This web app simulates six fundamental economic curves from Austrian and classical economics: (1) Armey Curve (government spending vs growth), (2) Laffer Curve (tax rates vs revenue), (3) J-Curve (economic shocks vs recovery), (4) Supply-Demand (market equilibrium), (5) Division of Labor (Adam Smith's Pin Factory), and (6) The Hockey Stick (capitalism's prosperity explosion). Adjust parameters to explore how policy, market forces, and economic freedom affect prosperity.
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.
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.
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.
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:
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).
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.
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:
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.
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.
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.
Consent to taxation rises from 0% to the optimal point for several interconnected reasons:
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.
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.
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.
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:
In practice, governments would see revenue approach zero rather than go negative, as the economy would collapse or transform into an untaxable shadow economy.
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.
The J-Curve shows what happens to an economy after a major policy change, like when a country suddenly devalues its currency or implements drastic budget cuts. The name comes from the "J" shape of the curve: things get worse before they get better. First, the economy dips down (the bottom of the J), then gradually recovers and eventually improves beyond where it started (the upward hook of the J).
Imagine the graph: time moves along the bottom (months or years), and economic health goes up and down on the side. When a big policy change happens, the economy starts at a normal level, immediately drops to a low point, then slowly climbs back up and eventually rises higher than before. This creates a shape that looks like the letter "J".
The curve follows this pattern:
Y(t) = Base + Slope · t - Depth · (1 - e^{-t / τ})
What this means in plain English:
When a government makes a big change, the negative effects hit immediately while the positive effects take time to materialize:
After communism fell in 1989, Poland implemented one of the first and most successful shock therapy programs under economist Leszek Balcerowicz. The country faced hyperinflation (600%), food shortages, and obsolete state-owned industries.
Unlike Russia's chaotic transition, Poland had several advantages: private agriculture still existed, democratic institutions were stable, privatization was more gradual and equitable, and the country maintained some social safety nets. This shows that shock therapy can work—but the details of implementation matter enormously.
In late 2023, Argentina implemented radical economic changes, creating a textbook J-Curve scenario:
By mid-2025, the recovery showed some fragility. The economy contracted again in Q2 2025, showing that J-Curves aren't always smooth. Global economic conditions and other factors can slow the recovery, reminding us that these reforms are risky and don't always work as planned.
Comparing successful cases (like Poland) with failures (like Russia in the 1990s) reveals critical factors that determine whether a J-Curve leads to recovery or becomes an L-shaped permanent decline:
What it means: Some functioning private sector, even if small (private farms, small businesses, market experience).
Why it matters: People need to know how markets work. Poland had private agriculture; Russia had complete state ownership. This gave Poles a head start in adapting to capitalism.
What it means: Legitimate government, rule of law, functioning courts, peaceful power transitions.
Why it matters: Without stability, reforms can't be properly implemented. Corruption flourishes, contracts aren't enforced, and public trust evaporates. Poland had stable democracy; Russia faced political chaos.
What it means: Careful transfer of state assets to private hands, avoiding concentration of wealth.
Why it matters: Rushed privatization (like Russia's voucher system) created oligarchs and extreme inequality. Poland's slower approach maintained public support and prevented wealth capture by a few insiders.
What it means: Unemployment benefits, food assistance, healthcare, pension protection during transition.
Why it matters: Cushions the blow for those who lose jobs. Reduces human suffering, prevents social unrest, and maintains political support for continuing reforms. Poland kept safety nets; Russia largely didn't.
What it means: Aid, loans, or investment from international institutions (IMF, World Bank, EU) or foreign investors.
Why it matters: Provides capital during the crisis, builds confidence, and helps stabilize currency. Poland had EU accession prospects; Argentina received $20B from the IMF.
What it means: Doing reforms in the right order: price liberalization first, then privatization; monetary stability before structural reforms.
Why it matters: Wrong sequence amplifies chaos. If you privatize before establishing market prices, assets get sold at arbitrary values, enabling corruption and wealth theft.
What it means: Government stays the course through 6-18 months of pain; population believes recovery will come.
