Economic Curves Simulator: Armey and Laffer

The Armey curve is an economic model that describes the relationship between the size of government spending and economic growth. It posits that as government spending increases from zero, economic growth initially rises because essential public goods—like infrastructure, the legal system, and education—boost prosperity and stability. However, after reaching an optimal point, further increases in government size lead to diminishing returns, inefficiency, and crowding out of private investment, so economic growth then declines.

Curve Structure

The Armey curve is typically illustrated as an inverted U-shaped (parabolic) graph. On the horizontal axis is government spending (often as a percentage of GDP), and on the vertical axis is economic growth or societal welfare. The "peak" of the curve represents the optimal government size for maximum growth. Empirical research commonly situates this optimum between 20% and 30% of GDP, though the exact figure varies by country and time period.

Mathematical Formulation

The curve is often expressed with a quadratic equation:

Y = β₀ + β₁ · G + β₂ · G²

where Y is economic growth, G is government spending, and coefficients β₁ > 0, β₂ < 0 capture the concave shape—positive effects at first, negative effects after the turning point.

Understanding the Intercept (β₀)

The positive intercept value β₀ represents the natural growth rate without government intervention. This is the baseline economic growth rate that would occur when G = 0 (zero government spending), driven purely by private sector activity including:

• Private sector dynamics: Businesses, entrepreneurs, and markets operating without government involvement
• Natural productivity gains: Technological innovation, efficiency improvements, and capital accumulation driven by private actors
• Spontaneous market coordination: Voluntary exchanges and market mechanisms that create value
• Population growth effects: Demographic changes that contribute to economic expansion

In essence, β₀ represents the growth rate of a purely laissez-faire economy where all economic activity comes from private initiative, without any government provision of public goods, infrastructure, regulation, or spending. The rest of the curve then shows how government spending can initially enhance this natural growth (through beneficial public goods) before eventually hindering it (through inefficiencies and crowding out).

Policy Implications

• An undersized government fails to provide basic public goods, resulting in low growth due to lack of order, security, and infrastructure.
• An oversized government leads to inefficiencies, less incentive for private enterprise, and lower economic growth due to excessive regulation, high taxation, and crowding out.

Historical Context

The Armey curve is named after U.S. Representative Dick Armey, who popularized it in the 1990s. Its logic parallels the Laffer curve, but focuses on the broader scope of government spending rather than just tax rates.

Understanding Negative Growth Rate Values

In the mathematical model, government spending levels beyond the optimal point can produce negative growth rate values. While sustained negative growth (economic contraction) is possible in reality, these extreme negative values represent theoretical scenarios of economic dysfunction:

• Economic stagnation: Excessive government spending (beyond the optimal point) creates such massive inefficiencies and crowding out of private investment that the economy contracts rather than grows.
• Regulatory capture: Oversized government becomes dominated by special interests and rent-seeking behavior, leading to resource misallocation and economic decline.
• Investment flight: Private businesses and entrepreneurs abandon the economy due to excessive bureaucracy, high taxes needed to fund large government, and lack of economic freedom.
• Productivity collapse: Innovation, entrepreneurship, and productive economic activity become so constrained by government interference that overall economic output shrinks.

In practice, extremely negative growth rates would signal economic crisis and would likely trigger policy corrections before reaching the theoretical extremes suggested by the mathematical model.

Comparative Findings

• Optimal government size can vary with factors like openness of the economy, population size, and national context.
• Empirical studies find different threshold values for optimal government size in countries such as the U.S., France, Kenya, Turkey, and Georgia—ranging from less than 20% up to 30% of GDP.

In summary, the Armey curve highlights that both too little and too much government spending can harm economic growth, and seeks to identify the optimal share that maximizes prosperity.

The Laffer curve illustrates the theoretical relationship between rates of taxation and the resulting levels of consent to taxation. It suggests that at a tax rate of 0%, consent is zero (no taxes means no need for consent), and at 100%, consent becomes -100% because there is no incentive to work or produce under complete confiscation. Between these extremes, consent rises to a maximum and then falls.

Curve Structure

The Laffer curve is depicted as an inverted U-shaped graph. The horizontal axis represents the tax rate (from 0% to 100%), and the vertical axis shows consent to taxation. The peak indicates the optimal tax rate that maximizes societal acceptance, beyond which higher rates reduce consent due to disincentives and resistance.

Important note on the y-axis: The y-axis doesn't measure literal revenue but rather the theoretical economic health and viability of the tax system. Thus, negative values serve as a warning of extreme disincentives and economic dysfunction rather than actual negative tax revenue.

Why Consent to Taxation Increases Initially

Consent to taxation rises from 0% to the optimal point for several interconnected reasons:

• Social contract fulfillment: As tax rates increase from zero, governments can provide essential public goods (defense, law enforcement, infrastructure) that citizens value, creating legitimate reasons for taxation.
• Economic productivity gains: Moderate taxation enables public investments that boost overall economic productivity, making the tax burden worthwhile as citizens see their living standards improve.
• Perceived fairness: When tax rates are reasonable and well-spent, citizens view taxation as a fair contribution to collective welfare rather than confiscation.
• Institutional trust: Effective government services funded by moderate taxes build public trust in institutions, increasing voluntary compliance and acceptance.
• Collective benefit recognition: Citizens understand that moderate taxation enables shared benefits (education, healthcare, infrastructure) that would be impossible to achieve individually.

Beyond the optimal point, consent declines as the marginal costs of higher taxes (reduced incentives, economic inefficiency) outweigh the marginal benefits of additional government services.

Mathematical Formulation

The curve can be modeled with a quadratic equation:

R = β₀ + β₁ · t + β₂ · t²

where R is tax revenue, t is the tax rate, β₀ is often 0, β₁ > 0, and β₂ < 0 to create the parabolic shape. More advanced models may use different functional forms.

Policy Implications

• If tax rates are below the peak, increasing them can boost revenue.
• If rates are above the peak, cutting taxes can stimulate economic activity, leading to higher revenue over time.
• It highlights the trade-off between tax rates and economic incentives, influencing debates on supply-side economics.

Historical Context

Popularized by economist Arthur Laffer in 1974 during a meeting with U.S. policymakers, the concept has roots in earlier thinkers like Ibn Khaldun (14th century) and John Maynard Keynes. It gained prominence in the 1980s under Reagan's tax cuts.

Understanding Negative Revenue Values

In the mathematical model, tax rates above 50% produce negative revenue values. While negative tax revenue is impossible in reality, these values represent theoretical economic collapse scenarios:

• Economic collapse: Extremely high tax rates (above 50% in this model) create such massive disincentives that economic activity collapses to the point where the government would theoretically need to pay people to engage in economic activity.
• Underground economy: People completely exit the formal, taxable economy and move to barter, black markets, or simply stop productive activity altogether.
• Capital flight: Businesses and wealthy individuals leave the country entirely, taking their economic activity with them.
• Disincentive effects: Work, investment, and entrepreneurship become so unprofitable that they essentially cease, creating a death spiral where the tax base evaporates.

In practice, governments would see revenue approach zero rather than go negative, as the economy would collapse or transform into an untaxable shadow economy.

Comparative Findings

• The consent-maximizing rate is often estimated around 25% for a balanced budget maximizing long term growth.

In summary, the Laffer curve demonstrates that tax rates and revenue are not linearly related, emphasizing the need to find the optimal rate that maximizes government income without stifling economic growth.