Unit 1 The supply side of the macroeconomy: Unemployment and real wages

1.7 The price-setting real wage (PS curve)

Figure 1.14 shows how the output per worker is divided between wages and profits.

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Figure 1.14

The price-setting curve, PS, is a horizontal line that shows the split in output per worker between wages and profits that arises when firms set their prices to maximize profits. This split will depend on the extent of competition in the markets for goods and services, and in the labour market. More market power for firms in either or both these markets reduces the real wage, which shifts the PS curve down.

If firms face little competition in the product market, it will be in their interest to set a higher price, with the result that their profit margins will be higher. The higher price pushes down the real wage. Likewise, if they face little competition for workers in the labour market, their wages will be low. In both cases, the PS curve is lower—workers get a lower share of output per worker and the real wage they get is lower. Equally, if competition in the economy is more intense, then profit margins will be lower and the real wage higher.

In this section, we analyse price setting and the PS curve more closely. Remember that we are making a simplifying assumption: the intensity of competition in both product and labour markets is constant (it does not depend on the firm’s output and employment).

A single firm’s price-setting decision

To understand the key ideas on which the price-setting real wage is based, think first of a single firm.

  • It employs many workers, paying them a nominal wage, W: The firm’s HR department determines the wage to be set, which depends on the desired number of employees (as explained in the previous section). Remember that all workers are equally productive and are paid the same wage.
  • It sells its product at a price, P: This is set by the firm to maximize profits.

How will the marketing department find the profit-maximizing price? First, it needs to consider the demand for the product: the price it chooses will determine how many items it can sell, and hence the firm’s revenue. Then, it can calculate the production costs by working out how many workers are needed to produce that amount of output, and—following the advice of HR—the wage that will be required. Putting all these things together, it can work out how much profit will be made at a given price. Then it can choose the price that maximizes profits.

Product demand

When the firm raises its price, demand for its product falls—in other words, fewer people will want to buy it. If it has competitors producing quite similar products to its own, demand will fall a lot when the price rises, because customers can easily find what they are seeking elsewhere.

In Section 7.5 of the microeconomics volume, we introduce in more detail the demand curve (the relationship between price and quantity) and the price elasticity of demand.

price elasticity of demand
The percentage change in demand that would occur in response to a 1% increase in price. We express this as a positive number. Demand is elastic if this is greater than 1, and inelastic if less than 1.
profit margin
The difference between the price of a product and its marginal production cost.

We measure how much demand falls when price rises by calculating the price elasticity of demand (PED): the percentage fall in the quantity demanded in response to a 1% increase in price. We say that demand is more elastic when the percentage fall is large, and less elastic if it is small. The more competition the firm faces from other firms selling similar products, the more elastic its demand will be.

When demand is more elastic, the firm will have to set a lower price, relative to the production cost of the product. Its profit margin will be low.

Marginal cost, the profit margin, and the markup

One way a firm can find the profit-maximizing price, P, is to think about whether it could make more profit than it does now if it produced one more unit of output. If the answer is yes, it should increase output, and continue doing so until profits no longer rise. If the answer is no, it should consider moving in the other direction.

marginal cost
The increase in total cost when one additional unit of output is produced. It corresponds to the slope of the total cost function at each point.
price markup
The price minus the marginal cost divided by the price. In other words, the profit margin as a proportion of the price. If the firm sets the price to maximize its profits, the markup is inversely proportional to the elasticity of demand for the good at that price.

In performing this calculation, it needs to know not only how much it will have to lower the price to sell an extra unit, but also how much its costs will rise. The additional cost of producing one more unit of output than it is currently producing is called the marginal cost (MC).

The firm’s profit margin is defined as the difference between the price and the marginal cost:

\[\text{profit margin} = P - \text{MC}\]

The profit margin as a proportion of the price is called the markup:

\[\text{markup} \ = \frac{P - \text{MC}}{P}\]

In some cases, the profit margin and markup may be zero, but a firm would never choose a price where its profit margin was negative. Note, too, that the markup will be less than 1.

The markup can also be defined as the profit margin as a proportion of the marginal cost. One definition can be written in terms of the other and the choice of which to use is a question of convenience. In the present context, the markup as a proportion of the price is used because it maps directly to the profit maximization problem.

Profit maximization is explained fully in Unit 7 of the microeconomics volume. We show there that when the firm sets its profit-maximizing price, the markup is inversely proportional to the PED:

\[\frac{P-\text{MC}}{P} = \frac{1}{PED}\]

If a firm’s PED is low (it faces little competition) we say that it has product market power: it can raise prices without losing many customers. Then it will set its price so that the profit margin and markup are high.

