---
title: "How and Why to Use Models"
author: "Joseph Mullins"
format:
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bibliography: ../references.bib
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---

# When You Don't Need a Model

## Causal Inference Without Deep Structure {.smaller}

- Many questions take the form: "What is the effect of $X$ on $Y$?"
- Often answerable with simple statistical models of causality (Potential Outcomes, Generalized Roy)
- No need to specify preferences, technology, or equilibrium
- But the [target parameter]{style="color: #2563eb"} must be precisely defined

- **Key requirement:** observed variation directly identifies the [counterfactual]{style="color: #2563eb"} of interest
- Three examples from our prototype models below

## Example: Social Security {.smaller}

- Consider the life-cycle savings model with a social security system
- The budget constraint:
$$ c_{t} + a_{t+1}/(1 + r) \leq (1-\tau)y_{t} + \mathbf{1}\{t\geq 65\}b $$
- Individuals receive benefit $b$ after age 65 and pay proportional taxes $\tau$

- Suppose eligibility is unexpectedly lowered from 65 to 60
- Repeated cross-sections of consumption by age and cohort are available

## Social Security: Target Parameter {.smaller}

- Let $t^*$ index cohorts by age at the time of the policy announcement
- $t^*$ indexes both a cohort *and* a treatment

- **Question:** *"What was the effect of this eligibility expansion on consumption at age $t$ for cohort $t^*$?"*
- $C_{t^*,t}(1)$: consumption under the announcement; $C_{t^*,t}(0)$: consumption absent the announcement
- Target parameter:
$$ \alpha_{t^*,t} = \mathbb{E}[C_{t^*,t}(1) - C_{t^*,t}(0)] $$
- Can we identify this without specifying the full model?

## Social Security: Difference-in-Differences {.smaller}

- Use the cohort aged 65 (unaffected) as a control group
- [**Parallel trends:**]{style="color: #2563eb"} for $t^*<65$, ages $t\geq t^*$ and $s < t^*$:
$$ \mathbb{E}[C_{t^*,t}(0)] - \mathbb{E}[C_{65,t}(0)] = \mathbb{E}[C_{t^*,s}(0)] - \mathbb{E}[C_{65,s}] $$
- The DD estimator constructs the counterfactual from observables:
$$ \alpha_{t^*,t} = \mathbb{E}[C_{t^*,t}(1)] - \mathbb{E}[C_{65,t}(0)] - \left(\mathbb{E}[C_{t^*,s}(0)] - \mathbb{E}[C_{65,s}(0)]\right) $$

## Social Security: Regression Specification {.smaller}

- The above was for a specific $(t^*, \text{control cohort}, s)$ triple
- With many cohorts and ages, the common regression:
$$ \mathbb{E}[C_{t^*,t}] = \gamma_{t^*} + \mu_{t} + \alpha_{t^*,t}\mathbf{1}\{t\geq t^*\} $$
- This layers in a stricter set of parallel trends assumptions

- No need to specify utility, discount factors, or savings technology
- Only need to defend parallel trends
- Farther-apart cohorts make this assumption stronger
- The regression cannot select "more ideal" controls for each treated cohort [@GOODMANBACON2021]

## Example: Firm Entry and Minimum Wage {.smaller}

- Panel data $(A_{m,t,1}, A_{m,t,2}, Z_{m,t})$ on firm activity across markets
- $A_{m,t,j}$: whether firm $j$ is active in market $m$ at $t$; entry when $A_{m,t+1,j} - A_{m,t,j} = 1$
- $X_m$: unobservable market-level profitability shifter
- $Z \in \{0,1\}$: local minimum wage policy indicator

- Payoff to entering under the policy:
$$ u_{1}(x,a,d') = \phi_{0} + \phi_{1}x - \phi_{2}d' - \phi_{3}(1-a) + \phi_{4}z $$
- $\phi_4$ captures the minimum wage effect on firm payoffs

