Comparative Statics Explained: How an Equilibrium Responds When Something Changes

Understanding comparative statics — simply put

An economic model describes an equilibrium: the price at which buyers and sellers agree, the output a firm chooses, the interest rate that balances saving and investment. That equilibrium depends on things the model takes as given, such as tax rates, incomes, technology or policy settings. Comparative statics asks what happens to the equilibrium when one of those given things changes. It compares the old equilibrium with the new one and measures the difference, without tracing the path the economy takes from one to the other.

The question sounds simple, and in a one-line model it is. The power of the method is that it still works when the equilibrium is defined by equations nobody can solve explicitly. As long as the equations are smooth, the implicit function theorem delivers the direction and size of the change from the slopes of those equations alone.

How does it actually help?

Take a market for a good with a sales tax. Shoppers pay the consumer price p, sellers receive the net price p/(1 + t), and the tax rate t is the parameter that policy changes. Demand depends on the consumer price, supply on the net price, and the market clears when the two are equal:

This equation defines the equilibrium price p as a function of t, even though no formula for that function is written down. Comparative statics asks for $dp/dt$, the effect of a change in the tax rate on the price shoppers pay. The implicit function theorem gives it as the ratio of two partial derivatives, $\tfrac{dp}{dt} = -\tfrac{\partial F/\partial t}{\partial F/\partial p}$. Worked through with elasticities, the percentage responses of quantity to price, the result is strikingly simple:

The left side is the pass-through rate: the share of a tax change that shows up in the consumer price. When supply is much more elastic than demand, the ratio is close to one and shoppers bear, or gain, almost the entire tax change. When demand is the more elastic side, sellers absorb more than half of it.

Why does this matter?

Nearly every policy question is a comparative statics question. Who pays a tax or gains from a tax cut? How much does a rise in the policy interest rate lower output in a simple macro model? How does a firm's optimal output respond to a higher wage? Tax authorities, central banks and competition regulators all answer such questions by differentiating an equilibrium condition. The method also disciplines the answer: it shows which slopes the result depends on, so a disagreement about the effect of a policy becomes a disagreement about measurable elasticities.

What makes comparative statics reliable?

The equilibrium has to exist, it has to move smoothly with the parameter, and the change has to be small enough that the slopes measured at the old equilibrium still describe the new one. A further condition, stability, connects the static comparison to real adjustment: if the market would not move toward the new equilibrium after a shock, comparing the two equilibria says little about what actually happens. The conditions, and why each one matters, are set out after the worked example.

The takeaway

Comparative statics turns "what if this changed?" into a derivative. The implicit function theorem supplies that derivative from the model's slopes, even when the model cannot be solved outright. The rest of this guide applies the method to a real tax cut, then sets out the mechanics.

Solving it by hand: a small market

Before the real data, the method can be worked through on a market small enough to check by hand. Demand is $D(p) = 100 - p$ and supply is $S(q) = 2q$, where $q = p/(1+t)$ is the net price sellers receive and the tax rate is $t = 0.2$ (illustrative numbers).

Step 1: solve for the equilibrium. $100 - p = 2p/1.2$ gives $p = 37.5$, a net price $q = 31.25$ and a quantity $Q = 62.5$.

Step 2: take the partial derivatives of excess demand. $\partial F/\partial p = -1 - 2/1.2 \approx -2.667$, which is not zero, so the theorem applies, and $\partial F/\partial t = 2p/1.2^2 \approx 52.08$.

Step 3: apply the implicit function theorem. $dp/dt = -(52.08)/(-2.667) \approx 19.53$.

Step 4: convert to elasticities. At the equilibrium $\varepsilon_S = 1$ and $\varepsilon_D = -p/Q = -0.6$, so the pass-through formula gives $1/1.6 = 0.625$. The same number comes from the derivative: $19.53 \times 1.2/37.5 = 0.625$.

Step 5: check. Solving outright gives $p(t) = 100(1+t)/(3+t)$, so $dp/dt = 200/(3+t)^2 = 19.53$ at $t = 0.2$, matching Step 3.

A real example: Germany's temporary VAT cut in 2020

On 1 July 2020, as part of its pandemic stimulus package, Germany cut its standard rate of value added tax (VAT) from 19% to 16% and its reduced rate from 7% to 5%. The cut was temporary and ended on 31 December 2020.

If shops passed the whole cut on, a product taxed at the standard rate would become cheaper by 1.16/1.19 − 1 = −2.52%, and a product at the reduced rate by 1.05/1.07 − 1 = −1.87%. How much of that maximum actually reaches shoppers is a comparative statics question, and the pass-through formula answers it in terms of the two elasticities.

Reading the result

The table shows the whole logic of tax incidence in five rows. When supply is perfectly elastic, sellers will not accept a lower net price, so the entire cut reaches shoppers. As supply becomes less elastic relative to demand, sellers keep a larger share of the cut as a higher net price, and the consumer price falls by less. In the last row, where demand is twice as elastic as supply, two-thirds of the cut stays with sellers.

