Reference

Symbols and terms

The notation and vocabulary of all six units on one page. Each entry links to the place where the course explains it.

The model

Parameters

θ, η and ρ are the same in every year. Unit 03 estimates σ, N and ζ for each year.

θElasticity across markets: How easily spending moves from one market to another. Each market is one sector. Fixed at 1.13125. 00 · AB08: demand
ηElasticity within a market: How easily buyers switch among the firms of one market; η > θ > 1. Fixed at 4.36875. It also sets the lowest possible markup, η/(η − 1) = 1.297, which a firm with a tiny market share charges. 05 · The markup floor
ρPersistence of productivity: How much of a firm’s log productivity carries over from period 1 to period 2. Fixed at 0.9. 02.3 · Parameters to moments
σProductivity dispersion: The standard deviation of the random shocks to log productivity. In period 1, log productivity has standard deviation σ/√(1 − ρ²). Estimated for each year. 02.3 · Parameters to moments
NPotential firms: The number of potential firms per market, a continuous number in the search. Estimated for each year. 03 · From a coordinate to an economy
MPotential firms in the solved economy: M = floor(N), the whole number of potential firms in each market. Entry and exit decide how many of them operate. 02.1 · Entry and exit
JMarkets: The number of markets in one economy: 10,000. Firm arrays have M rows and J columns.
ζFixed-cost control: Before any firm exits in period 1, ζ picks φ from the firms’ break-even values. Estimated for each year. 02.3 · Parameters to moments
φFixed labor: The labor each operating firm needs before it produces anything, in model labor units; the pages show it in millionths. It stays the same in every exit round and in period 2. 02.3 · Parameters to moments
ψ_SGA, ψ_COGSLabor shares of costs: The shares of labor in SGA costs and in the cost of goods sold, both 0.33. They turn wage bills into costs: SGA = Wφ/ψ_SGA. 02.3 · Build the three moments
γ, α_COGSTechnology: Returns to scale, 1.0252, and the elasticity of output with respect to the cost of goods sold, 0.8365. The elasticity with respect to capital is their difference. 02.3 · Build the three moments
φ̄, ϕLabor supply: The scale and the elasticity of household labor supply, 1 and 0.25: households supply φ̄W^ϕ units of labor at the wage W. 02.2 · Cournot and equilibrium

Solving one economy

Firms and markets

What the solver finds for given parameters and random draws.

sMarket share: A firm’s share of the sales in its own market. 02.2 · Cournot and equilibrium
εElasticity a firm faces: 1/ε = s/θ + (1 − s)/η. A firm with a tiny share faces about η; a firm with the whole market faces θ. 02.2 · Cournot and equilibrium
μMarkup: Price over marginal cost, μ = ε/(ε − 1). It rises with the firm’s market share. 00 · AB08: competition
CournotCournot competition: Firms choose quantities, and prices follow from demand. A firm with a larger share has more pricing power, so shares, markups and prices must be solved together: the share fixed point. 00 · AB08: competition
W, YWage and output: The wage and the economy’s output; the price of output is 1. They adjust until the labor firms use equals the labor households supply. 02.2 · Cournot and equilibrium
n, eLabor and employment: n is a firm’s production labor. Its employment is e = n + φ while it operates and 0 otherwise. 02.3 · Build the three moments
OperatingOperating, entering, exiting: Of the M potential firms in a market, the operating firms produce in a period. Between the two periods a firm can continue, enter (operate only in period 2) or exit (operate only in period 1). 02.1 · Entry and exit
RoundExit round: One pass of the entry and exit calculation: solve the economy, compare each firm’s profit with a cutoff for the whole economy, and remove the firms at or below it. The rounds stop when fewer than three exits are proposed. Rounds are solver steps, not dates. 02.1 · Entry and exit
PeriodTwo periods: Each economy is solved for two periods. Firms keep their identity and φ; productivity changes. The two periods give the employment reallocation moment; they are not two calendar years. 05 · Firms across years

What the model is fitted to

The three moments

Computed the same way in the model and in the data.

