Unit 8 Supply and demand: Markets with many buyers and sellers

8.6 Changes in supply and demand

Quinoa is a cereal crop grown on the Altiplano, a high barren plateau in the Andes of South America. It is a traditional staple food in Peru and Bolivia. At the beginning of the twenty-first century, as its nutritional properties became known, there was a huge increase in demand from richer, health-conscious consumers in Europe and North America. Figures 8.13a and 8.13b show that the price of quinoa trebled and production almost doubled in 10 years. Figure 8.13c indicates the strength of the increase in demand: spending on imports of quinoa rose from just $2.4 million to $43.7 million.

In this bar chart, the horizontal axis shows years from 2001 to 2011, and the vertical axis shows production of quinoa in thousands of tons, ranging from 0 to 90. Data for Ecuador, Peru, and Bolivia are shown. From 2001 to 2011, quinoa production increased from 45 thousand tons to 80 thousand tons, while the proportion of quinoa produced by each country stayed fairly constant, with Ecuador producing around 1%, and Bolivia and Peru dividing the remaining percentage roughly equally between them.
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Figure 8.13a The production of quinoa.

Jose Daniel Reyes and Julia Oliver. 2013. ‘Quinoa: The Little Cereal That Could’. The Trade Post. 19 March. Underlying data from Food and Agriculture Organization of the United Nations. FAOSTAT Database.

For the producer countries, these changes were a mixed blessing. While their staple food became expensive for poor consumers, farmers—who are among the poorest—benefitted from the boom in export sales.

In this line chart, the horizontal axis shows years ranging from 2001 to 2010, and the vertical axis shows the price of quinoa in dollars per ton, ranging from 0 to 1,400. Two lines show the price of quinoa over time in Bolivia and Peru, respectively. From 2001 to 2007, the price of quinoa stayed fairly constant in both countries, at around 450 for Bolivia and 350 for Peru. From 2007 to 2010, the price of quinoa increased dramatically to 1,300 for Bolivia and 1,200 for Peru.
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Figure 8.13b Quinoa producer prices.

Jose Daniel Reyes and Julia Oliver. 2013. ‘Quinoa: The Little Cereal That Could’. The Trade Post. 19 March. Underlying data from Food and Agriculture Organization of the United Nations. FAOSTAT Database.

How can we explain the rapid increase in the price of quinoa? In this section and the next, we analyse the effects of changes in demand and supply using simple examples. At the end of the next section, you can apply the analysis to the real-world case of quinoa.

In this bar chart, the horizontal axis shows years from 2001 to 2011, and the vertical axis shows the import demand for quinoa in millions of dollars, ranging from 0 to 50. Bars show the import demand in four regions: EU-27 countries, Canada, the United States, and all other countries. From 2001 to 2011, global import demand for quinoa increased from 2 million to nearly 45 million dollars. In all years, more than half of the total demand for quinoa comes from EU-27 countries, followed by Canada and the United States, then all other countries.
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Figure 8.13c Global import demand for quinoa.

Jose Daniel Reyes and Julia Oliver. 2013. ‘Quinoa: The Little Cereal That Could’. The Trade Post. 19 March.. Underlying data from Food and Agriculture Organization of the United Nations. FAOSTAT Database.

An increase in demand

Imagine a market in which hat sellers make and sell hats to consumers. Figure 8.14 shows the competitive equilibrium of this market. At point A, the equilibrium price equalizes the number of hats demanded by consumers and supplied by hat sellers. At this point, no one can benefit by offering or charging a different price, given the price everyone else is offering or charging—it is a Nash equilibrium.

Suppose that hat-wearing becomes more fashionable. More people want to buy hats. Follow the steps in Figure 8.14 to analyse the effects of this increase in the demand.