What happened in Germany? The Federal Statistical Office (Destatis) reported that consumer price inflation fell from +0.9% in June 2020 to −0.1% in July 2020, a fall of 1.0 percentage point. The consumer price index fell 0.5% from June to July, and goods prices fell 1.9% over the month. Destatis noted that the extent of pass-through was very difficult to determine, but that a model calculation assuming full and immediate pass-through would have lowered the July inflation rate by 1.6 percentage points.

The observed 1.0-point fall is smaller than the 1.6-point full pass-through benchmark. That gap is consistent with partial pass-through, but a change in the annual rate also reflects price movements a year earlier (base effects) and price changes unrelated to the tax. Food prices, for example, fell 2.7% over the month, partly because most food is taxed at the reduced rate and partly for seasonal reasons. The headline rate alone cannot separate the VAT effect from those changes. Measuring pass-through properly requires comparing the prices of taxed products with a control group over the same months, a difference-in-differences design.

Product-level data can separate those effects. A study for the Federal Ministry of Justice and Consumer Protection by the ZEW institute tracked the retail prices of individual products and estimated that about 85% of the July 2020 cut was passed on for the goods it covered: about 96% for durable goods and about 76% for fast-moving consumer goods such as groceries, about 70% of which are taxed at the reduced rate. Services were not covered. When the higher rates returned in January 2021, the reversal was close to symmetric for durable goods and somewhat smaller for non-durables.

The formula turns that estimate back into a statement about elasticities. A pass-through of 0.85 means $\varepsilon_S/(\varepsilon_S - \varepsilon_D) = 0.85$, so $\varepsilon_S \approx 5.7\,\lvert\varepsilon_D\rvert$. For the goods covered, supply behaved as if it were close to six times as elastic as demand over the months of the cut, which places the estimate between the second and third rows of the table. By type the spread is wide: 96% for durables implies a ratio of about 24, and 76% for non-durables about 3. These are as-if values: they assume a competitive market, and a cut announced as lasting six months can draw a different response from a permanent one.

The reversal offers a check in the other direction. Inflation rose from −0.3% in December 2020 to +1.0% in January 2021, when the higher rates returned. Destatis stated that the higher rates had been passed on to consumers, but it also cited the CO2 charge on fuels introduced at the start of 2021, so the January rise is not a clean measure of reverse pass-through. Comparative statics treats a cut and a rise symmetrically; whether real pass-through is symmetric is an empirical question.

The assumptions

The equilibrium exists and is locally unique. Comparative statics compares two equilibria, so both must exist. If several equilibria exist near each other, a small change in the parameter can make the economy jump between them, and the derivative of "the" equilibrium is not defined.

The equilibrium conditions are continuously differentiable. The implicit function theorem needs smooth functions, because it works by approximating each equation with a straight line near the equilibrium. Kinks, such as a price ceiling or a capacity limit that binds exactly at the equilibrium, break the approximation, and the effect of a change can differ depending on its direction.

The relevant derivative is not zero. For one equation, $\partial F/\partial p$ must be non-zero; for a system, the Jacobian matrix of the equations with respect to the endogenous variables must have a non-zero determinant. That derivative appears in the denominator of the comparative statics result. If it is zero, the equations do not pin down how the equilibrium moves, and the formula divides by zero.

Changes are small. The result is a derivative, accurate for marginal changes. For a large change, such as a 3-point VAT cut, the slopes at the old equilibrium may differ from the slopes along the way. The worked example handles this by assuming constant elasticities, which keeps the formula exact in percentage terms; with other functional forms the prediction is an approximation.

The equilibrium is stable. Comparative statics says where the new equilibrium is, not whether the economy goes there. Samuelson's correspondence principle links the two: assuming the equilibrium is stable under a plausible adjustment process often fixes the sign of the denominator, and therefore the sign of the comparative statics result.

The formula

The derivations below are optional for anyone who only needs the intuition, but they are the version used in graduate courses and research.

One equation, one endogenous variable. Let $F(y, \theta) = 0$ define the equilibrium value of an endogenous variable y as a function of a parameter $\theta$. Differentiating the identity $F(y(\theta), \theta) = 0$ with respect to $\theta$ gives the implicit function theorem result:

The tax example. With $F(p,t) = D(p) - S\big(p/(1+t)\big)$, the two partial derivatives are:

Substituting into the implicit function theorem and multiplying both sides by (1+t)/p:

Because $\varepsilon_S > 0$ and $\varepsilon_D < 0$, the pass-through rate lies between 0 and 1. Since $\ln q = \ln p - \ln(1+t)$, the elasticity of the net price with respect to (1+t) is the pass-through rate minus one, $\varepsilon_D/(\varepsilon_S - \varepsilon_D)$, which is negative: a tax cut raises the net price sellers receive.

Several equations: the general case. Many models have several endogenous variables $y = (y_1, \dots, y_n)$ determined by n equations $F(y, \theta) = 0$. Differentiating the whole system with respect to $\theta$ gives a linear system:

If $\det J \neq 0$, the solution is $dy/d\theta = -J^{-1}(\partial F/\partial \theta)$. Each derivative can be computed with Cramer's rule, which replaces one column of J with the right-hand side and divides the two determinants. Solving systems like this is the core use of linear algebra in economics.

