Modeling the Tariff Pass‑Through as a First‑Order Lag



When a tariff is imposed, its full effect is not felt immediately. Instead, prices adjust gradually as inventories turn over, contracts are renegotiated, and consumers become aware of the new cost. One common way to model this gradual adjustment is to use an exponential lag model. For a good with an initial price P0P_0 and a tariff that ultimately raises the price by ΔP\Delta P (expressed in absolute terms or as a percentage), we can write the time‑dependent price as:

P(t)=P0+ΔP (1−e−λt),P(t) = P_0 + \Delta P\,\Bigl(1 - e^{-\lambda t}\Bigr),

where:

  • P(t)P(t) is the price at time tt (with tt measured in months, for example),
  • P0P_0 is the initial price before the tariff,
  • ΔP\Delta P is the full price increase that the tariff would induce (e.g. if a 10% tariff on a $100 product, then ΔP=$10\Delta P = \$10),
  • λ\lambda is a positive parameter (per month) that describes the speed of the adjustment (the higher λ\lambda is, the faster the market passes the tariff onto final prices).

At t=0t = 0, we have P(0)=P0P(0) = P_0. As t→∞t \to \infty, the exponential term e−λt→0e^{-\lambda t} \to 0, so P(t)→P0+ΔPP(t) \to P_0 + \Delta P; that is, the full tariff effect is realized.

2. Determining the "Full Impact" Time

We want to know how long it takes for the tariff’s impact to be nearly complete. A common threshold is when 99% of the full impact is passed on, meaning:

1−e−λt∗=0.99.1 - e^{-\lambda t^*} = 0.99.

Solving for t∗t^*:

e−λt∗=0.01,e^{-\lambda t^*} = 0.01, −λt∗=ln⁡(0.01),-\lambda t^* = \ln(0.01), t∗=−ln⁡(0.01)λ≈4.6052λ.t^* = \frac{-\ln(0.01)}{\lambda} \approx \frac{4.6052}{\lambda}.

This t∗t^* is the time required for the price to reach 99% of its full post‑tariff level.

Examples:

  • Fast adjustment: If λ=0.5\lambda = 0.5 per month, then t∗≈4.60520.5≈9.21 months.t^* \approx \frac{4.6052}{0.5} \approx 9.21\text{ months}.
  • Slower adjustment: If λ=0.25\lambda = 0.25 per month, then t∗≈4.60520.25≈18.42 months.t^* \approx \frac{4.6052}{0.25} \approx 18.42\text{ months}.

Thus, with a higher adjustment speed (λ=0.5\lambda = 0.5), the market feels nearly the full tariff impact in about 9 months, whereas a slower adjustment (λ=0.25\lambda = 0.25) can take more than 18 months.

3. Linking Price Changes to Demand (Optional Extension)

The tariff affects prices, which in turn affect demand. Suppose that without tariffs, demand is given by a function Q=f(P)Q = f(P). When the tariff passes through, the evolving price P(t)P(t) leads to a time‐dependent demand:

Q(t)=f(P0+ΔP (1−e−λt)).Q(t) = f\Bigl(P_0 + \Delta P\,(1 - e^{-\lambda t})\Bigr).

If you have an estimate of the price elasticity of demand, EdE_d, you could approximate the percentage change in quantity demanded due to the price increase. For small changes, the relative change in QQ might be approximated by:

ΔQ(t)Q0≈Ed⋅ΔP (1−e−λt)P0.\frac{\Delta Q(t)}{Q_0} \approx E_d \cdot \frac{\Delta P\,(1 - e^{-\lambda t})}{P_0}.

This shows that as tt increases, the fraction (1−e−λt)(1 - e^{-\lambda t}) grows toward 1, and the full demand response is eventually realized.

4. What to Expect: A Full Run‑Down

a. Immediate Aftermath (tt near 0):

  • Price: At t=0t = 0, P(0)=P0P(0) = P_0. The tariff is announced or implemented, but prices remain at their initial levels.
  • Demand: No immediate change because the new price hasn’t fully taken effect.

b. Intermediate Period (0 < tt < t∗t^*):

  • Price Increase: Prices start rising gradually following P(t)=P0+ΔP (1−e−λt)P(t) = P_0 + \Delta P\,(1 - e^{-\lambda t}). The pace of increase is steep initially and then tapers off.
  • Consumer Adjustment: As prices slowly rise, consumers gradually perceive higher costs. The impact on demand will depend on the price elasticity:
    • For elastic goods, even a small price increase can lead to a significant drop in quantity demanded.
    • For inelastic goods, demand might not fall much until nearly the full tariff impact is felt.
  • Supply Chains: Firms adjust supply chain contracts, pass through costs to retailers, and eventually, inventory turnover reflects the new pricing.