In this diagram, the horizontal axis shows the quantity of hats, in thousands, ranging from 0 to 60, and the vertical axis shows the price in dollars, ranging from 0 to 25. Coordinates are (quantity, price). There are three lines. The first is an upward-sloping line that starts at (0, 2) and is labelled Supply. The second is a downward-sloping line that connects the points (0, 20) and (40, 0), and is labelled Original demand. These two lines intersect at the point A (24, 8). The third line is labelled New demand. It lies above the original demand curve at all points, is flatter than the original demand curve, and intersects the supply curve at point C (32, 10). It passes through points B (24, 14) and D (37, 8).
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Figure 8.14 An increase in demand for hats.

The initial equilibrium: In this diagram, the horizontal axis shows the quantity of hats, in thousands, ranging from 0 to 60, and the vertical axis shows the price in dollars, ranging from 0 to 25. Coordinates are (quantity, price). There are two lines that intersect at the point A (24, 8). The first is an upward-sloping line that starts at (0, 2) and is labelled Supply. The second is a downward-sloping line that connects the points (0, 20) and (40, 0), and is labelled Original demand.
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The initial equilibrium

At point A, the market is in competitive equilibrium. 24,000 hats are sold at a price of $8 each. The supply curve is the marginal cost curve, so the marginal cost of producing a hat at point A is $8.

An increase in demand: In this diagram, the horizontal axis shows the quantity of hats, in thousands, ranging from 0 to 60, and the vertical axis shows the price in dollars, ranging from 0 to 25. Coordinates are (quantity, price). There are three lines. The first is an upward-sloping line that starts at (0, 2) and is labelled Supply. The second is a downward-sloping line that connects the points (0, 20) and (40, 0), and is labelled Original demand. These two lines intersect at the point A (24, 8). The third line is labelled New demand. It lies above the original demand curve at all points, and is flatter than the original demand curve.
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An increase in demand

Hat-wearing becomes more fashionable. More people want to buy hats at each possible price. The demand curve shifts to the right.

Excess demand: In this diagram, the horizontal axis shows the quantity of hats, in thousands, ranging from 0 to 60, and the vertical axis shows the price in dollars, ranging from 0 to 25. Coordinates are (quantity, price). There are three lines. The first is an upward-sloping line that starts at (0, 2) and is labelled Supply. The second is a downward-sloping line that connects the points (0, 20) and (40, 0), and is labelled Original demand. These two lines intersect at the point A (24, 8). The third line is labelled New demand. It lies above the original demand curve at all points, is flatter than the original demand curve, and passes through point D (37, 8). The horizontal distance between points A and D is excess demand.
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Excess demand

If the price remained at $8, there would be excess demand for hats—that is, demand would exceed supply (point D).

Raising price and quantity: In this diagram, the horizontal axis shows the quantity of hats, in thousands, ranging from 0 to 60, and the vertical axis shows the price in dollars, ranging from 0 to 25. Coordinates are (quantity, price). There are three lines. The first is an upward-sloping line that starts at (0, 2) and is labelled Supply. The second is a downward-sloping line that connects the points (0, 20) and (40, 0), and is labelled Original demand. These two lines intersect at the point A (24, 8). The third line is labelled New demand. It lies above the original demand curve at all points, is flatter than the original demand curve, and intersects the supply curve at point C (32, 10). It passes through points B (24, 14) and D (37, 8).
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Raising price and quantity

With excess demand, hat sellers who observe more customers realise that they could make higher profits both by raising the price and by producing more hats. They adjust prices and quantities until a new equilibrium emerges at point C.

The increase in demand moves the equilibrium from A to C, where more hats are sold at a higher price. Since the price is higher, additional hats are produced at higher marginal cost. Although more people buy hats at point C compared to point A, those with WTP between $8 and $10 (between points C and D) no longer want to buy.

We have described the scenario in which hats become more fashionable as an ‘increase in demand’. It’s important to understand exactly what we mean:

  • Demand is higher at each possible price, so the demand curve has shifted.
  • In response to this shift, there is a change in the equilibrium price. At the current price, sellers find that they can sell more hats than before.
  • Sellers respond to this price message by increasing the quantity supplied, along the supply curve.
  • But the supply curve itself has not shifted (the marginal costs of hat sellers have not changed): instead the equilibrium quantity supplied has increased because of the price change.

