Model long-run behavioural change.

Many generations coexist

The economy is not a single "representative agent." Many generations coexist — new labour-force entrants, workers at peak earnings, retirees drawing savings and pensions — each with a different remaining lifespan, accumulated wealth, and response to a policy change.

Age cohorts

OG-Core models S economically active age cohorts (typically 80, ages 21 to 100). Each period a new cohort is born and the oldest dies with some probability; the age distribution evolves with fertility, mortality, and immigration, reaching a stationary distribution in the long run.

Ability types

Each cohort splits into J ability types with distinct permanent labour productivity. High-ability types earn more per hour and typically save more, so reforms hit the income distribution, not just the average.

Three decisions each period

Each period, every household chooses how much to consume, work, and save to maximise expected lifetime utility, subject to a budget constraint.

Consumption utility

Consumption utility is CRRA (constant relative risk aversion); the risk aversion parameter governs the preference for smooth consumption. A Stone-Geary Cobb-Douglas form first aggregates the individual goods, allowing for minimum subsistence levels.

Labour disutility

Labour disutility is elliptical rather than a standard power function: the marginal cost of work is zero at zero hours and infinite at the upper bound, so solutions stay interior without costly occasionally-binding constraint methods.

Bequest utility

Bequest utility is the "warm glow" of leaving wealth at death, weighted by age-specific mortality. Its strength varies by ability type, calibrated to match the observed wealth distribution.

Budget constraint and Euler equations

Each period, savings returns, labour earnings, bequests received, government transfers, and pensions must cover consumption (including consumption taxes), income and wealth taxes, and savings carried forward.

Two Euler equations characterise the optimum: work until the after-tax wage equals the marginal disutility of labour; save until the discounted, after-tax return justifies forgoing consumption today.

CES technology

Production comprises M industries of perfectly competitive firms combining private capital, public capital (government infrastructure), and labour through a constant elasticity of substitution (CES) technology.

Substitution and productivity

The elasticity parameter sets how easily firms substitute capital for labour; at one, the function reduces to Cobb-Douglas. Total factor productivity varies by industry and over time, capturing sector-specific technological change.

First-order conditions

Profit maximisation yields the standard conditions: the wage equals the marginal product of labour; the rental rate equals the after-tax marginal product of capital, net of depreciation, tax deductions, and investment tax credits.

Public capital rents

A distinctive OG-Core feature: public capital generates economic rents. Firms cannot deduct the cost of government infrastructure, so its returns flow to private capital owners through an augmented rate of return.

Revenue and spending

The government taxes through five instruments — individual income, consumption, wealth, corporate income, and bequests — and spends on public goods, infrastructure, pensions, lump-sum transfers, and universal basic income. Debt finances the difference.

Income tax: Gouveia-Strauss functions

OG-UK uses the Gouveia-Strauss specification: the effective tax rate is a smooth, monotonically increasing function of income, the marginal rate its analytical derivative. Estimation runs on PolicyEngine-UK output from the Enhanced FRS — the Family Resources Survey enhanced with HMRC's Survey of Personal Incomes, the Living Costs and Food Survey, and the Wealth and Assets Survey.

Other tax instruments

The wealth tax is a progressive three-parameter function, spanning zero taxation to smoothly increasing marginal rates. Consumption taxes are linear rates by good, covering VAT and excise. Corporate income taxes are flat rates by industry.

Fiscal closure rule

Debt cannot grow without bound, so a fiscal closure rule activates after a specified period, bringing debt-to-GDP gradually to a target via government spending, transfers, or both — the modeller's choice.

Supply equals demand

Market clearing conditions tie households, firms, and government together: in equilibrium, supply equals demand in every market simultaneously.

Labour and capital markets

Labour: efficiency-weighted supply across ages and ability types must equal firm demand across industries. Capital: total domestic and foreign savings must equal firm capital demand plus government debt. The model is an open economy: parameters for foreign capital and foreign debt holdings set the degree of capital mobility.

Goods and debt markets

Each industry's output must equal consumption demand; the final industry clears residually, absorbing investment, government purchases, and remaining demand. Government debt splits between domestic and foreign holders.