## Firm Entry: Target and Identification {.smaller}

- $Z_{m,1} = 0$ initially; a subset adopt permanently at $t^*$, unanticipated
- **Question:** *"What is the effect of the minimum wage on firm entry?"*
- $N_{m,t} = D_{m,t,1} + D_{m,t,2}$: number of active firms
- $N_{\tau,m}(z)$: potential outcome of $N$ in market $m$, $\tau$ periods after adoption
- Dynamic treatment effect on the treated:
$$ \alpha_{\tau} = \mathbb{E}[N_{\tau}(1) - N_{\tau}(0) \mid Z = 1] $$
- Effects differ across markets --- [heterogeneity]{style="color: #2563eb"} matters
- Selection into adoption: $\mathbb{E}[\Delta_{\tau}|Z=1] \neq \mathbb{E}[\Delta_{\tau}]$

## Firm Entry: Parallel Trends {.smaller}

- [Parallel trends]{style="color: #2563eb"} justifying the event study:
$$ \mathbb{E}[N_{t}(0)|Z=1] - \mathbb{E}[N_{t}(0)|Z=0] = \text{constant} $$
- If markets are in the ergodic distribution of the MPE, $N$ is stationary absent intervention and parallel trends holds

- Event-study regression:
$$ D_{m,t} = \gamma_{m} + \mu_{t} + \mathbf{1}\{t\geq t^*\}\alpha_{t-t^*} + \epsilon_{m,t} $$
- Robustly identifies the average effect among adopters (uniform timing)
- With staggered adoption and heterogeneous effects, regression delivers a hard-to-interpret weighted average [@GOODMANBACON2021]

## Example: Tax Reform Experiments {.smaller}

- Three states run different experiments with taxes and transfers
- Let $\mathcal{Y}_j$ be the net income function in state $j$:
\begin{align}
\mathcal{Y}_{A}(W,H) &= b_{A} + (1-\tau_{A})WH \\
\mathcal{Y}_{B}(W,H) &= WH + \sum_{k=0}^{5}\tau_{k}(WH-\overline{E}_{k})\mathbf{1}\{WH>\overline{E}_{k}\} \\
\mathcal{Y}_{C}(W,H) &= WH(1-\tau_{C}) + \mathbf{1}\{H>20\}b_{C}
\end{align}
- Participants don't anticipate assignment; the experiment is temporary (3 periods)
- $Z_j \in \{0,1\}$: assignment to treatment or control in state $j$

## Tax Reform: Identification {.smaller}

- **Question:** *"What is the effect of each unannounced, temporary reform on labor supply?"*
- Compare treatment and control means in each state:
$$\mathbb{E}[H \mid Z_j = 1, j] - \mathbb{E}[H \mid Z_j = 0, j]$$
- [Random assignment]{style="color: #2563eb"} identifies the ATE --- no further assumptions needed

# Reasons to Use a Model

## When the Question Requires One {.smaller}

- The most obvious reason: the research question cannot be articulated without a model
- The most useful modeling insights concern efficiency, market failures, and policy design

- What does the [optimal]{style="color: #2563eb"} tax and transfer system look like?
- What is the cheapest way to incentivize competition between firms?
- What are the welfare costs of incomplete insurance markets?

- These involve [counterfactual]{style="color: #2563eb"} policy environments never observed
- They require specifying objectives, constraints, and behavioral responses

## Welfare Calculations {.smaller}

- [**Revealed preference:**]{style="color: #2563eb"} treating individuals as knowing what they like lets us infer preferences and evaluate policy

- How do individuals value the social security program? How would they value a different $(\tau, b)$?
- What is the willingness to pay for a lump-sum transfer $b$ financed by taxes $\tau$? (We return to this below)