Figure 8.14 shows that if the supply curve were steeper (more inelastic), price would rise more and quantity would increase less. Conversely, if the supply curve were quite flat (elastic), then the price rise would be smaller and the quantity increase larger.

exogenous shock
An exogenous shock (for example a demand shock or a supply shock) is a change in one or more of the exogenous variables in a model—that is, variables that are othewise held constant by the modeller.
exogenous
Exogenous means ‘generated outside the model’. In an economic model, a variable is exogenous if its value is set by the modeller, rather than being determined by the workings of the model itself. See also: endogenous.

Shifts in demand (or supply) are often referred to as exogenous shocks in economic analysis. We start by specifying an economic model and find the equilibrium. Then we analyse how the equilibrium changes when something changes—the model receives a shock. The shock is called exogenous because the model doesn’t explain why it happened: it shows the consequences, not the causes.

Market equilibration through rent-seeking

How does the hat market adjust from A to C? At the original competitive equilibrium, the price of a hat was $8, and all buyers and sellers were acting as price-takers. When demand increases, they do not immediately know that the equilibrium price has risen to $10. If everyone were to remain a price-taker, the price would not change. But when demand shifts, some of the buyers or sellers will realise that they can benefit by being a price-maker, and decide to offer or charge a different price from the others.

For example, when a hat seller notices that every day there are customers wishing to buy hats, but none left on the shelf, she realises that some of them would be happy to pay more than the going price. And that some who had bought hats at the going price would have been willing to pay more. So she will raise her price the next day—price-taking is no longer her best strategy, and she becomes a price-maker. She does not know exactly where the new demand curve is, but she realises that now there are people who want to buy hats, but go home disappointed.

By raising the price, she raises her profit rate. If she was making normal profit in the original equilibrium, she is now earning an economic rent (at least temporarily)—that is, higher profits than are necessary to keep her hat business going.

Moreover, because hats are now being sold at prices above the marginal cost in the hat industry, some sellers will produce and sell more hats. As a result of the rent-seeking behaviour of hat sellers, a new equilibrium is reached at point C in Figure 8.14. At this point, the market again clears and none of the sellers or buyers can benefit from charging a price different from $10. They all return to being price-takers, until the next shock comes along.

disequilibrium rent
The economic rent that arises when a market is not in equilibrium, for example when there is excess demand or excess supply in a market for some good or service. In contrast, rents that arise in equilibrium are called equilibrium rents.

When a market is not in equilibrium, both buyers and sellers can act as price-makers, transacting at prices different from the previous equilibrium price and earning disequilibrium rents. In the opposite case of a fall in demand for hats, there would be excess supply at the original equilibrium price of $8. A customer at the hat shop might say to the hat seller: ‘You have quite a few unsold hats piling up on your shelf. I’d be happy to buy one of those for $7.’ To the buyer this would be a bargain. But it may also be a good deal for the seller, if at the reduced level of sales $7 is still greater than the marginal cost of producing a hat.

An increase in supply due to improved productivity

As an example of an exogenous increase in supply (a supply shock), think again about the case of the bread market. The supply curve represents the marginal cost of producing bread. Suppose that bakeries discover a new technique that allows workers to make bread more quickly. This will reduce the marginal cost of a loaf at each level of output. The marginal cost curve of each bakery shifts down, and so does the market supply curve.

Figure 8.15 shows the original supply and demand curves for the bakeries, and what happens when marginal costs fall.

In this diagram, the horizontal axis shows the quantity of loaves, denoted Q, and ranges from 0 to 10,000. The vertical axis shows the price in euros, denoted P, and ranges from 0 to 5. Coordinates are (quantity, price). An upward-sloping, convex curve starting from point (0, 1) is labelled original supply (marginal cost). A downward-sloping, convex curve starting from point (0, 4.75) and passing through point (10,000, 0.5) is labelled demand. The demand and original supply curves intersect at point A (5,000, 2). Another upward-sloping, convex curve is labelled new supply (marginal cost), lies below the original supply curve at all points, and intersects the demand curve at point B (6,100, 1.5).
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Figure 8.15 An increase in the supply of bread: a fall in marginal cost.

