first, then compare your approach with the complete solution and the reasoning behind it.
Differentiate a reduced-form equilibrium
Equilibrium price is p* = (a − c)/(b + d). If b = 2 and d = 3, find ∂p*/∂a.
p* = (a − c)/(b + d)
In plain English: The equilibrium price equals the gap between the demand and supply intercepts divided by the sum of their price slopes.
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*
- Equilibrium price: The price at which quantity demanded equals quantity supplied.How to read: equilibrium price
- a
- a Demand: The relationship between a good's price and the quantity buyers are willing and able to purchase at each price, holding other relevant factors fixed.-Intercept: A point where a graph meets a coordinate axis. For y = mx + b, (0, b) is the y-intercept; an x-intercept satisfies f(x) = 0.How to read: a
- c
- a Supply: The relationship between a good's price and the quantity sellers are willing and able to offer at each price, holding other relevant factors fixed.-Intercept: A point where a graph meets a coordinate axis. For y = mx + b, (0, b) is the y-intercept; an x-intercept satisfies f(x) = 0.How to read: c
- b, d
- positive slope Parameter: A value treated as fixed while a model is solved, although it may be changed to compare different scenarios.How to read: b and d
Solution
- Differentiate the numerator a − c with respect to a: the derivative is 1.
- The denominator b + d is constant with respect to a.
- Therefore ∂p*/∂a = 1/(b + d).
- Substitute b = 2 and d = 3: ∂p*/∂a = 1/5 = 0.2.
Visual check
A parameter shift changes equilibrium
The graph uses changes, so the origin represents no change in either variable.
- Equilibrium response
How to read this graph
- Horizontal axis: Change in a, Δa
- Vertical axis: Change in p*, Δp*
- Curves and lines: Equilibrium response
- Marked points: Open the exact-coordinate table below to read their values.
What the graph shows: Every one-unit increase in the demand intercept changes equilibrium price by 0.2.
Read the marked points as data
| Feature | Change in a, Δa | Change in p*, Δp* |
|---|---|---|
| No parameter change | 0 | 0 |
Logic and solving tips
- A reduced form makes comparative statics direct: differentiate the solved endogenous variable with respect to a parameter.
- Differentiate the reduced form while holding every other parameter fixed and preserve the denominator.
Where economists use it
A policy analyst differentiates a solved market model to estimate how a tax, subsidy, demand shift, or production-cost change moves equilibrium price and quantity.
Key idea: A reduced form makes comparative statics direct: differentiate the solved endogenous variable with respect to a parameter.
Applied case study
Comparative statics after a wheat-demand shock
Scenario
Economists differentiate solved equilibria to predict responses to nearby shocks. USDA found that wheat-specific supply and demand shocks were the dominant drivers of the 2008 wheat price spike.
Problem
Let qᴰ = a - 2p and qˢ = 10 + 3p. Find ∂p*/∂a, then estimate the price effect when a rises from 60 to 70.
p* = (a - 10)/5 ∂p*/∂a = 1/5
In plain English: The same unknown values must satisfy every equation in the system at once, so the equations are solved jointly.
Worked solution
- Set demand equal to supply: a - 2p = 10 + 3p.
- Solve for price: p* = (a - 10)/5.
- Differentiate: ∂p*/∂a = 1/5 = 0.2.
- For Δa = 10, Δp* ≈ 0.2(10) = 2; direct prices are 10 and 12.
Economic interpretation: In this teaching market, a 10-unit outward demand shift raises the equilibrium price index by 2 units as both sides of the market absorb the shock.
Source: USDA Economic Research Service, Deconstructing Wheat Price Spikes The 2008 episode and USDA finding are sourced. The linear equations and shock size are teaching assumptions, not USDA estimates of the historical price change.
Reasoning habits that help
- A reduced form makes comparative statics direct: differentiate the solved endogenous variable with respect to a parameter.
- Implicit differentiation tracks how an endogenous solution must move to keep an equilibrium equation satisfied.
- The envelope theorem values a parameter change using its direct effect at the optimum, without first solving for the change in the optimizer.