c. Longer Term (t ≥ t∗t^*):

  • Full Impact: Prices have essentially reached P0+ΔPP_0 + \Delta P. Nearly 99% of the tariff’s effect is in place.
  • Demand: Consumers have fully adjusted to the new price level. Depending on the elasticity, total quantity demanded has decreased (if the good is elastic) or decreased only slightly (if inelastic).
  • Market Dynamics: Over this period, the tariff’s effect has percolated through all channels—from manufacturers to importers to retailers—resulting in the full economic impact.

d. Considerations and Calibration:

  • Parameter Estimation: In a real macro analysis, you would estimate λ\lambda using historical data on how quickly previous tariffs or similar shocks affected prices.
  • Heterogeneity: Different sectors or product categories might have different λ\lambda values. For instance, high-tech products with just‑in‑time supply chains might adjust faster than bulky consumer durables.
  • Contractual Lags: Pre‑existing contracts or long‑term pricing agreements might delay the pass‑through further, effectively lowering the observed λ\lambda.

5. Summary

  • Exponential Lag Model: P(t)=P0+ΔP (1−e−λt)P(t) = P_0 + \Delta P\,(1 - e^{-\lambda t}) captures the gradual pass‑through of tariff-induced price increases.
  • Full Impact Time: The time to reach 99% of the new price is t∗≈4.6052λt^* \approx \frac{4.6052}{\lambda}.
  • Example Calculations: With λ=0.5\lambda = 0.5, expect about 9 months; with λ=0.25\lambda = 0.25, about 18 months.
  • Demand Dynamics: As the price gradually increases, demand shifts according to the price elasticity of the product.
  • Practical Application: Use historical tariff data to calibrate λ\lambda for different industries, then apply the model to forecast how long it will take for the tariff effect to be fully reflected in prices and demand.

This framework provides a detailed mathematical understanding of the tariff‐effect lag. While real‑world complexities (e.g., contractual constraints, multi‑channel pricing, and dynamic consumer behavior) add layers of nuance, the basic exponential lag model offers a valuable starting point for estimating the timeline over which a tariff will fully affect market prices and, consequently, demand.


Below are several product‐by‐product examples that use our simple exponential lag model to illustrate how long it takes for the tariff effect to fully “hit” each market. Recall the model:

P(t)=P0+ΔP (1−e−λt),P(t) = P_0 + \Delta P \, \bigl(1 - e^{-\lambda t}\bigr),

and the “full effect” time (e.g. 99% pass‑through) is approximately

t∗≈4.605λ.t^* \approx \frac{4.605}{\lambda}.


Examples 

Example 1: Next‑Generation VR Headset (Consumer Electronics)

  • Baseline price: P0=$500P_0 = \$500
  • Tariff: 10% tariff → ΔP=$50\Delta P = \$50
  • Adjustment speed: Assume fast adjustment with λ=0.5\lambda = 0.5 per month (reflecting agile tech supply chains and rapid price updates).

Full Effect Time:

t∗≈4.6050.5≈9.21 monthst^* \approx \frac{4.605}{0.5} \approx 9.21 \text{ months}

Interpretation:
For the VR headset, nearly the entire tariff‐induced price increase will be passed through in about 9 months. In this period, the price will gradually rise from $500 to roughly $550, and the corresponding demand will adjust based on the product’s price elasticity.

Example 2: Gasoline (Commodity)

  • Baseline price: P0=$3.00P_0 = \$3.00 per gallon
  • Tariff: Suppose a smaller tariff (or duty) adds 5% → ΔP=$0.15\Delta P = \$0.15
  • Adjustment speed: Gasoline prices are often sticky due to long‐term supply contracts and refining logistics. Let’s assume a slower pass‑through with λ=0.2\lambda = 0.2 per month.

Full Effect Time:

t∗≈4.6050.2≈23.0 monthst^* \approx \frac{4.605}{0.2} \approx 23.0 \text{ months}

Interpretation:
For gasoline, the full price impact of the tariff takes nearly two years to materialize. Prices will gradually increase from $3.00 to about $3.15 per gallon over this period. Because gasoline is typically price inelastic, the demand reduction may be modest even after full pass‑through.

Example 3: Flat‑Screen Television (Consumer Durable)

  • Baseline price: P0=$800P_0 = \$800
  • Tariff: A 10% tariff results in ΔP=$80\Delta P = \$80
  • Adjustment speed: With a mix of supply chain factors (e.g., inventory turnover, contract renegotiations), assume λ=0.4\lambda = 0.4 per month.

