Hicksian demand function
Based on Wikipedia: Hicksian demand function
The Economist Who Taught Us to Separate Two Kinds of Pain
When the price of coffee doubles, you buy less of it. That seems obvious enough. But here's a puzzle that consumed some of the brightest economic minds of the twentieth century: why do you buy less?
Is it because coffee now looks expensive compared to tea, so you switch? Or is it because you're suddenly poorer—the same paycheck buys less stuff than it used to—and when people get poorer, they cut back on luxuries?
The answer, maddeningly, is both. And disentangling these two effects turns out to be crucial for understanding everything from how taxes distort behavior to why some goods seem to defy the laws of economics entirely.
Enter John Hicks, a British economist who shared the 1972 Nobel Prize in Economics. Hicks gave us a mental tool—a thought experiment, really—that lets us isolate one of these effects in pure form. The tool is called the Hicksian demand function, and while the name sounds forbidding, the idea is surprisingly intuitive once you see it.
Two Ways to Think About Buying Less Coffee
Let's stay with that coffee example. Before the price hike, you were happily buying three cups a day. The price doubles. Now you buy one cup.
The substitution effect is what happens when coffee becomes relatively more expensive compared to other things you enjoy. Tea used to cost the same as coffee; now it's half the price. Even if you had all the money in the world, the new price ratio might make tea look more attractive. You substitute away from the now-expensive thing toward the now-relatively-cheap thing. This effect always pushes you to buy less of the thing whose price rose. Always. No exceptions.
The income effect is different. When coffee prices double, your paycheck doesn't grow to match. In a very real sense, you've become poorer. Your dollars don't stretch as far as they used to. And when people become poorer, they change their consumption patterns. Usually, they buy less of most things. But not always—and this is where the story gets interesting.
The Imaginary Compensation
Here's Hicks's clever trick. He asked: what if we could observe the substitution effect in isolation? What if, after the price of coffee doubled, some benevolent fairy godmother slipped extra money into your wallet—exactly enough extra money to keep you precisely as happy as you were before?
Not happier. Not sadder. Exactly as satisfied with life as you were when coffee was cheap.
This imaginary compensation is the key to the Hicksian demand function. It tells us how much coffee you'd buy at the new, higher price if someone ensured you could still afford the same overall level of satisfaction you enjoyed before.
Economists call this "compensated demand" because your income has been compensated for the price change. They also sometimes call it the pure substitution effect, because the income effect has been artificially neutralized.
In contrast, the ordinary demand function—the one that shows what you actually buy at different prices given your fixed salary—is called Marshallian demand, named after Alfred Marshall, another giant of nineteenth and twentieth century economics.
What's the Difference in Practice?
The Hicksian demand curve always slopes downward. When something gets more expensive, you buy less of it—full stop. The substitution effect never reverses.
But the Marshallian demand curve, the one that shows actual purchasing behavior with fixed income, can sometimes do strange things. It usually slopes downward too. But occasionally, for certain peculiar goods, it slopes upward.
Giffen Goods: When Poverty Forces Perverse Choices
In nineteenth century Ireland, potatoes were a staple food for the poor. The story goes that when potato prices rose, poor Irish families actually bought more potatoes, not fewer. How could this be?
Before the price increase, a poor family might have divided their tiny food budget between mostly potatoes (cheap and filling) and a little bit of meat (expensive but nutritious). When potato prices rose, the family became effectively poorer. The income effect kicked in hard. And when very poor people become even poorer, they can no longer afford even that small amount of meat. They have to abandon it entirely and spend everything on the cheapest calories available—which means buying more potatoes than before.
This is a Giffen good, named after the Scottish statistician Robert Giffen who supposedly first observed this behavior. For Giffen goods, the income effect is so powerful that it overwhelms the substitution effect. The Marshallian demand curve slopes upward.
But even for Giffen goods, the Hicksian demand curve still slopes downward. If we compensated those Irish families enough to keep them at the same utility level after the potato price increase, they would certainly buy fewer potatoes and more of other things. The substitution effect, in isolation, always works the same way.
Normal Goods, Inferior Goods, and the Slope of Demand
This brings us to an elegant way of categorizing goods based on how people react to becoming richer or poorer.
A normal good is something you buy more of when your income rises. Restaurant meals, vacations, organic produce—these are normal goods for most people. When you get a raise, you upgrade.
An inferior good is something you buy less of when your income rises. Instant noodles, bus tickets, used clothing—these are things you move away from as you prosper. They're not inferior in quality necessarily; they're just things that people substitute away from when they can afford alternatives.
For normal goods, the income effect reinforces the substitution effect. When the price rises, you buy less because it's relatively expensive (substitution effect) and also because you're effectively poorer and you buy less of normal goods when you're poorer (income effect). Both effects push in the same direction.
For inferior goods, the effects work against each other. When the price rises, you buy less due to substitution, but the fact that you're now effectively poorer means you might buy more of this inferior good (since poorer people consume more inferior goods). The net effect depends on which force is stronger.
For Giffen goods, the income effect wins, and you see the paradox of upward-sloping demand.
Reading the Curves
Here's a practical insight. If you could somehow observe both the Hicksian demand curve and the Marshallian demand curve for the same good, you could immediately tell whether it's normal or inferior.
If the Hicksian curve is steeper than the Marshallian curve, the good is normal. The income effect is reinforcing the substitution effect, making the Marshallian response (which includes both effects) more dramatic than the pure substitution response.