REALLEmployment reallocation: (JC + JD)/[½(E₁ + E₂)]: the jobs created and destroyed between the two periods, over average employment. 02.3 · Build the three moments
E_t, JC, JDEmployment and job flows: E_t is total employment in period t. Job creation JC adds up firms’ increases in employment; job destruction JD adds up their decreases. JC − JD = E₂ − E₁. 02.3 · Build the three moments
μ̄Sales-weighted markup: The average markup of the operating firms in period 1, each weighted by its share w of all sales in the economy. 02.3 · Build the three moments
SGASelling, general and administrative costs: Overhead costs. In the model they are the wage bill of fixed labor divided by ψ_SGA. 02.3 · Build the three moments
SGA / TCSGA share of total cost: Each operating firm’s SGA over its total cost in period 1, averaged with sales weights. It is not total SGA over total cost. 02.3 · Build the three moments
TargetsTargets: The data’s values of the three moments in a year. 03 · Read the targets

Units 01, 03 and 05

Estimation

Choosing parameters so that the model’s moments match the data’s.

MomentMoment: A statistic computed the same way in the data and in the model, such as a variance or an average. 01 · Moment matching
GMMGeneralized method of moments: Choose the parameters whose model moments, calculated directly, come closest to the data’s. 01 · Moment matching
SMMSimulated method of moments: The same, with the model’s moments computed from S simulated data sets. A larger S reduces the simulation error, not the sampling error of the data. 01 · Moment matching
CRNCommon random numbers: The same random draws for every parameter candidate, so that differences between candidates come from the parameters, not from new draws. 01 · Keep the random draws
BoxSearch box: Each year’s lower and upper limits for σ, N and ζ. 03 · Read the targets and ranges
SobolSobol points: A fixed sequence of points that spreads evenly over a box. The course maps the same 512 three-dimensional points into each year’s box to get 512 candidates for (σ, N, ζ). 03 · From a coordinate to an economy
LossWeighted loss: The sum of the three squared gaps between model and data, each multiplied by a weight: the inverse of that moment’s sample variance over the 44 years. The candidate with the lowest loss is the year’s estimate. 03 · Turn the gaps into one score
UntargetedUntargeted moment: A statistic the estimation did not use. Comparing it with the data tests the model. 05 · Untargeted moment
SplitWithin, reallocation, net entry: The three parts of the change in the sales-weighted markup between two years: markups changing inside continuing firms, sales moving among them, and firms entering and exiting. 05 · Split the change

Unit 04

Computing

Making the same calculation take less time.

WorkerWorker process: A separate program that solves its own economies. In the course each worker uses one linear-algebra thread. 04.01 · Parallel work
ShardShard: A share of the candidates that one process solves; no two shards share a candidate. 04.01 · Parallel work
SpeedupSpeedup and efficiency: With W workers, the speedup is T(1)/T(W) and the efficiency is T(1)/(W × T(W)), where T is the time to finish the same work. 04.01 · Parallel work
VectorizeVectorization: One array operation over all firms and markets instead of a loop that calls a function once per market. 04.02 · Vectorization
B × M × JBatch: B economies with the same M, stacked into one array so that the GPU works on all of them at once. 04.03 · GPU batching
KernelKernel: A function that runs on the GPU; each call is a launch. Fusing kernels combines several operations into one launch, so that intermediate arrays need not be written to memory. 04.04 · Kernel fusion
ResidentResident loop: An iteration that runs on the GPU together with its stopping checks, without returning to the host after each step. 04.05 · Resident iteration
ProfileProfiling: Measuring where the computing time goes before changing the code. 04.06 · Profiling
FP32, FP64Single and double precision: Floating-point numbers with about 7 and about 16 significant digits. The course checks its answers in FP64. 04.07 · Mixed precision
MixedMixed precision: FP32 only for a first guess of the market shares; the correction and the rest of the economy stay in FP64. 04.07 · Mixed precision
TritonTriton: A language for writing GPU kernels in Python. The optional lessons of unit 04 use it. 04 · Contents
Warm-upFair comparison: Time the same checked work with the same clock, after a warm-up, and keep every repeated time. 04.08 · Fair comparison