Why it matters: If government caves to protests too early or reverses reforms, the pain was for nothing. Both Poland and Argentina maintained commitment despite protests.
What it means: Lower initial debt, younger population, intact trade relationships, moderate (not extreme) crisis.
Why it matters: Easier to recover from a shallow hole than a deep one. Countries with multiple compounding crises (debt + demographics + trade collapse) face much harder paths.
No single factor guarantees success, but countries with more of these advantages tend to achieve genuine J-Curves. Those lacking them (like Russia in the 1990s) often experience L-curves—a dip without recovery. Implementation details matter as much as the policies themselves.
When the curve dips below zero, it represents a genuine economic crisis: businesses closing, people losing jobs, money fleeing the country, and widespread hardship. These negative values show the real human cost of economic shocks before recovery begins.
Bottom line: The J-Curve teaches us that major economic reforms often cause significant pain before delivering benefits. It's like economic surgery - the recovery period is tough, but the goal is to emerge healthier than before.
The supply and demand model is the cornerstone of economics, illustrating how buyers (demand) and sellers (supply) interact to determine price and quantity in a market. Demand slopes downward: as price falls, more is bought. Supply slopes upward: as price rises, more is sold. Their intersection yields equilibrium—the price where quantity demanded equals quantity supplied, and markets clear without surplus or shortage.
The visual representation of supply and demand evolved over more than 50 years through the contributions of several pioneering economists:
Historical Timeline: 1838 (Cournot - demand curve) → 1870 (Jenkin - supply curve & combination) → 1890 (Marshall - popularization). This gradual development shows how economic tools evolve through the collaborative efforts of thinkers building on each other's insights.
This simulator uses curved supply and demand lines, which is more realistic than the straight lines often shown in textbooks. Here's why:
Straight lines are simpler for teaching basics, but curved lines better represent how real markets actually work. You'll see these curved shapes in actual economic data and professional analysis.
The supply and demand model reveals one of economics' most profound insights: market prices and resource allocation emerge spontaneously through voluntary exchange, without central planning or coordination. This phenomenon, called "spontaneous order," demonstrates that complex economic coordination doesn't require a guiding authority—it arises naturally from individuals pursuing their own interests.
Markets are not static equilibrium points but rather dynamic discovery processes where entrepreneurs and consumers constantly adapt to changing conditions:
This spontaneous order only functions when governments refrain from interfering with voluntary exchange. Price controls, production quotas, licensing restrictions, and other interventions short-circuit the discovery process:
The supply and demand model teaches epistemic humility: no individual, committee, or government possesses the knowledge scattered across millions of people to centrally plan an economy. Consumer preferences, production technologies, resource availabilities, and individual circumstances change constantly—far too fast and complex for any planning agency to track. The market doesn't require anyone to know everything; it only requires freedom for individuals to act on their local knowledge through voluntary exchange.
When left free to operate, supply and demand coordinates the actions of strangers who never meet, allocates resources to their highest-valued uses as judged by consumers, incentivizes innovation and efficiency, and creates prosperity—not because anyone designed it to do so, but because voluntary exchange naturally aligns individual incentives with social benefit. Government interference, however well-intentioned, disrupts this delicate coordination mechanism, creating shortages, surpluses, misallocations, and stifled innovation.
The graph plots price (vertical axis) against quantity (horizontal axis). The demand curve descends from left to right; the supply curve ascends. Shifts in either curve (due to taxes, subsidies, preferences, or costs) alter equilibrium, revealing surpluses (excess supply) or shortages (excess demand).
It's crucial to understand the difference between moving along a curve and shifting the entire curve:
This happens when the price changes but the curve itself stays in place. You're just moving to a different point on the same line. For example:
The entire demand curve shifts right or left when something OTHER than price changes consumer behavior:
The entire supply curve shifts right or left when something OTHER than price changes producer behavior:
You can simulate these shifts by adjusting the parameters:
Try it: Change 'a' from 120 to 140 and click Simulate. You'll see the demand curve shift right, creating a higher equilibrium price and quantity!