Full Effect Time:

t∗≈4.6050.4≈11.51 monthst^* \approx \frac{4.605}{0.4} \approx 11.51 \text{ months}

Interpretation:
In the TV market, it might take around 11–12 months for the full tariff impact (an $80 increase) to be reflected in retail prices. Retailers and importers adjust gradually, so consumers will start noticing a progressive price rise over roughly a year. Depending on the TV’s price elasticity, this may lead to a moderate decline in quantity demanded.

Example 4: Wheat (Agricultural Commodity)

  • Baseline price: P0=$5.00P_0 = \$5.00 per bushel
  • Tariff: Suppose a tariff adds 20% → ΔP=$1.00\Delta P = \$1.00
  • Adjustment speed: Agricultural products often have intermediate lag times due to storage, seasonal contracts, and slower market adjustments. Assume λ=0.3\lambda = 0.3 per month.

Full Effect Time:

t∗≈4.6050.3≈15.35 monthst^* \approx \frac{4.605}{0.3} \approx 15.35 \text{ months}

Interpretation:
For wheat, it takes about 15 months for nearly all of the tariff’s impact to be passed on, raising the price from $5.00 to about $6.00 per bushel. This longer lag reflects the slower turnaround in agricultural markets. Since wheat tends to be price inelastic (it’s a staple commodity), the quantity adjustment might be limited despite the price increase.

Final Thoughts

Each example demonstrates how the parameter λ\lambda (adjustment speed) plays a critical role in determining the lag between tariff implementation and full price impact. While consumer electronics like VR headsets can adjust quickly (around 9 months), more complex supply chains or long‑term contracts—such as in gasoline or agricultural markets—can result in a full pass‑through time stretching from about 15 to 23 months.

Keep in mind that this model is a simplified framework. In practice, calibration would involve detailed historical data, consideration of multi‑channel pricing effects, and other market dynamics. Nonetheless, these examples illustrate the math and intuition behind the tariff effect lag across different sectors.

Popular posts from this blog

The Lewis Ratchet: Forward Asymmetric Momentum Transfer

Image
  The Lewis Ratchet: Forward Asymmetric Momentum Transfer I figured out how to make a model of this:  https://www.youtube.com/watch?v=A5x0Yd7kjSc Imagine you’re floating on a small raft in a calm lake—a bit like Huck Finn’s raft, but with a twist. Instead of a motor or engine, your raft relies on an ingenious idea: by moving an internal mass back and forth in an asymmetric way, you can slowly “scooch” the raft forward. This is the essence of the Lewis Ratchet. The Basic Idea In an ideal, frictionless world, any internal motion—like shaking a heavy bar or throwing a medicine ball—cancels out. When you push one way, you push the other equally. But if you can arrange for the forward push to be strong and quick, and the return motion to be gentle and slow, the momentum transferred during the fast “kick” isn’t completely undone by the slow “reset.” In a real-world setting, friction (or water drag) then “locks in” that net gain, causing the raft to move forward incrementally with ea...
Read more

The Lewis Ratchet Redeux: The Lewis Ratchet in Space

Image
  The Lewis Ratchet Revisited: Two Perspectives, One Principle https://www.youtube.com/watch?v=OxHIMhTpFYU When we first imagine the Lewis Ratchet —the idea that you can move a platform by throwing and retrieving a mass inside it—it’s often easiest to picture a raft on water. You throw a medicine ball forward (the “kick”), then gently walk it back (the “reset”), and thanks to friction or drag, the raft creeps ahead a little more each cycle. That initial, Huck Finn–style image might suggest that friction is absolutely essential. After all, friction is what lets you walk without slipping and prevents the raft from drifting back during the slow reset. But in a more general, frictionless environment—say, a sealed box floating in the vacuum of space—the same fundamental process happens via normal forces , collisions , and hooks or latches that allow internal masses to push off the box’s walls. This is an example of the idea.  The Vpython Script launches 4 masses and then the Spac...
Read more

Exploring momentum redirection - Understanding Inertial Displacement

Image
 Foundations, NASA’s Helical Engine, and Newton’s Legacy 1. Introduction: The Quest for Reactionless Propulsion  Today we embark on an in‐depth exploration of advanced propulsion systems that—at first glance—seem to defy conventional intuition. According to Sir Isaac Newton, every action has an equal and opposite reaction. In his Philosophiæ Naturalis Principia Mathematica , Newton established that the momentum of an isolated system must be conserved. Later, Emmy Noether elegantly formalized this idea: every differentiable symmetry of the action of a physical system has a corresponding conservation law. For example, spatial translation symmetry guarantees conservation of linear momentum. Yet here we stand, contemplating systems that propose to generate thrust by only rearranging internal momentum—without expelling mass. How can a vehicle “move” if, by Newton’s and Noether’s words, its center of mass must remain unchanged in the absence of an external force? Today, we analyz...
Read more