Where economists use these methods
- A policy analyst differentiates a solved market model to estimate how a tax, subsidy, demand shift, or production-cost change moves equilibrium price and quantity.
- When an equilibrium condition cannot be neatly solved for price or output, implicit differentiation still reveals the direction and size of its response to income or policy changes.
- A welfare analyst uses the envelope theorem to value a small tax or price change from the existing optimum without resolving every induced choice from the beginning.
Chapter vocabulary
Study 24 key terms
Say what the term means before opening it. Then compare your explanation with the definition, notation, and example.
Endogenous variable
A variable whose equilibrium or optimal value is determined by the model being analyzed.
Example: Equilibrium income is endogenous in the simple Keynesian model.
Textbook reference: Chapter 14, Section Chapter introduction, printed page 529Exogenous variable
A variable whose value is taken as given from outside the model and whose change may shift the model's solution.
Example: Planned investment is exogenous in the chapter's simple income-determination model.
Textbook reference: Chapter 14, Section Chapter introduction, printed page 529Comparative statics
Analysis of how an equilibrium or optimum changes when an exogenous variable or parameter changes, comparing the old and new solutions without tracing the adjustment path.
Example: It can determine how an output tax changes a monopoly's profit-maximizing quantity.
Textbook reference: Chapter 14, Section Chapter introduction, printed page 529Keynesian multiplierdY*/dI = 1/(1-c)
The change in equilibrium income generated by a one-unit change in autonomous expenditure in the simple Keynesian model.
How to read: the change in equilibrium income per unit change in investment
Textbook reference: Chapter 14, Section 14.1 Introduction to Comparative Statics, printed page 531Fundamental equation
An equilibrium or optimality equation reduced to the endogenous variable of interest, the relevant exogenous variable, and model parameters.
Example: A monopoly's first-order condition can serve as the fundamental equation for studying how a tax changes optimal output.
Textbook reference: Chapter 14, Section 14.1 Introduction to Comparative Statics, printed pages 537, 539Implicit functionF(x*, alpha) = 0
An equation that defines an endogenous solution as a function of exogenous variables without algebraically isolating it.
How to read: F of equilibrium x and alpha equals zero
Textbook reference: Chapter 14, Section 14.1 Introduction to Comparative Statics, printed pages 531, 539Explicit solution functionx* = x(alpha)
A solved expression giving an endogenous equilibrium or optimum directly as a function of exogenous variables and parameters.
How to read: equilibrium x is a function of alpha
Textbook reference: Chapter 14, Section 14.1 Introduction to Comparative Statics, printed pages 531, 539Comparative-statics derivativedx*/dalpha
The derivative of an equilibrium or optimal value with respect to an exogenous variable, measuring the local response of the solution.
How to read: the change in equilibrium x with respect to alpha
Textbook reference: Chapter 14, Section 14.1 Introduction to Comparative Statics, printed pages 531, 539General comparative-statics method
A procedure that differentiates all equilibrium equations with respect to an exogenous variable and solves the resulting linear system for the responses of endogenous variables.
Example: The method can find how income and the interest rate jointly respond to fiscal or monetary changes in an IS-LM system.
Textbook reference: Chapter 14, Section 14.2 General Comparative-Statics Analysis, printed pages 540, 552Nonsingular equilibrium systemdet(F) != 0
An equilibrium system whose derivative matrix has a nonzero determinant, allowing local solution responses to be determined.
How to read: the determinant of F is nonzero
Textbook reference: Chapter 14, Section 14.2 General Comparative-Statics Analysis, printed pages 548, 552Normal goodpartial x_i/partial m > 0
A good whose ordinary demand rises when consumer income rises, holding prices fixed.
How to read: demand for good i increases with income
Textbook reference: Chapter 14, Section 14.2 General Comparative-Statics Analysis, printed pages 550-551Strictly inferior goodpartial x_i/partial m < 0
A good whose ordinary demand falls when consumer income rises, holding prices fixed.
How to read: demand for good i decreases with income
Textbook reference: Chapter 14, Section 14.2 General Comparative-Statics Analysis, printed pages 550-551Weakly inferior goodpartial x_i/partial m = 0
In the chapter's classification, a good whose ordinary demand is unchanged by a small increase in income at the point considered.
