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Micro to Macro Fluctuations

Understanding the Transition from Micro to Macro Fluctuations

This section provides an overview of how micro-level fluctuations aggregate into macro-level fluctuations, both in linear and log terms. The scheme in Figure 1 illustrates the relationships and equations discussed below.

Scheme for the equations describing differences and variances of linear combinations, in linear and log scale.

Key Components

  • Firms:

    • Large fluctuations are represented by 10Fk10^{F_{k}}.
    • Group of firms defined by specific equations:
      • Quantile Sum: Refer to Eq quantile_sum.
      • Power Sum: Refer to Eq power_sum__narrow_bin.
    • Mean levels:
      • E[sD]E[s^*_D]: See Eqs E_N, E_L.
      • E[log(sD)]E[\log(s^{*_D})]: See Eqs Elog_N, Elog_L and Figures Elog_mu0_fnq_0, Elog_sigma0_fmu_0, Elog_mu0_fsigma_0.
    • Variance of means:
      • var[sD]var[s^*_D]: See Eqs alpha_1, var_N, var_L and Figures var_mu0_fsigma_0, fig:var_mu0_fnq_0.
      • var[log(sD)]var[\log(s^{*_D})]: See Eqs var_log_N, var_log_L.
  • Sector and Aggregate:

    • Level differences:
      • ΔSp\Delta S_p and ΔX\Delta X.
      • Equations: ΔSp\Delta S_p to ΔX\Delta X (Eq eqa), ΔSp\Delta S_p to FpΔlog(Sp)F_{p} \equiv \Delta \log(S_p) (Eq Ft_def).
    • Variance:
      • cov(Si,Sj)cov(S_{i}, S_{j}) and var(X)var(X).
      • Equations: cov(Si,Sj)cov(S_{i}, S_{j}) to var(X)var(X) (Eq var_cov_sum_def), cov(Fi,Fj)cov(F_{i}, F_{j}) to var(log(X))var(\log(X)) (Eq var_log_shocks).

Equations and Relationships

  • Growth Rate Equation:
    • ΔX\Delta X to Δlog(X)\Delta \log(X) (Eq growth_rates).
  • Variance in Terms of Deltas:
    • ΔX\Delta X to var(X)var(X) (Eq var_delta).
  • Logarithmic Relationships:
    • FpΔlog(Sp)F_{p} \equiv \Delta \log(S_p) to Δlog(X)\Delta \log(X) (Eq parts_linear_approx).
    • var(X)var(X) to var(log(X))var(\log(X)) (Eq var_log).

This structured approach helps in understanding how individual firm-level fluctuations can aggregate into sectoral or macroeconomic fluctuations, emphasizing the importance of both linear and logarithmic transformations.