A worked example: fiscal policy in IS-LM. Take a simple IS-LM model in which output Y and the interest rate r are the endogenous variables, written as two linear equations:

The Jacobian has rows (1 − c, b) and (k, −h), so $\det J = -h(1-c) - bk$, which is negative. To find the effect of G, replace the first column of J with the derivative of the right-hand side with respect to G, which is (1, 0):

With illustrative values c = 0.6, b = 20, k = 0.5 and h = 40, $\det J = -26$ and $dY/dG = 40/26 \approx 1.54$. The simple Keynesian multiplier $1/(1-c) = 2.5$ is larger because it ignores the rise in the interest rate. Replacing the second column instead gives $dr/dG = k/(h(1-c) + bk) = 0.5/26 \approx 0.019$, the crowding-out channel that fiscal multipliers depend on.

Comparative statics of an optimum. When y is chosen to maximize an objective f(y, $\theta$), the first-order condition $\partial f/\partial y = 0$ plays the role of F. The implicit function theorem then gives:

The second-order condition for a maximum makes the denominator negative, so the sign of the response equals the sign of the cross-partial derivative. If a higher $\theta$ raises the marginal return to y, the optimal y rises. For example, a firm choosing labor L to maximize $p\cdot 2\sqrt{L} - wL$ has cross-partial $1/\sqrt{L} > 0$ with respect to the output price p, so $dL^*/dp > 0$; solving gives $L^* = (p/w)^2$ and $dL^*/dp = 2L^*/p$. The envelope theorem handles the companion question, how the optimal value itself changes with $\theta$. With constraints, the same steps are applied to the first-order conditions of the Lagrangian, which is why the bordered Hessian appears in comparative statics results.

When this isn't the right tool

Large or discrete changes. A derivative describes a small change. For a large shock or a switch between regimes, the model has to be solved at both parameter values, or the comparison done numerically.

Adjustment paths. Comparative statics compares end points. How long prices take to adjust, whether they overshoot, and what happens in between require a dynamic model. In the VAT case, shops may have repriced gradually, or not at all for a cut known to be temporary.

Estimating the slopes. The formula is only as good as the elasticities fed into it. Demand and supply elasticities are estimated from data, which requires methods that separate the two curves, such as instrumental variables.

Market power. The pass-through formula assumes competitive markets. Under monopoly or oligopoly, pass-through depends on the curvature of demand and can exceed 100%, so the competitive formula can understate or overstate the effect of a tax change.

Practice problems

Problem 1. Demand is $Q = 100 - 2p$, where p is the price buyers pay. Supply is $Q = -20 + 4(p - T)$, where T is a per-unit tax. Use the implicit function theorem to find $dp/dT$, and say who bears the tax.

Solution. Write $F(p,T) = 100 - 2p + 20 - 4(p - T) = 0$. Then $\partial F/\partial p = -6$ and $\partial F/\partial T = 4$, so $dp/dT = -4/(-6) = 2/3$. Buyers bear two-thirds of the tax and sellers one-third. Solving directly gives p = 20 + (2/3)T, which confirms it.

Problem 2. In the IS-LM model above, let c = 0.75, b = 25, k = 0.5 and h = 50, with the price level fixed at P = 1. Use Cramer's rule to find $dY/dG$ and $dY/dM$.

Solution. $\det J = -50(0.25) - 25(0.5) = -25$. For G, the right-hand side derivative is (1, 0), so $dY/dG = (-50)/(-25) = 2$. For M it is (0, 1); replacing the first column gives the determinant 0 × (−50) − 25 × 1 = −25, so $dY/dM = (-25)/(-25) = 1$. Check: with $dY/dG = 2$ and $dr/dG = 0.5/25 = 0.02$, the IS equation gives 0.25 × 2 + 25 × 0.02 = 1 and the LM equation gives 0.5 × 2 − 50 × 0.02 = 0, as required.

Further reading

• Optimization and Lagrange Multipliers Explained — internal
• Linear Algebra for Economists Explained — internal
• Supply and Demand Explained — internal
• Understanding Inflation — internal
• Difference-in-Differences Explained — internal
• Instrumental Variables Explained — internal
• Fiscal Policy and Multipliers — internal
• Destatis, consumer prices in July 2020 — external reference
• Destatis, consumer prices in January 2021 — external reference
• ZEW for the Federal Ministry of Justice and Consumer Protection, consumer prices after the VAT cut (2021) — external reference

The VAT rates, dates and inflation figures are from Destatis press releases of 13 August 2020 and 10 February 2021. The 85% pass-through estimate and its split by type are from the ZEW study for the Federal Ministry of Justice and Consumer Protection (June 2021, updated October 2021); the implied elasticity ratio is an EconoLens calculation. The elasticities in the worked example are illustrative, and the pass-through and price figures are EconoLens calculations from the comparative statics formula, not estimates of the actual German pass-through.