The profit-maximizing price

At the profit-maximizing price, the markup is equal to the inverse of the price elasticity of demand, \(1/\text{PED}\). Our assumption that the intensity of competition faced by the firm in the product market is constant means that the PED does not vary with the firm’s output. So the profit-maximizing markup is a constant too. We will use the Greek letter \(\mu\) (pronounced ‘mu’) for the profit-maximizing value of the markup: \(\mu = 1/\text{PED}\). Then we can say that the firm sets its price so that:

\[\frac{P-\text{MC}}{P}=\mu\]

Rearranging this expression:

\[P = \frac{1}{1 - \mu} \text{MC}\]

This tells us that the profit-maximizing price is proportional to the marginal cost of production.

The factor \(1/(1-\mu)\) is a number greater than 1. If the firm faces little competition in the product market, the demand elasticity is low, so the markup, \(\mu\), is high, and the price is a higher multiple of marginal cost.

The marginal cost

average cost
The total cost of producing the firm’s output divided by the total number of units of output produced.

In our model, labour is the firm’s only cost, and output per worker, \(λ = Y/N\), is constant (it doesn’t depend on employment, N). This means that the average cost of a unit of output, AC, is proportional to the wage:

\[\text{AC} = \frac{\text{total cost of labour}}{\text{units of output}} = \frac{WN}{Y} = \frac{W}{\lambda}\]

However, the wage (determined by HR) depends on the number of workers the firm wants to employ. To increase output, the firm will not only have to increase employment, but also pay a higher wage to all its workers. Remember that the reservation wage curve (and therefore also the no shirking wage curve) is upward sloping. As we explain further in the extension to this section, the marginal cost (MC, the additional cost of producing one more unit of output) is given by:

\[\text{MC} = (1+\eta)\frac{W}{\lambda}\]

In this expression, the term \(η\) (pronounced ‘eta’) is a measure of how much wages have to be increased, which depends on competition in the labour market. Our assumption that the intensity of labour market competition is constant means that \(η\) is constant and the firm’s marginal cost is proportional to the wage.

In Unit 6 of the microeconomics volume, we explain more fully why firms facing little competition in the labour market have the power to keep their wages low.

An example of labour market power comes from the early days in Silicon Valley when there were few employers of animation engineers. Two companies, Pixar and Lucasfilm, collaborated to keep down wages from the mid-1980s. This ended only in 2008, with payouts to the affected engineers and wage increases elsewhere in the industry.1

wage markdown
When employers have labour market power they ‘mark down’ the wage. In our model with constant output per worker, the markdown is equal to the difference between the marginal cost of output and the average wage cost of a unit of output, as a proportion of the average wage cost of a unit of output.

We can call \(η\) ‘the markdown’. Just as the markup, \(\mu\), indicates how far the price is ‘marked up’ above the marginal cost, the wage markdown, \(η\), indicates how far the wage is marked down so that the wage cost per unit, \(W/λ\), is below the marginal cost. And both depend on the degree of competition. The markup, \(\mu\), is high when product market competition is low; likewise the markdown, \(η\), is high when labour market competition is low. In the extreme case of one seller, a monopoly firm sets a higher price than when there are competing sellers; similarly, in the case of a single firm in a labour market (think of a ‘company town’), the firm sets a lower wage than when jobseekers have more choice of employers. This is the case of monopsony.

The price-setting real wage

The profit-maximizing price, P, is proportional to MC, and MC is proportional to the nominal wage, W.

We can deduce that—whatever the level of output and employment—the firm will set a price proportional to its nominal wage, W. In other words, the ratio between W and P—the price-setting real wage—is constant.

In the extension, we show how to rearrange the equations above for P and MC to get a simple expression for the real wage. It is:

\[\frac{W}{P} = (1 - \sigma) \lambda\]

where \(σ\) (pronounced ‘sigma’) is a constant between 0 and 1.

This equation tells us that the firm sets its price so that each worker is paid a real wage that is a constant share (\(1 – σ\)) of the output per worker, \(λ\). The firm gets the remaining share of \(λ\)—that is, it makes a profit of \(σλ\) on each worker.

We also show in the extension that the profit share, \(σ\), depends on competition in the product and labour markets (that is, on \(\mu\) and \(η\)). If both markets are highly competitive, the profit share, \(σ\), will be low and the wage share (\(1 – σ\)) will be close to 1. If there is less competition in either market, the firm will get a higher share of output and wages will be lower.

The economy’s price-setting curve

Since all firms are identical and make the same choices, the aggregate price-setting real wage, \(w = W/P\), is a constant share of output per worker, whatever the level of aggregate employment:

\[w = (1 - \sigma) \lambda\]

Figure 1.22 shows the price-setting curve, and the shares of output that go to workers and firms.