What defines an equilibrium

An equilibrium is a set of prices — interest rate, wage, goods prices — at which households and firms optimise, the government budget holds, and all four markets clear: a large-scale nonlinear fixed-point problem.

Two-stage approach

The model solves in two stages: first the long-run steady state, then the transition path from today's economy to it.

Stage 1: Steady state

The steady state — all variables constant after removing trend growth — comes from a nested fixed-point algorithm. The outer loop guesses aggregate prices and quantities; the inner loop solves the Euler equations for every ability type and age, in parallel since types are independent. Aggregation, firm demands, and the government budget check yield new guesses; iterate to convergence.

Stage 2: Transition path (TPI)

Time Path Iteration (TPI) then solves the economy's path to the steady state over T periods: guess entire time paths for prices and quantities, solve every cohort's lifetime decisions given those paths (with rational expectations about future prices), check the implied paths against the guesses, iterate. This produces the year-by-year projections.

From theory to UK data

OG-Core supplies the theory; the OG-UK calibration layer anchors every parameter to official UK data: depreciation from ONS capital-stocks data (6.5%/yr), potential-output growth from the OBR EFO (1.1%/yr), the discount factor matched to the ONS household saving ratio, the world interest rate to Bank of England gilt yields, and a Frisch labour-supply elasticity of 0.35–0.4 (Blundell, MaCurdy & Meghir). Demographics — population by age, mortality, immigration — come from UN data; the state pension age matches current UK rules.

Tax function estimation

Taxes are not stylized wedges. OG-UK runs PolicyEngine-UK on household microdata under baseline and reform, computes marginal rates by perturbation — add £1 of employment (or dividend) income per adult, rerun, read the change in net income — and fits three-parameter Gouveia–Strauss schedules to the effective-rate microdata for each year of a three-year budget window. Marginal-rate schedules are analytical derivatives of the same fitted parameters, so ETRs and MTRs are consistent by construction; the model sees the reform's true shape across the income and age distribution.

Real-world mapping

The model works in abstract units. OG-UK anchors steady-state GDP to the ONS figure — live series with cached fallbacks — and scales all other variables proportionally, so results read as changes in billions of pounds.

Two complementary views

Paired with a static microsimulation engine, OG-UK gives two views of a UK tax or benefit reform. Static analysis shows which households gain or lose, and by how much. OG-UK shows GDP, investment, government revenue, interest rates, and wages as the economy adjusts over time.

Why both views matter

The two can tell different stories. A reform that raises revenue statically may raise less in practice as higher rates reduce labour supply and investment; one that costs revenue upfront might pay for itself through extra activity. OG-UK captures both dynamics.

Year-by-year transition paths

Rather than comparing two steady states, the transition path shows the year-by-year adjustment — how GDP dips in the first years, how interest rates respond, when revenue stabilises — on paths that map onto OBR forecast horizons.

What the model does not capture

OG-UK inherits the simplifications of structural macro models. Caveats for reading its outputs:

• Smoothed tax functions, not statutory bands. A single Gouveia–Strauss function fitted to PolicyEngine-UK output folds income tax and National Insurance together. Reforms changing average liability flow through cleanly; reforms whose mechanism is the kink itself (a new threshold, an allowance taper change) register only as far as the smoothed function shifts.
• Permanent ability types, no earnings risk. With J = 7 deterministic ability types and no idiosyncratic earnings shock within a type, precautionary savings and earnings-risk-driven inequality are not channels here — the same deliberate trade-off the OBR's own UK OLG model (compared on the Validation tab) makes.
• UK as a single entity. Calibration is to UK-wide aggregates, with no England / Scotland / Wales / Northern Ireland breakdown; devolved tax differentials and explicitly regional reforms sit below the model's granularity.
• Reforms are step changes. A PolicyEngine reform sets parameter values from a start date, holding thereafter; phased introductions, sunset clauses or year-on-year indexation changes need explicit scripting.
• Truncated horizon. The default 60-period transition assumes the steady state arrives by year 60; longer-run effects collapse into the steady-state anchor.

These are tractability choices, not bugs — worth knowing when reading the Showcase charts.