The initial equilibrium point: In this diagram, the horizontal axis shows the quantity of loaves, denoted Q, and ranges from 0 to 10,000. The vertical axis shows the price in euros, denoted P, and ranges from 0 to 5.In this diagram, the horizontal axis shows the quantity of loaves, denoted Q, ranging from 0 to 10,000. The vertical axis shows the price in euros, denoted P, and ranges between 0 and 5. Coordinates are (quantity, price). An upward-sloping, convex curve starting from point (0, 1) is labelled original supply (marginal cost). A downward-sloping, convex curve starting from point (0, 4.75) and passing through point (10,000, 0.5) is labelled demand. The demand and original supply curves intersect at point A (5,000, 2).
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The initial equilibrium point

The city’s bakeries start out at point A, producing 5,000 loaves and selling them for €2 each.

A fall in marginal costs: In this diagram, the horizontal axis shows the quantity of loaves, denoted Q, and ranges from 0 to 10,000. The vertical axis shows the price in euros, denoted P, and ranges from 0 to 5.In this diagram, the horizontal axis shows the quantity of loaves, denoted Q, ranging from 0 to 10,000. The vertical axis shows the price in euros, denoted P, and ranges between 0 and 5. Coordinates are (quantity, price). An upward-sloping, convex curve starting from point (0, 1) is labelled original supply (marginal cost). A downward-sloping, convex curve starting from point (0, 4.75) and passing through point (10,000, 0.5) is labelled demand. The demand and original supply curves intersect at point A (5,000, 2). Another upward-sloping, convex curve is labelled new supply (marginal cost) and lies below the original supply curve at all points.
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A fall in marginal costs

The supply curve corresponds to the bakeries’ marginal costs. When marginal costs fall, the supply curve shifts down.

An increase in supply: In this diagram, the horizontal axis shows the quantity of loaves, denoted Q, and ranges from 0 to 10,000. The vertical axis shows the price in euros, denoted P, and ranges from 0 to 5.In this diagram, the horizontal axis shows the quantity of loaves, denoted Q, ranging from 0 to 10,000. The vertical axis shows the price in euros, denoted P, and ranges between 0 and 5. Coordinates are (quantity, price). An upward-sloping, convex curve starting from point (0, 1) is labelled original supply (marginal cost). A downward-sloping, convex curve starting from point (0, 4.75) and passing through point (10,000, 0.5) is labelled demand. The demand and original supply curves intersect at point A (5,000, 2). Another upward-sloping, convex curve is labelled new supply (marginal cost) and lies below the original supply curve at all points.
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An increase in supply

The supply curve has shifted down. But another way to think of this change is to say that the supply curve has shifted to the right. Since costs have fallen, the amount that bakeries supply at each price is greater—an increase in supply.

Excess supply when the price is €2: In this diagram, the horizontal axis shows the quantity of loaves, denoted Q, and ranges from 0 to 10,000. The vertical axis shows the price in euros, denoted P, and ranges from 0 to 5.In this diagram, the horizontal axis shows the quantity of loaves, denoted Q, ranging from 0 to 10,000. The vertical axis shows the price in euros, denoted P, and ranges between 0 and 5. Coordinates are (quantity, price). An upward-sloping, convex curve starting from point (0, 1) is labelled original supply (marginal cost). A downward-sloping, convex curve starting from point (0, 4.75) and passing through point (10,000, 0.5) is labelled demand. The demand and original supply curves intersect at point A (5,000, 2). Another upward-sloping, convex curve is labelled new supply (marginal cost) and lies below the original supply curve at all points. The horizontal distance between point A and the new supply curve is excess supply.
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Excess supply when the price is €2

At the original price, there is more bread than buyers want (excess supply). The bakeries could benefit from lowering their prices and selling more bread.

