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Econ-440 · economics · Economics

Declining business dynamism

LLM status Open unverified

Statement

Original subject group: Growth and productivity

Audit classification: Explicit published open question. Reconciling aggregate and industry patterns is an explicit research gap; the four-block decomposition is editorial. Verification concerns the cited version, not first priority or proof that this exact editorial specification remains unsolved on October 8, 2026.

Definitions and precise research target

Editorial mathematical specification.
Let $D_t$ be a predeclared business-dynamism index built from firm entry, exit, and employment reallocation in a fixed population. Consider a structural firm-dynamics model with parameter blocks $\theta=(\theta^N,\theta^C,\theta^I,\theta^R)$, representing demographics and labor supply, competitive conditions, intangible technology and diffusion, and regulation. Specify how each block affects entry costs, incumbent innovation, and firm growth; a residual trend must not silently receive a causal label. For initial and final parameter vectors $\theta_0,\theta_1$, let $D(\theta)$ be the equilibrium index. Decompose $D(\theta_1)-D(\theta_0)$ into block contributions using a declared Shapley ordering average over counterfactual combinations. Identify those contributions from moments containing both aggregate trends and cross-industry variation, augmented by credible policy or technological instruments. In particular, test whether the same parameter changes explain industry-level relationships among markups, entry, and reallocation rather than only matching aggregate comovement. Report observationally equivalent decompositions and parameter uncertainty, distinguish measured markups from primitive market power, and assess robustness to firm-population coverage. A successful result reconciles time-series decline with cross-sectional evidence and predicts held-out changes. The target is a quantitatively disciplined explanation for a stated period and economy, not a presumption that a single driver must dominate universally.

One precise index is $D_t=\omega_1 E_t+\omega_2 X_t+\omega_3J_t$, where $E_t$ and $X_t$ are firm entry/exit fractions in a fixed covered population, $J_t$ is a predeclared employment-reallocation rate, and nonnegative fixed weights sum to one. Do not mix establishment entry with firm entry without a mapping. For the four parameter blocks $K$, define $v(S)=D(\theta_1^S,\theta_0^{K\setminus S})$; the Shapley contribution for each block uses the same factorial weights as standard finite-set allocation. Every mixed-parameter economy must have a defined selected equilibrium; otherwise that decomposition is not defined. The allocation is convention-dependent, while identification requires data restrictions distinguishing parameter blocks. A measured markup is a price-to-estimated-marginal-cost ratio and does not by itself identify the underlying competitive primitive.

Variants and assumption changes

The following are editorial variants, not additional published open conjectures.
  1. Population mechanism. Vary labor-force growth and age composition while holding entry technology and competition parameters fixed. Require the resulting model to match age-specific firm entry and industry reallocation, beyond the aggregate decline.
  2. Industry reconciliation. Estimate a competition/entry mechanism that predicts both aggregate time trends and cross-industry markup-entry patterns. Compare alternative markup measures and public/private firm coverage under the same model restrictions.

References and source audit

Reconciling aggregate and industry patterns is an explicit research gap; the four-block decomposition is editorial.
  1. Brian C. Albrecht; Ryan A. Decker. 2024. Rising Markups and Declining Business Dynamism: Evidence From the Industry Cross Section. Section2, Related literature, final paragraph on reconciliation; dated March8,2024. https://www.federalreserve.gov/econres/notes/feds-notes/rising-markups-and-declining-business-dynamism-evidence-from-the-industry-cross-section-20240308.html.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-440-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-440

Business dynamism: exact attribution formulas and an identification obstruction

Problem ID: L26-436. Status: Unresolved. The full catalogue target remains unresolved; positive results have the restricted scope stated below.

An attribution rule is not an identification strategy

The canonical target decomposes a specified business-dynamism change into demographic, competition, intangible-technology and regulation blocks, with selected equilibria defined for all mixed-parameter economies. It further requires reconciling aggregate trends with cross-industry evidence. This attempt proves the exact Shapley interaction formula, constructs observationally equivalent decompositions, and gives a simple entry economy in which cross-sectional and time-series markup relationships have opposite signs. It does not determine the empirical contribution of any driver.

Let $K=\{N,C,I,R\}$. For initial and final primitive vectors define \[ v(S)=D(\theta_1^S,\theta_0^{K\setminus S}),\qquad S\subseteq K. \] Assume every one of these 16 economies has a finite, defined selected-equilibrium index; otherwise the displayed attribution is not well defined. The Shapley contribution is \[ \phi_j=\sum_{S\subseteq K\setminus\{j\}} \frac{|S|!(4-|S|-1)!}{4!}\{v(S\cup\{j\})-v(S)\}. \tag{1} \] This averages the marginal contribution over the 24 block orderings. It is a chosen attribution convention, not a causal assumption that the blocks can actually be randomized independently.

Proposition (Every interaction is divided among its participants): For $T\subseteq K$, let $\delta_T=\sum_{S\subseteq T}(-1)^{|T|-|S|}v(S)$. Then \[ v(S)=\sum_{T\subseteq S}\delta_T,\qquad \phi_j=\sum_{T\ni j}\frac{\delta_T}{|T|},\qquad \sum_j\phi_j=v(K)-v(\varnothing). \tag{2} \]
Proof: Substitute the definition of $\delta_T$ into its sum over $T\subseteq S$. For a fixed $v(U)$ the coefficient is $\sum_{T:U\subseteq T\subseteq S}(-1)^{|T|-|U|}$, equal to one if $U=S$ and zero otherwise by the binomial identity. This proves the first formula. For the component $\delta_T{\mathbf1}\{T\subseteq S\}$, block $j\in T$ contributes $\delta_T$ precisely when it appears last among the members of $T$ in a uniformly random ordering. That event has probability $1/|T|$ by symmetry. Blocks outside $T$ contribute zero. Linearity yields the second formula, and summing over $j$ counts each nonempty interaction once, giving the third. ∎