In this diagram, the horizontal axis shows employment, N, and the vertical axis shows output per worker (also called productivity, labelled lambda) and real wage labelled w. There is a vertical line at the end of the horizontal axis to denote the labour force. There are two horizontal lines intersecting the vertical axis, the price-setting curve intersects the vertical axis at w, and the output per worker line intersects the vertical axis at lambda and is above the price-setting curve. The difference between w and the origin on the vertical axis is the real wage, (W/P = (1-sigma)lambda), and the difference between lambda and w is the real profit per worker (lambda*sigma).
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Figure 1.22 Determinants of the PS real wage.

Question 1.5 Choose the correct answer(s)

Suppose the nominal wage (W) is $10, the markup (\(μ\)) is 0.5, the markdown (\(η\)) is 0.25, and the marginal cost is $5. Based on this information and the discussion in this section, read the following statements and choose the correct option(s).

  • The elasticity of demand is 4.
  • The marginal cost is equal to the price.
  • The real wage (\(W/P\)) is 1.
  • The average product of labour (\(λ\)) is 2.5.
  • The elasticity of demand is the inverse of the markup, so is \(1 / 0.5 = 2\).
  • With a markup of 0.5, the marginal cost is equal to half of the price (\(\text{MC} = (1 − 0.5)P\)).
  • The marginal cost ($5) is half of the price, so the price P is $10. The nominal wage is also $10, so the real wage is 1.
  • Use the formula \(\text{MC} = (1+η)\frac{W}{λ}\) and substitute MC = $5, \(η\) = 0.25, and W = $10 to get \(λ\) = 2.5.

Question 1.6 Choose the correct answer(s)

Suppose the nominal wage (W) is $20, the markup (\(μ\)) is 0.4, the markdown (\(η\)) is 0.5, and the marginal cost is $15. Assume that the firm’s only costs are labour costs. Based on this information and the discussion in this section, read the following statements and choose the correct option(s).

  • The real wage (\(W/P\)) is $1.25.
  • The average product of labour (\(λ\)) is 0.5.
  • The firm’s profit share is 0.6.
  • The average cost (AC) is $10.
  • The price P is $25 (= \(\frac{1}{1-\mu}\text{MC}\)), so the real wage W/P is 0.8.
  • Use the formula \(\text{MC} = (1+η)\frac{W}{λ}\) and substitute \(\text{MC} = \$15\), \(η = 0.5\), and \(W = \$20\) to get \(λ = 2\).
  • Using the equation for the price-setting curve \(\frac{W}{P} = (1 - \sigma) λ\), substitute 0.8 for the real wage and \(λ = 2\) to get \(\sigma = 0.6\).
  • The average cost is the wage divided by the average product of labour, which is \(20/2 = $10\).

What shifts the price-setting curve?

Both the diagram in Figure 1.22 and the equation show that the price-setting real wage will increase if productivity increases or if the profit share, \(σ\), falls. But what mechanisms bring this about?

  • Higher labour productivity: Higher productivity reduces costs and firms cut their prices. The result is a higher real wage.
\[\begin{align*} \uparrow \text{productivity} \text{ }\Rightarrow \text{ } \downarrow \text{ }\text{MC} \text{ }\Rightarrow \text{ }\downarrow P \text{ }\Rightarrow \text{ }\uparrow \frac{\displaystyle W}{\displaystyle P} \end{align*}\]
  • More competition for goods resulting in a lower markup: More intense competition among sellers in the economy reduces the markup firms can incorporate in their prices; the share of profits will be lower and real wages will rise.
\[\begin{align*} \downarrow \text{markup}\text{ } \Rightarrow \text{ }\downarrow P \text{ }\Rightarrow\text{ } \uparrow \frac{\displaystyle W}{\displaystyle P} \text{ } \Rightarrow \text{ } \downarrow \sigma \end{align*}\]
  • More competition for labour resulting in a lower markdown: More intense competition among employers seeking to hire workers reduces the markdown in the wages firms set: the share of profits is lower and real wages rise.
\[\begin{align*} \downarrow \text{markdown}\text{ } \Rightarrow \text{ }\uparrow W \text{ }\Rightarrow\text{ } \uparrow \frac{\displaystyle W}{\displaystyle P} \text{ } \Rightarrow \text{ } \downarrow \sigma \end{align*}\]

Exercise 1.4 Shifts in the price-setting curve

Suppose the nominal wage (W) is $15, the marginal cost is $7.5, the average product of labour (\(λ\)) is 2, and the markup is 25%.

  1. What is the real wage? Draw a diagram like Figure 1.22 to illustrate your answer.

For each of the following situations, explain (using your diagram from Question 1) what happens to the price-setting curve, and calculate the new real wage.

  1. The average product of labour (\(λ\)) doubles from 2 to 4.

  2. The firm’s profit share changes to 0.2.

What determines the height of the price-setting curve?