The new equilibrium point: In this diagram, the horizontal axis shows the quantity of loaves, denoted Q, and ranges from 0 to 10,000. The vertical axis shows the price in euros, denoted P, and ranges from 0 to 5. Coordinates are (quantity, price). An upward-sloping, convex curve starting from point (0, 1) is labelled original supply (marginal cost). A downward-sloping, convex curve starting from point (0, 4.75) and passing through point (10,000, 0.5) is labelled demand. The demand and original supply curves intersect at point A (5,000, 2). Another upward-sloping, convex curve is labelled new supply (marginal cost), lies below the original supply curve at all points, and intersects the demand curve at point B (6,100, 1.5).
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The new equilibrium point

The new market equilibrium is at point B, where more bread is sold and the price is lower. The demand curve has not shifted, but the fall in price has led to an increase in the quantity of bread demanded, along the demand curve.

The improvement in the technology of breadmaking leads to:

  • an increase in supply (the supply curve shifts)
  • a fall in the price of bread
  • a rise in the quantity sold.

The demand curve does not shift, but the quantity demanded rises along the demand curve in response to the price change.

The supply curve would also shift if the number of firms in the market and their capacity for producing bread were to change. In the next section, we consider why and how this might happen, and how the equilibrium would change.

Exercise 8.5 Prices, shocks, and revolutions

Historians usually attribute the wave of revolutions in Europe in 1848 to long-term socioeconomic factors and a surge of radical ideas. But a poor wheat harvest in 1845 lead to food shortages and sharp price rises, which may have contributed to these sudden changes.1

The table shows the average and peak prices of wheat from 1838 to 1845, relative to silver. There are three groups of countries: those where violent revolutions took place, those where constitutional change took place without widespread violence, and those where no revolution occurred.

  1. Explain, using supply and demand curves, how a poor wheat harvest could lead to price rises and food shortages.
  2. Find a way to present the data to show that the size of the price shock, rather than the price level, is associated with the likelihood of revolution. (An Excel file containing this data is available for download).
  3. Do you think this is a plausible explanation for the revolutions that occurred?
  4. A journalist suggests that similar factors played a part in the Arab Spring in 2010. Read the post. What do you think of this hypothesis?
Avg. price
(1838–45)
Max. price
(1845–48)
Violent revolution (1848) Austria 52.9 104.0
Baden 77.0 136.6
Bavaria 70.0 127.3
Bohemia 61.5 101.2
France 93.8 149.2
Hamburg 67.1 108.7
Hessedarmstadt 76.7 119.7
Hungary 39.0 92.3
Lombardy 88.3 119.9
Mecklenburgschwerin 72.9 110.9
Papal states 74.0 105.1
Prussia 71.2 110.7
Saxony 73.3 125.2
Switzerland 87.9 146.7
Württemberg 75.9 128.7
Immediate constitutional change (1848) Belgium 93.8 140.1
Bremen 76.1 109.5
Brunswick 62.3 100.3
Denmark 66.3 81.5
Netherlands 82.6 136.0
Oldenburg 52.1 79.3
No revolution (1848) England 115.3 134.7
Finland 73.6 73.7
Norway 89.3 119.7
Russia 50.7 44.1
Spain 105.3 141.3
Sweden 75.8 81.4

Helge Berger and Mark Spoerer. 2001. ‘Economic Crises and the European Revolutions of 1848.’ The Journal of Economic History 61 (2): pp. 293–326.

Exercise 8.6 Cotton prices and the American Civil War

Read Section 8.1 and the ‘Great economists’ box about Friedrich Hayek. Use the supply and demand model to represent the following events described in the reading. In each case, indicate which curve(s) shift and explain the result.

  1. The increase in the price of US raw cotton (show the market for US raw cotton, a market with many producers and buyers).
  2. The increase in the price of Indian cotton (show the market for Indian raw cotton, a market with many producers and buyers).
  3. The reduction in textile output in an English textile mill (show a single firm in a competitive product market).