For example, a local response representation $D=D_0+n+c+i+r+\lambda nc$, comparing all-zero changes with fixed $(n,c,i,r)$, allocates $\phi_N=n+\lambda nc/2$ and $\phi_C=c+\lambda nc/2$, with $\phi_I=i$, $\phi_R=r$. At $D_0=0.5,n=c=-0.1,i=r=0,\lambda=1$, the total change is $-0.19$ and the two active contributions are each $-0.095$. This arithmetic check shows why reporting only the isolated one-block effects $-0.1,-0.1$ fails to sum to the actual change. The polynomial is a response illustration, not a substitute for proving equilibrium existence.

Identical observed trends, opposite mechanism allocations

Suppose the available moments, including every observed industry's entry index, depend on two primitive blocks only through their sum: \[ M_j(\theta)=g_j(\theta^N+\theta^C),\qquad j=1,\ldots,J. \] Consider baseline $(\theta^N,\theta^C)=(0,0)$ and two final explanations: $(a,0)$ and $(0,a)$. Every observed moment agrees in the two explanations, regardless of the number of industries. If $D(\theta)=g(\theta^N+\theta^C)$, Shapley assigns the entire change $g(a)-g(0)$ to $N$ under the first explanation and to $C$ under the second. Mixed counterfactuals differ, although actual endpoints and all specified observations agree.

This is exact nonidentification, not sampling uncertainty. Adding more moments of the same sum cannot repair it. Locally, a moment map $M(\theta)$ with a rank-deficient derivative has directions the linearized moment equation cannot distinguish; global injectivity requires still stronger restrictions. In a linear benchmark $M=A\theta$, full column rank identifies $\theta=(A^TA)^{-1}A^TM$, while a nonzero vector in $\ker A$ gives observationally equivalent parameter vectors. External instruments matter only if they add restrictions separating the relevant columns or otherwise break this equivalence.

A small economy reconciling two opposite correlations

Let industries differ in a persistent opportunity shifter $a_j$. At date $t$, let $c_t$ be a common competitive or access-cost primitive. There is a unit mass of potential entrants in each industry with independent project quality $\theta\sim\operatorname{Unif}[0,1]$. A project enters if its net private surplus $\theta+a_j-c_t$ is nonnegative. Restrict the parameter range to $0<c_t-a_j<1$. The unique entrant fraction is \[ E_{jt}=1+a_j-c_t. \tag{3} \] Let incumbent price divided by true unit marginal cost be an observed markup index $m_{jt}=m_0+a_j+c_t$, with $m_0$ chosen so this ratio exceeds one. This pricing rule can be rationalized by an isoelastic demand elasticity $\epsilon_{jt}=m_{jt}/(m_{jt}-1)>1$ and monopolistic price setting with unit marginal cost: maximizing $(p-1)p^{-\epsilon}$ gives $p=\epsilon/(\epsilon-1)=m_{jt}$. Demand shifters and the entry opportunity specification are stipulated primitives of this partial-equilibrium example.

Across industries at a fixed date, ${\operatorname{Cov}}_j(m_{jt},E_{jt})={\operatorname{Var}}_j(a_j)>0$. Within an industry over dates with changing $c_t$, ${\operatorname{Cov}}_t(m_{jt},E_{jt})=-{\operatorname{Var}}_t(c_t)<0$. These equalities follow by substituting (3), with constants dropping out. Thus a common change can raise aggregate markups and lower entry even though high-markup industries have higher entry in the cross section. No measurement error is required for this sign reversal.

This construction does not establish that the named competition mechanism caused a historical dynamism decline. The choice of how $c_t$ changes demand elasticity and entry costs is a maintained structural link. Its value is to show why aggregate comovement and industry covariance are separate moment restrictions and need not share a sign. Measured markups based on estimated marginal costs add another inference problem absent from the benchmark.

Uncertainty in counterfactual evaluations propagates transparently

Suppose every estimated hybrid equilibrium value has deterministic error at most $\epsilon$: $|\widehat v(S)-v(S)|\le\epsilon$ for all $S$. Equation (1) then gives \[ |\widehat\phi_j-\phi_j|\le2\epsilon. \tag{4} \] Indeed, each marginal difference has error at most $2\epsilon$, and the nonnegative ordering weights sum to one. A simultaneous confidence event for all 16 values therefore transfers to simultaneous attribution error bounds. Pointwise intervals for separate values cannot simply be treated as simultaneous without multiplicity control. Correlation across counterfactual errors can make (4) conservative, but never invalid on its stated event.

A successful empirical answer still needs firm rather than establishment population definitions, identified transition primitives, selected equilibria for all hybrids, reliable markup measures and held-out predictions. Shapley bookkeeping and a sign-reconciliation example are useful consistency checks; they neither identify parameter histories nor prove a universal driver of declining business dynamism.

Completion Estimate

35%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt derives exact Shapley interaction allocations, observationally equivalent mechanism histories, and a small entry model reconciling opposite time-series and industry correlations. Identified four-block firm dynamics, reliable markup measurement, counterfactual equilibria, and held-out historical predictions remain absent.

References

  • [fed] B. C. Albrecht and R. A. Decker (2024), Rising Markups and Declining Business Dynamism: Evidence From the Industry Cross Section, FEDS Notes, 8 March. Federal Reserve research note, 8 March 2024. The checked primary note motivates reconciling aggregate and industry evidence. The entry economy and attribution calculations here are illustrative, not estimated explanations of its data.

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