How to read: the partial derivative of demand for good i with respect to income equals zero
Textbook reference: Chapter 14, Section 14.2 General Comparative-Statics Analysis, printed pages 550-551Slutsky equation
A decomposition of an ordinary price effect on demand into a utility-compensated substitution effect and an income effect.
Example: It explains why a price increase usually reduces demand and identifies the exceptional Giffen case.
Textbook reference: Chapter 14, Section 14.2 General Comparative-Statics Analysis, printed pages 550-551Substitution effect
The change in demand caused by a change in relative prices when utility is held fixed.
Example: For the good whose own price rises, the compensated substitution effect is nonpositive under standard preferences.
Textbook reference: Chapter 14, Section 14.2 General Comparative-Statics Analysis, printed page 550Income effect
The part of a price effect caused by the change in the consumer's purchasing power at fixed money income.
Textbook reference: Chapter 14, Section 14.2 General Comparative-Statics Analysis, printed pages 550-551Giffen goodpartial x_i/partial p_i > 0
A strictly inferior good for which the adverse income effect of a price increase is strong enough to outweigh the substitution effect, causing demand to rise with its own price.
How to read: demand rises as the good's own price rises
Textbook reference: Chapter 14, Section 14.2 General Comparative-Statics Analysis, printed page 551Implicit function theorem
A theorem giving conditions under which equilibrium equations locally define endogenous variables as differentiable functions of exogenous variables.
Example: A nonzero determinant of the equilibrium derivative matrix is the key local condition in the general system.
Textbook reference: Chapter 14, Section 14.2 General Comparative-Statics Analysis, printed pages 552-553Value functionV(alpha) = f(x*(alpha), alpha)
A function giving the maximized or minimized objective value as a function of the problem's exogenous variables.
How to read: optimized value V as a function of alpha
Example: A cost function is the value function of a cost-minimization problem.
Textbook reference: Chapter 14, Section 14.3 The Envelope Theorem, printed page 555Envelope theoremdV/dalpha = partial L/partial alpha at the optimum
At an optimum, the derivative of the value function with respect to an exogenous variable equals the direct partial derivative of the Lagrange function with respect to that variable.
How to read: the value effect equals the direct Lagrangean effect at the optimum
Example: It finds how maximized profit changes with a tax without differentiating every optimal choice first.
Textbook reference: Chapter 14, Section 14.3 The Envelope Theorem, printed pages 554, 557Constraint constant
An exogenous term that shifts a constraint while leaving its functional relationship among choice variables otherwise unchanged.
Example: Total available labor is a constraint constant in an economywide labor-allocation problem.
Textbook reference: Chapter 14, Section 14.3 The Envelope Theorem, printed page 557Lagrange multiplier as a value effect
For a constraint constant, the appropriately signed optimal multiplier equals the marginal change in the optimized value from relaxing the constraint.
Example: A nonbinding constraint has a zero multiplier because a small relaxation does not change the optimized value.
Textbook reference: Chapter 14, Section 14.3 The Envelope Theorem, printed page 557Cost-curve envelope
The relationship in which the long-run cost curve is tangent to and no higher than each short-run cost curve generated by a fixed input level.
Example: At the tangency output, the selected fixed capital level is also optimal in the long run.
Textbook reference: Chapter 14, Section 14.3 The Envelope Theorem, printed pages 558-559Shadow wage ratepartial V/partial L^0 = lambda*
The marginal increase in optimized national output value from one more unit of available labor in a planning problem.
How to read: the marginal value of available labor equals lambda star
Textbook reference: Chapter 14, Section 14.3 The Envelope Theorem, printed pages 559, 561End-of-chapter problems
Practice all 6 Chapter Review Exercises
The index covers every numbered Review Exercise in this chapter of Mathematics for Economics, 3rd edition. Each reference opens the textbook exercise itself, including every listed subpart, with MFE hints, a symbol guide, a worked explanation, progress tracking, and similar-problem practice.
Practice end-of-chapter problems