 

Question 8.8 Choose the correct answer(s)

Figure 8.14, reproduced here, shows the hat market before and after a demand shift. Based on this information, read the following statements and choose the correct option(s).

In this diagram, the horizontal axis shows the quantity of hats, in thousands, ranging from 0 to 60, and the vertical axis shows the price in dollars, ranging from 0 to 25. Coordinates are (quantity, price). There are three lines. The first is an upward-sloping line that starts at (0, 2) and is labelled Supply. The second is a downward-sloping line that connects the points (0, 20) and (40, 0), and is labelled Original demand. These two lines intersect at the point A (24, 8). The third line is labelled New demand. It lies above the original demand curve at all points, is flatter than the original demand curve, and intersects the supply curve at point C (32, 10). It passes through points B (24, 14) and D (37, 8). The horizontal distance between points A and D is excess demand.
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  • After demand increases, sellers will initially sell more hats at $8.
  • The adjustment to the new equilibrium is driven by the rent-seeking behaviour of the buyers and the sellers.
  • While the market adjusts, some buyers may pay more for a hat than others.
  • The new equilibrium price may be anywhere between A and B.
  • Sellers would not increase sales beyond A at $8, because their marginal cost would be higher than $8.
  • Adjustment requires prices and quantities to be changed. Rent-seeking provides incentives for buyers and sellers to make these changes.
  • Until the new equilibrium is reached, buyers and sellers may find opportunities to benefit from transactions at different prices.
  • In the new equilibrium, supply must equal demand (at the new demand curve). The new equilibrium price will be $10.

Extension 8.6 Changes in supply and demand

We have analysed the effects on equilibrium price and quantity of changes in market conditions by examining what happens in a diagram when the supply or demand curve shifts. In this extension, we carry out similar analyses algebraically, using calculus (differentiation) to determine whether equilibrium prices and quantities increase or decrease. This can be done even in cases where it is not possible to calculate explicit expressions for the equilibrium values.

If we know the demand and supply curve for a market, we can find the competitive equilibrium price and quantity by solving the two equations simultaneously for \(P\) and \(Q\). With a demand curve, \(Q=D(P)\), and supply curve, \(Q=S(P)\), the equilibrium price equalizes demand and supply:

\[D(P)=S(P)\]

If we can solve this equation to find \(P\), we can then find the corresponding equilibrium quantity by substituting back into the supply or demand curve. But what happens when one of these functions changes?

To model a demand or supply shock mathematically, we introduce parameters into the supply and demand curve to represent things other than price that affect the market for a good. Analysing how an equilibrium changes when a parameter changes is known in economics as comparative statics.

Linear supply and demand

Consider first the case that you analysed in Exercise E8.2, of a market with linear supply and demand functions:

\[D(P)=a-bP, \quad S(P)=c+dP\]

where \(a,\ b,\ c,\ d\) are constants. Assume that they are all positive, and that \(a>c\). This ensures that there is a single equilibrium with a positive price, \(P^*\), and positive quantity, \(Q^*\):

\[P^*=\frac{a-c}{b+d} \quad Q^*= a-bP^* = \frac{ad+bc}{b+d}\]

A demand shock

Suppose that in this market, there is a positive demand shock—that is, the quantity demanded is now higher at any given price. We can model a demand shock as an increase in the parameter, \(a\); if you think about the diagram, this represents a parallel rightward shift of the demand curve. It is similar to the demand shock in the market for hats shown in Figure 8.14 in the main part of this section, except that the slope of the curve doesn’t change.

One method for working out the effect is to find the new equilibrium when the demand curve is \(D(P)=a+\Delta a - bP\), and the supply curve is unchanged; \(\Delta a > 0\) represents the increase in \(a\). Then you can compare the new equilibrium price and quantity with the original ones.