## Making Sense of Puzzling Data {.smaller}

- The most fundamental causal question: what is the effect of price on quantities?
- A silly question --- but only because supply and demand theory is so fundamental
- Without theory, all you see is a cloud of points:

```{julia}
#| echo: false
#| label: fig-identification
#| fig-cap: "Shifting supply and demand curves trace out equilibrium points"

using Plots, Random, Logging
Random.seed!(123)

eq_prices = Float64[]
eq_quantities = Float64[]

anim = @animate for i in 1:60
    a_d = 10 + 2*randn()
    b_d = -1.0
    a_s = 2 + 2*randn()
    b_s = 1.0

    Q_eq = (a_d - a_s) / (b_s - b_d)
    P_eq = a_d + b_d * Q_eq

    push!(eq_quantities, Q_eq)
    push!(eq_prices, P_eq)

    Q_range = range(0, 8, length=100)
    p = plot(Q_range, a_d .+ b_d .* Q_range,
             color=:blue, lw=2,
             xlim=(0, 8), ylim=(0, 12),
             xlabel="Quantity", ylabel="Price",
             legend=false)
    plot!(Q_range, a_s .+ b_s .* Q_range,
          color=:red, lw=2)
    scatter!(eq_quantities, eq_prices,
             color=:black, ms=4, alpha=0.6)
    scatter!([Q_eq], [P_eq],
             color=:green, ms=8, marker=:star)
end

with_logger(NullLogger()) do
    gif(anim, fps=1)
end
```

## Making Sense of Puzzling Data {.smaller}

- **With** supply and demand, each point is the equilibrium outcome of two structural relationships
- Wright proposed [instrumental variables]{style="color: #2563eb"} to disentangle supply from demand
- Perhaps the earliest estimated structural model

## When Variation Doesn't Identify the Counterfactual {.smaller}

- In the examples above, we defined the "treatment" very specifically
- Models help articulate what can and cannot be identified by observed variation
- They outline assumptions under which related counterfactuals can be forecast, even without exact replication
- Often framed as [*internal vs external validity*]{style="color: #2563eb"} --- but the "external validity" column is too rich to leave undiscussed

:::{.callout-note}
## @HeckmanVytlacil2005
Estimands from simple statistical models designed to infer causal effects (IV, DD, RD) are rarely parameters of exact policy interest
:::

## Social Security: A Concrete Case {.smaller}

To see what DD misses, simplify the savings model:

- Each individual faces a *known* income sequence $\{y_{n,t}\}_{t=1}^{T}$
- Agents face a *natural borrowing constraint*:
$$ \sum_{s=t}^{T}q_{s-t}c_{s} \leq a_{t} + \sum_{s=t}^{T}q_{s-t}(1-\tau)y_{n,s} + \sum_{s\geq 65}^{T}q_{s-t}b $$
where $q_{\tau} = 1/(1+r)^{\tau}$ is the price of consumption $\tau$ periods ahead
- $\beta(1+r)=1$, $\psi = 0$: agents perfectly smooth consumption to NPV of net income

## Social Security: Consumption Effect {.smaller}

- Mean consumption effect for cohort $t^*$ at any $t \geq t^*$ equals the NPV effect of the announcement:
$$ \Delta C_{t^*,t} = \sum_{s=60}^{64}q_{s-t^*}b + \sum_{s=t^*}^{T}(\tau - \tau')\overline{y}_{t^*,t} $$
where $\overline{y}_{t^*,t}$ is average income for cohort $t^*$ at age $t$
- Parallel trends holds: cross-cohort differences are constant with age ($\mu_t$ normalizable to zero)
- DD robustly identifies $\alpha_{t^*,t}$
- The model implies $\alpha_{t^*,t}$ is constant in $t$
- Each effect depends on [**how cohort $t^*$ expects the expansion to be financed**]{style="color: #2563eb"} ($\tau' - \tau$), which we haven't specified

## Social Security: What DD Doesn't Tell Us {.smaller}

- The $\alpha_{t^*,t}$ are informative but don't tell us about *future* policy changes
- They don't even tell us what would happen if the expansion were repealed
- Extrapolation requires more theory

DD does **not** identify:

- The total effect of social security (not just the expansion)
- Effects of additional changes in the eligibility age
- Effects of changes in $b$ and $\tau$ at different ages and anticipation horizons
- Effects on cohort $t^*$ if they learned at any age other than $t^*$ (one-to-one mapping between cohort and treatment)
- Effects for cohorts who anticipated the policy their whole life
- Effects under alternative financing arrangements