Curved Demand (Realistic Model):
P = a / (1 + bQ)^0.5
This creates a convex curve
showing diminishing marginal utility—each additional unit
provides less value to consumers.
Curved Supply (Realistic Model):
P = c + d·Q^1.3
This creates a convex curve
showing increasing marginal costs—each additional unit becomes
more expensive to produce.
Equilibrium: Found where demand price equals supply price (solved numerically)
You might notice the equilibrium shows decimal quantities like "55.75 units" instead of whole numbers. This isn't a mistake—it's how economic models work in practice:
Bottom line: While buying 0.75 of an individual apple doesn't make sense, an economy producing or consuming 55.75 million apples per year, or a market clearing at 55.75 tons of apples, is perfectly reasonable. The model represents aggregate market behavior, not individual purchases.
a - Demand Scale Factor:
This controls the overall level of the demand curve—how high it sits on the price axis. A higher 'a' means consumers are willing to pay more at any given quantity. Think of it as the "value intensity"—how much consumers value this product. For luxury goods or essentials, this would be high. For low-value items, it would be low. For example, if a = 120, consumers place high value on the product and are willing to pay premium prices.
b - Demand Curvature (Diminishing Utility Rate):
This controls how quickly consumer willingness to pay decreases as they buy more. A higher 'b' means the curve drops faster—each additional unit loses value more rapidly (strong diminishing marginal utility). A lower 'b' means consumers maintain their willingness to pay longer as quantity increases. For example, if b = 2, consumer valuations drop moderately with each additional purchase. Think: the first ice cream is worth $5, the second $3, the third $1.
c - Minimum Production Cost (Supply Base):
This is the base cost of production—the minimum price producers need even for small quantities. It represents fixed costs and baseline variable costs. If the market price falls below this, producers won't supply anything because they'd lose money on every unit. For example, if c = 20, producers need at least $20 per unit to cover their basic costs of materials, labor, and overhead.
d - Marginal Cost Growth Rate:
This controls how rapidly production costs accelerate as output increases. A higher 'd' means costs rise very quickly with more production (steep curve)—you quickly hit capacity constraints, need expensive overtime, less efficient equipment, etc. A lower 'd' means costs rise more gradually. For example, if d = 2, costs accelerate moderately. The curve shape (Q^1.3) means producing the 100th unit costs much more than the 10th unit—capturing the reality of diminishing returns and capacity constraints.
The realistic curved model captures key economic realities that linear models miss:
This curved model is more realistic than straight lines because it reflects how people actually behave: diminishing marginal utility for consumers and increasing marginal costs for producers.
In the mathematical model, you may see negative prices on the chart at very high quantities. While negative prices are unusual in real markets, they can occur in special circumstances and help illustrate important economic concepts:
When the demand curve drops below zero, it means consumers would need to be paid to take the product. Real-world examples include:
When the supply curve goes below zero at low quantities, it represents a theoretical scenario where producers would pay to produce. While rare, this can happen when:
In most real markets, producers simply stop supplying when prices fall too low, and consumers stop demanding at very high prices. Negative prices represent extreme theoretical scenarios that help us understand the mathematical model. For educational purposes, they demonstrate that these are linear projections—in reality, markets don't operate at these extreme quantities.
Rooted in Adam Smith's "invisible hand" (1776), formalized by Alfred Marshall in 1890. It underpins modern microeconomics, guiding antitrust, trade, and regulation.
In summary, supply and demand reveals the invisible forces that balance markets—or disrupt them under princely decree.
The Division of Labor & Market Size curve, based on Adam Smith's famous "Pin Factory" example, demonstrates one of the most powerful insights in economics: the division of labor—limited by the extent of the market—is the primary engine of productivity growth and prosperity.
In The Wealth of Nations (1776), Adam Smith observed that a single worker making pins alone could produce perhaps 20 pins per day. But when the process was divided into 18 specialized tasks shared among 10 workers, each worker could produce an astonishing 4,800 pins per day—a 240-fold increase in productivity! This wasn't just efficiency; it was a complete transformation of economic output.