But an easier way of working out what happens to price and quantity when a parameter changes is to think of \(P^*\) and quantity \(Q^*\) as functions of the parameters, and partially differentiate with respect to the parameter that changes. So for an increase in \(a\):

\[P^*=\frac{a-c}{b+d} \Rightarrow \frac{\partial P^*}{\partial a}= \frac{1}{b+d}\]

The derivative is positive; hence an increase in \(a\) results in an increase in \(P^*\). Similarly \(\frac{\partial Q^*}{\partial a}= \frac{d}{b+d}>0\), so \(Q^*\) increases too.

Of course, the derivatives tell us the effects of a very small increase in \(a\). But since they are positive wherever the equilibrium lies, we can deduce that a rise in \(a\), whether big or small, will always raise the equilibrium price and quantity.

Also, \(\frac{\partial Q^*}{\partial a}\) is less than 1. So the increase in equilibrium quantity is smaller than the increase in \(a\). The rise in \(a\) tells us how much the quantity demanded would increase if the price stayed the same. But \(P^*\) rises, so consumers buy less than they would have done without the price change.

In this example, you can easily obtain the same results diagrammatically by drawing an outward shift of the demand curve. But in other economic models, it can be difficult to be sure that your diagram captures all the possibilities, while using an algebraic method allows you to consider all the possible cases systematically.

Exercise E8.4 A negative supply shock

For the linear supply and demand functions described in this extension, \((D(P)=a-bP, \quad S(P)=c+dP)\), use derivatives to analyse what happens to equilibrium price and quantity when there is a negative supply shock (a decrease in \(c\)). Draw a supply–demand diagram to verify your answer.

The non-linear case

If the demand and supply curves are non-linear, it can be difficult to find an explicit solution for the equilibrium price and quantity. But it is still possible to model the effect of a shock that shifts one of the curves, and work out how it affects the equilibrium. We did this diagrammatically for the bread market example. Here we do the same thing algebraically.

We will write the demand curve for bread as \(Q=D(P,a)\), and the supply curve as \(Q=S(P,c)\). Introducing the two parameters, \(a\) and \(c\), enables us to model shifts in the curves—demand and supply shocks.

Think first about the demand curve. The quantity demanded decreases with the price, \(P\), as before; but it also depends on a parameter, \(a\), which captures consumer tastes. A high value of \(a\) represents a situation in which consumers have a strong liking for bread, so will purchase a high quantity at any given price. When \(a\) is low, less bread is demanded at each price. The dependence of demand on both \(P\) and \(a\) can be described using the partial derivatives:

\[\frac{\partial D}{\partial P} < 0; \quad \frac{\partial D}{\partial a} > 0\]

Then a positive demand shock can be represented as an increase in \(a\). It has the effect of increasing the quantity demanded at each price. When the demand curve is drawn in \((Q,P)\) space, as usual, it is drawn for a fixed value of \(a\). An increase in \(a\) shifts the demand curve to the right. Similarly, a fall in \(a\) represents a negative demand shock and shifts the curve to the left.

In the same way, a change in the parameter, \(c\), shifts the supply curve. We can think of \(c\) as representing technology. A rise in \(c\) corresponds to a technological improvement, which lowers the marginal cost of producing bread, and hence implies that bakeries will supply more bread at any given price. So both partial derivatives are positive:

\[\frac{\partial S}{\partial P} > 0; \quad \frac{\partial S}{\partial c} > 0\]

Equivalently, since the supply curve is the marginal cost curve, it shifts the supply curve down. This is the case illustrated in Figure 8.15 in the main part of this section.

An increase in \(c\) shifts the supply curve to the right.

For any given \(a\) and \(c\), the demand curve is downward sloping and the supply curve is upward sloping in \((Q,P)\) space. Hence, there is at most one equilibrium price \(P^*\), and corresponding equilibrium quantity \(Q^*\).