## Tax Reform: What the Model Reveals {.smaller}

- Effect of reform $A$ when $\beta(1+r) = 1$ (stationary consumption):
$$\Delta H_{n,t} = \psi\log(1-\tau) - \psi\sigma\Delta \log\left(C^*_{n}\right) $$
- $\Delta C^*_n$ embodies the income effect; solved by plugging optimal labor supply into the intertemporal budget constraint

- The ATE depends on income effects, which depend on the [*perceived persistence*]{style="color: #2563eb"} of the reform
- Substantial heterogeneity in treatment effects
- Different distributions of wages or work costs across populations imply different impacts

## Tax Reform: What Experiments Don't Tell Us {.smaller}

Each experiment robustly identifies the effect of reform $j$ on population $j$. It does **not** identify:

- Effects of reforms with different persistence --- experimental findings must be interpreted carefully regarding perceived duration
- The effect of reform $A$ on population $B$, or vice versa
- The distribution of individual treatment effects
- Effects of partly anticipated reforms
- Effects at scale, where general equilibrium responses may appear
- The model can invert out wage and work cost distributions from panel data, then return to the experimental data to estimate ATEs along latent dimensions

## Entry-Exit: What the Event Study Doesn't Tell Us {.smaller}

The model-free approach does not identify:

- The effect when the minimum wage introduction is [anticipated]{style="color: #2563eb"}
- The effect on markets that do *not* adopt
- The effect of repealing the minimum wage
- The effect of nominal changes to the minimum wage

- Each of these is arguably a more useful or compelling policy calculation

## Interpolating Variation {.smaller}

- Each tax reform is related: each takes the form $\mathcal{Y}(W,H)$ over wages and hours
- Estimating labor supply as a nonparametric function of $\mathcal{Y}$ would require implausible amounts of policy variation

- Models [**interpolate**]{style="color: #2563eb"} related --- but not functionally identical --- variation
- In the labor supply model, any $\mathcal{Y}$ enters through the budget constraint --- different reforms are comparable without new parameters
- See @Mullins2026 for a similar exercise with welfare reform experiments

## Articulated Variation {.smaller}

- [**Articulated variation:**]{style="color: #2563eb"} when a policy's effect can be modeled without additional parameters
- Typically involves *a priori* known changes to prices and endowments
- With structural parameters in hand, any observed variation in $\mathcal{Y}$ can be interpreted through the model

- **Counterexample:** the minimum wage in the entry-exit model
- Embodied as a reduced-form parameter $\phi_4$ because within-period production is not modeled
- The policy is *not* well-articulated: further nominal wage changes cannot be studied

# Marschak's Maxim

## The Principle {.smaller}

- We've explored the boundaries of causal inference and where modeling adds value
- Less instructive for *how* to do economic modeling
- More art than science --- but [Marschak's Maxim]{style="color: #2563eb"} is a useful guiding principle

:::{.callout-tip}
## Marschak's Maxim [@Marschak1953; @HeckmanVytlacil2007]
Researchers should specify the minimal set of model ingredients needed to answer their research question of interest
:::

## Marschak's Maxim: In Practice {.smaller}

- This may seem obvious --- the hard part is figuring out what the [minimal set]{style="color: #2563eb"} is
- Surprisingly easy to forget once deep inside your model
- *"Is this ingredient essential to my question?"* --- ask repeatedly

- Your research question is the mast you tie yourself to
- Decide what it is as early as possible

## Marschak's Original Example: Setup {.smaller}

- A monopolist chooses quantities to maximize profit; the government taxes quantities
- Demand: $p = \alpha_0 - \alpha_1 q$
- Profits (constant marginal cost $c$):
$$ \Pi(q) = (\alpha_0 - c - \tau)q - \alpha_1 q^2 $$
- Optimal quantity and tax revenue:
$$q^* = \frac{\alpha_0 - c - \tau}{2\alpha_1}, \qquad R = \frac{(\alpha_0 - c - \tau)\tau}{2\alpha_1}$$

## Marschak's Original Example: Three Problems {.smaller}

Suppose $q$ varies exogenously across markets. Three problems, three information requirements:

1. **Firm maximizing profit:** observe $\Pi = aq - bq^2$, set $q^* = a/2b$
2. **Government forecasting revenue:** only needs $\tau$ and observed quantities
3. **Government optimizing taxes:** extract $a = \alpha_0 - c - \tau$ from the reduced form, set $\tau^* = (\alpha_0 - c)/2$

- The setup is unconventional --- we typically assume agents already act optimally

## Lessons from Marschak's Example {.smaller}

Two insights:

1. Each question requires only certain *combinations* of parameters, each observable in [reduced-form]{style="color: #2563eb"} relationships
2. Once agents optimize, reduced-form relationships no longer suffice for forecasting policy effects

- Point (2) is the @lucas1976 critique, articulated two decades earlier
- Point (1): answers to some questions are invariant to particular model ingredients
- In practice we can't always guarantee this, but the insight motivates seeking a minimal set of ingredients

## Two-Stage Budgeting {.smaller}

- Utility over $K$ commodities $x$ with price vector $p$:
$$V_t(p,a,y) = \max_{x,a'} \left\{ v(x) + \beta\, \mathbb{E}\, V_{t+1}(p',a',y') \right\}$$
subject to:
$$ p \cdot x + \frac{1}{1+r}a' \leq y_t,\qquad a \geq \underline{a} $$
- Suppose the question concerns savings accumulation and intertemporal policy counterfactuals

## Two-Stage Budgeting: The Result {.smaller}

- If $v$ is [homothetic]{style="color: #2563eb"}, there exists a price index $P(p)$ and indirect utility $u$ such that [@Gorman1959]:
$$V(p,X) = u(X / P(p))$$
- Consumers first allocate aggregate expenditure across periods, then allocate within periods separately

- Classic application of Marschak's Maxim: if your question is about intertemporal allocation and counterfactuals don't affect intratemporal margins, no need to model intratemporal consumption under regularity conditions
- All you need is a reasonable price index
- @Gorman1959 and others sought assumptions permitting broader expenditure categories in demand systems

## Sufficient Statistics: Setup {.smaller}

- Static labor supply for $i \in [0,1]$: $H_i = \arg\max_h\; C - v(h)$, with $C = b + (1-\tau)hW_i$
- Transfer financed through taxes: $b = \tau \int W_i H_i\, di$
- Utilitarian welfare objective:
$$ V(\tau) = \int (C_i - v(H_i))\, di $$
- What is the welfare effect of a marginal expansion in $b$, financed through $\tau$?

## Sufficient Statistics: Derivation {.smaller}

- Applying the envelope theorem:
$$ \frac{dV}{d\tau} = -\int W_iH_i\,di + \int W_iH_i\,di + \tau \int \frac{d(W_iH_i)}{d\tau}\, di $$
- Assuming a constant elasticity of taxable income $Z_i = W_iH_i$ w.r.t. $\tau$:
$$\frac{dV}{d\tau} = \bar{Z}\; \varepsilon_{Z,\tau}$$
- The willingness to pay for the transfer **exactly cancels** the mechanical tax cost
- Only the [**distortionary cost**]{style="color: #2563eb"} of behavioral adjustments remains [@Harberger1954]

## Sufficient Statistics: Scope and Limits {.smaller}

- A larger literature on [*sufficient statistics*]{style="color: #2563eb"} in public finance derives approximations to local policy effects
- Welfare formulas depending on empirical quantities and elasticities
- Less reliance on specific functional forms

- As models grow more elaborate, required elasticities can become intractable [@kleven]
- Sufficient statistics remain a useful guide to understanding policy fundamentals
- A great example of Marschak's Maxim!

# Summary

## Takeaways {.smaller}

- Quasi-experimental methods are powerful when observed variation directly identifies the [counterfactual]{style="color: #2563eb"}
- Models become essential for [**welfare**]{style="color: #2563eb"}, [**optimal policy**]{style="color: #2563eb"}, or [**out-of-sample counterfactuals**]{style="color: #2563eb"}
- Models [interpolate]{style="color: #2563eb"} related variation through [articulated]{style="color: #2563eb"} policy channels --- connecting different reforms through the same structural parameters
- [**Marschak's Maxim:**]{style="color: #2563eb"} always aim for the minimal set of ingredients your question requires

## References