This model demonstrates the fundamental importance of increasing returns to scale—the true driver of the 100-fold increase in per capita GDP over the last 250 years. Key insights include:
The model shows productivity as a function of specialization enabled by market size:
Productivity = Base × (1 + Workers × Specialization Factor)Returns Parameter
Where:
Government intervention—through regulation, trade barriers, taxes, and restrictions—limits market size and prevents specialization. This destroys the very mechanism that creates prosperity. Excessive regulations can be a primary cause of economic decline.
The division of labor requires secure private property rights and voluntary exchange. Without property rights, there's no basis for specialized production and trade. This is why socialism—which attacks private property—inevitably leads to poverty.
Contrary to Malthusian pessimism, more people mean larger markets, more specialization, and higher productivity for everyone. The world population multiplied 7x from 1810 to 2000, yet per capita output grew 9.2x—productivity grew faster than population!
Protectionism artificially limits market size, forcing people back toward self-sufficiency (like the solitary pin-maker producing only 20 pins). Free trade expands markets and enables the specialization that creates wealth.
From year 0 to 1800, per capita GDP grew only 48% (0.02% annually) under conditions of limited trade and small markets. From 1800 to 2000, with expanding free trade and market integration, per capita GDP grew 820% (multiplied by 9.2x). The difference? Increasing returns from expanding division of labor in larger markets.
Adjust the parameters below to see how market size, specialization depth, and the degree of increasing returns affect productivity. Notice how small increases in market size (enabling more workers/specialization) create exponential productivity gains—not linear ones. This is Adam Smith's insight and the foundation of modern prosperity.
"The Hockey Stick" demonstrates the most dramatic economic transformation in human history, showing a stunning visual pattern: for 1,800 years, per capita GDP barely moved—then suddenly exploded after 1800 when capitalism emerged.
When you plot per capita GDP from year 0 to present, the curve looks exactly like a hockey stick: flat for millennia (the handle), then shooting upward exponentially (the blade). This isn't a gradual improvement—it's a complete rupture with all prior human history.
Here are the precise numbers that demonstrate capitalism's transformative impact:
Bottom line: From 1800 to present, per capita GDP multiplied by more than 15 times globally. This happened in just 200 years after 1,800 years of near-total stagnation. The growth rate is still accelerating!
The Hockey Stick isn't just about wealth—it's about lifting humanity from misery:
"Capitalism lifted 90% of the world's population out of extreme poverty—and continues to do so at an accelerating pace."
This curve illustrates that the explosion occurred precisely when humanity adopted:
The timing is too perfect to be coincidence. For 1,800 years under feudalism, monarchy, and central planning, humanity was trapped in poverty. Then came Adam Smith's Wealth of Nations (1776), the American Revolution (1776), and the unleashing of market forces—and suddenly the hockey stick begins.
This simulator lets you explore different scenarios by adjusting the "capitalism adoption year." See what happens if capitalism had emerged earlier—or later. The model uses actual historical data to show:
"How can anyone demonize an economic system that not only lifted 90% of the world's population out of extreme poverty, but continues to drive unprecedented prosperity? There has never been, in all of human history, a time of greater prosperity than the one we live in today."
The Hockey Stick is visual proof that capitalism isn't just efficient—it's one of the greatest humanitarian achievements in human history.
The simulator uses a piecewise function matching historical phases. In the two centuries since capitalism's adoption (typically 1800), per capita GDP has multiplied by approximately 15-40× depending on the region, lifting billions out of poverty. Today's world average is around $25,000 PPP per capita (2023 IMF data), up from ~$600-700 in year 0 / ~$650 in 1800.
GDP(year) = { Pre-Capitalism: Base × (1.0002)(year - 0)
}
{ Post-Capitalism: Accelerating Exponential Growth }
Where growth rates shift through five distinct historical phases.
Unlike neoclassical models that struggle to explain the Hockey Stick (Solow-Swan only accounts for 15% of growth via capital accumulation), the Austrian approach emphasizes:
The name comes from the unmistakable shape: lay a hockey stick on its side, and that's what history looks like. This term is widely used because it's the most intuitive way to grasp the magnitude of capitalism's transformative achievement.