In the example of linear supply and demand, we knew the exact form of the supply and demand functions, and we were able to solve the equations explicitly for the equilibrium price and quantity, in terms of the parameters. Here, that is not possible. But we know that the equilibrium price \(P^*\) (if it exists) satisfies:

\[D(P^*, a)=S(P^*, c)\]

And that the quantity, \(Q^*\), is then given by:

\[Q^*= S(P^*, c)\]

These two equations implicitly determine \(P^*\) and \(Q^*\). They will depend on the two parameters; we can think of them as functions of \(a\) and \(c\):

\[P^* = P^*(a, c), \quad Q^* = Q^*(a, c)\]

Now, we can use what we know about \(P^*\) and \(Q^*\) to work out how they change when \(a\) or \(c\) changes. Specifically, we can use the technique of implicit differentiation to find expressions for their partial derivatives with respect to \(a\) and \(c\).

First consider a change in the parameter, \(a\) (a demand shock). We differentiate the equilibrium equation with respect to \(a\), remembering that \(P^*\) depends on \(a\):

\[\begin{align*} D(P^*, a)&=S(P^*, c) \\ \Rightarrow \frac{\partial D}{\partial P} \frac{\partial P^*}{\partial a}+ \frac{\partial D}{\partial a}&=\frac{\partial S}{\partial P} \frac{\partial P^*}{\partial a} \end{align*}\]

This equation can be rearranged to write \(\partial P^*/\partial a\) in terms of the other partial derivatives:

\[\frac{\partial P^*}{\partial a} =\frac{\frac{\partial D}{\partial a}} {\frac{\partial S}{\partial P}-\frac {\partial D}{\partial P} }.\]

The denominator of this fraction is positive, because we know that \(\partial S/\partial P \gt 0\) and \(\partial D/\partial P \lt 0\). The numerator is positive too, from the way we specified the demand function, above. Hence, we can conclude that \(\partial P^*/\partial a \gt 0\). Therefore, a positive demand shock (an increase in \(a\)) leads to an increase in the equilibrium price.

To find \(\partial Q^*/\partial a\), we can use the equation:

\[Q^*=S(P^*, c)\]

again remembering that \(P^*\) is a function of \(a\). Differentiating with respect to \(a\):

\[\frac{\partial Q^*}{\partial a} =\frac{\partial S}{\partial P^*} \frac{\partial P^*}{\partial a}\]

From this expression, since \(\partial S/\partial P^*\gt 0\) and we have just shown that \(\partial P^*/\partial a\gt0\), we deduce that \(\partial Q^*/\partial a\gt0\) too. Therefore, a positive demand shock (an increase in \(a\)) leads to an increase in both the equilibrium price and the equilibrium quantity, and a negative demand shock has the opposite effects.

This result is very general. We have demonstrated that the qualitative effects of a demand shock (any shock that increases demand at each price) on the equilibrium price and quantity are the same as the ones we obtained diagrammatically in the market for hats, whatever the precise form of the supply and demand functions—provided that they have the standard properties. That is, the demand curve slopes downward, and the supply curve slopes upward.

Work through Exercise E8.5 to perform the same analysis for a supply shock.

Exercise E8.5 Analysing the effects of a supply shock

Follow the steps to analyse the effects of a positive supply shock.

  1. Find the partial derivatives of \(P^*\) and \(Q^*\) with respect to \(c\). (Hint: Use implicit differentiation.)
  2. Use the equation for the supply curve and/or the equation for the demand curve to determine the sign of these partial derivatives. (Hint: Check the sign of the numerator and/or denominator.)
  3. Use a diagram to verify your mathematical answer.

Read more: Section 15.1 (on implicit differentiation), and Section 15.2 and the first two paragraphs of Section 15.3 (on comparative statics), of Malcolm Pemberton and Nicholas Rau. Mathematics for Economists: An Introductory Textbook (4th ed., 2015 or 5th ed., 2023). Manchester: Manchester University Press.

  1. Helge Berger and Mark Spoerer. 2001. ‘Economic Crises and the European Revolutions of 1848.’ The Journal of Economic History 61 (2): pp. 293–326.