If the Hicksian curve is less steep than the Marshallian curve, the good is inferior. The income effect is partially offsetting the substitution effect.
Why Economists Love This Distinction
You might reasonably ask: why does any of this matter? We can't actually observe the Hicksian demand function directly. We can't see what people would buy in a world where fairy godmothers perfectly compensate them for every price change. We can only observe what people actually do buy—the Marshallian demand.
The answer is that the Hicksian framework is enormously useful for thinking about welfare, taxation, and policy.
Measuring the True Cost of Price Changes
When prices rise, how much worse off are people really? If we just look at their spending patterns, we might get confused. They're buying different stuff now. Are they suffering, or are they cleverly substituting toward better deals?
The Hicksian framework gives us a clean answer. The true cost of a price increase is the amount of extra money you'd need to give someone to keep them at their original utility level. This is called the compensating variation, and it's directly connected to the Hicksian demand function.
Alternatively, we might ask: how much money could we take away from someone before a price change to make them as badly off as they'll be after the price change? This is called the equivalent variation. It's another welfare measure based on the same Hicksian logic.
Tax Policy and Deadweight Loss
When the government taxes a good, it raises its price. People respond by buying less—but why? Because of the substitution effect (the taxed good is now relatively expensive) and possibly the income effect (they have less real purchasing power).
The pure efficiency cost of a tax—the deadweight loss, in economists' jargon—comes from the substitution effect alone. People are shifting their purchases toward less-taxed goods not because those goods are genuinely better for them, but simply to avoid the tax. That shift represents wasted value.
The income effect, by contrast, doesn't create deadweight loss in the same way. It's a pure transfer: the government has your money instead of you having it. That might be good or bad depending on what the government does with it, but it's not an efficiency loss in itself.
Understanding the Hicksian demand function lets economists isolate the substitution effect and measure exactly how much efficiency is lost to taxation.
The Mathematical Elegance
There's another reason economists are fond of the Hicksian approach: it's mathematically nicer to work with.
When you write down the consumer's problem in Hicksian terms, you're minimizing expenditure subject to achieving a fixed level of satisfaction. The thing you're minimizing—total spending—is just prices multiplied by quantities. That's a linear function of the quantities, which makes the math considerably cleaner than the Marshallian alternative.
In the Marshallian world, you're maximizing utility subject to a budget constraint. Utility functions can be curved, kinked, or otherwise complicated. The Hicksian formulation sidesteps some of that complexity.
The two approaches are mathematically equivalent—they're described as "duals" of each other, meaning they're two sides of the same coin. But depending on what you're trying to prove or calculate, one formulation might be far more convenient than the other.
The Expenditure Function Connection
The Hicksian demand function is intimately connected to something called the expenditure function. The expenditure function tells you the minimum amount of money needed to achieve a given level of satisfaction at given prices.
A beautiful mathematical result called Shephard's lemma tells us that if you take the derivative of the expenditure function with respect to a particular price, you get the Hicksian demand for that good. In other words, the rate at which minimum required spending increases as a price rises tells you exactly how much of that good the compensated consumer would buy.
This relationship is more than a curiosity. It provides a practical way to derive Hicksian demands and to check that economic models are internally consistent.
The Slutsky Equation: Connecting the Two Worlds
The relationship between Hicksian and Marshallian demand is formalized in the Slutsky equation, named after the Russian economist Eugen Slutsky who derived it in 1915.
The Slutsky equation says, roughly: the total effect of a price change on quantity demanded (the Marshallian response) equals the substitution effect (the Hicksian response) plus the income effect.
This might sound like mere accounting, but it's actually a powerful decomposition. It tells you exactly how to break down any observed demand response into its component parts. And it provides testable implications. For instance, if you know someone's Marshallian demands at various price-income combinations, you can use the Slutsky equation to infer the underlying substitution patterns.
A Property That Might Surprise You
The Hicksian demand function has a peculiar property called homogeneity of degree zero in prices. This is jargon, but the idea is simple.
Suppose all prices double. Not just coffee, but tea, bread, rent, everything—uniformly twice as expensive. How would the Hicksian demand for coffee change?
It wouldn't change at all.
Why? Because the problem the consumer is solving is: "What's the cheapest bundle that gives me this level of satisfaction?" If all prices double, the relative prices stay the same. Coffee is still twice as expensive as tea, just like before. The same bundle that minimized spending before still minimizes spending now—it just costs twice as much in absolute terms.
This makes intuitive sense when you think about it. The Hicksian demand isolates the effect of relative prices, and a uniform scaling of all prices leaves relative prices unchanged.
Beyond the Textbook
John Hicks developed these ideas in the 1930s, building on earlier work by Irving Fisher, Vilfredo Pareto, and Alfred Marshall. The framework became a cornerstone of modern microeconomics, taught in every graduate economics program in the world.
But it's worth noting what the framework assumes. Consumers are imagined to have well-defined preferences that can be represented by a utility function. They're assumed to be rational in the sense that they minimize spending to achieve their goals. And the framework works best when we're talking about small changes to prices—large upheavals might push people into entirely different modes of thinking and choosing.
Still, as a tool for disciplined thinking about how prices affect behavior, the distinction between Hicksian and Marshallian demand remains invaluable. It reminds us that every price change has two faces: the pure logic of substitution toward better deals, and the raw reality of having more or less purchasing power than before.
The fairy godmother who perfectly compensates for price changes doesn't exist, of course. But imagining her helps us understand the world as it actually is.