We now summarize what determines the height of the price-setting curve. In Unit 2, we will analyse how government policy can affect this.

  • Labour productivity: For any given markup and markdown, the level of labour productivity—how much a worker produces in an hour—determines the real wage on the price-setting curve. The greater the level of labour productivity (or equivalently, the average product of labour, called lambda, \(λ\)), the higher the real wage that is consistent with the given profit share.

Higher productivity pushes the price-setting curve upwards. Shares stay the same; the price-setting real wage is higher.

labour market power
A firm has labour market power (sometimes called monopsony power) if it can reduce the wage it needs to pay its workers by lowering the number of workers that it employs.
  • Competition: You know that the intensity of competition facing firms determines the extent to which they can profit by charging a price that exceeds their costs, that is, their markup. The more intense the competition, the lower the markup. Since this leads to lower prices across the whole economy, it implies higher real wages, which pushes the price-setting curve upwards. Similarly, if employers have less labour market power (a lower markdown) then the wage is higher and the price-setting curve shifts upward.

More competition means workers get a bigger share; the price-setting real wage is higher.

Extension 1.7 Deriving the price-setting real wage

In the main part of this section, we provide an outline of the derivation of the price-setting real wage set by the firm. In this extension, we work through the derivation in more detail, to explain where each of the results comes from. We make more use of equations, but calculus is not required to understand them.

If you are familiar with wage setting, price setting, and elasticity, from Units 6 and 7 of the microeconomics volume, this extension will be easier to understand. But even if you are not, the extension will still provide some additional insight into where the macroeconomic PS curve comes from.

We derive the price-setting real wage in three steps, exactly as we did in the main part of the section. Most of the additional explanation comes at steps two and three.

Step 1: The profit-maximizing price

We assume for our model that the intensity of competition in the product market, as measured by the price elasticity of the demand (PED) faced by the firm, is constant. Then, as explained before, the firm’s profit-maximizing price is proportional to the marginal cost of producing output:

\[P = \frac{1}{1-\mu}\text{MC}\]

where \(\mu\) is the profit-maximizing markup, which is a constant equal to \(1/\text{PED}\). Since the intensity of competition is constant, PED and \(\mu\) are constant, too. We know that \(\mu\) will be a number between 0 and 1; we can think of it as representing the firm’s product market power; when \(\mu\) is high (close to 1), price is a higher multiple of marginal cost.

Step 2: The marginal cost

The next step is to work out the marginal cost—that is, the additional costs the firm would incur if it produced one more unit of output.

Labour is the firm’s only input, and only cost. With employment N, output \(Y = λ N\), where output per worker, \(λ\), is constant (it doesn’t depend on employment). To increase output, the firm has to increase employment. And to recruit and motivate more workers, it has to raise the wage, W, along its no-shirking wage curve.

Since \(N = Y/λ\), the firm’s cost function—the total cost of producing output Y—is:

\[\begin{align*} C &= WN \\ &=\left( \frac{W}{\lambda} \right) Y \end{align*}\]

\(W/λ\) is the wage cost per unit of output—that is, the average cost. To simplify the algebra, we will call this average cost a, so that the cost function is:

\[C = aY\]

But we have to remember that a increases when output rises, because the wage increases.

The mathematical symbol \(\Delta\), pronounced ‘delta’, is often used as a shorthand for the ‘the change in’. This calculation of marginal cost is really an approximation, but it is a good one provided that a one-unit change in Y is small relative to total output. (We could calculate marginal cost exactly using calculus methods.)

The calculation is similar to the one used in Section 7.5 of the microeconomics volume to calculate marginal revenue. There we draw a rectangle to represent total revenue, and consider how the rectangle changes when output rises by one unit. If you like, you can do the same here by drawing a rectangle to represent total costs, \(C = aY\).

To find the marginal cost, suppose that output Y is increased by one unit. We will write \(Δa\) for the corresponding change in a. The increase in costs, MC, is the cost of the extra unit, plus the extra costs of all the other units:

\[\text{MC} = a+ Y \Delta a\]

We can rearrange this expression to get:

\[\begin{align*} \text{MC} &= a \left( 1 + Y \left(\frac{\Delta a}{a} \right) \right) \\ &= a \left( 1 + \frac{\Delta a/a}{1/Y} \right) \end{align*}\]

\(1/Y\) is the proportional increase in output, and, since \(Y = λ N\), it is equal to the proportional increase in employment. Likewise, \(\Delta a/a\) is the proportional increase in a, and since \(a = W/ λ\), it is equal to the proportional increase in the wage. So the expression for marginal cost can be written as:

\[\text{MC} = a\left(1 + \frac{\text{proportional increase in } W}{\text{proportional increase in } N} \right)\]

Although we used an approximation to obtain it, this expression is exactly the same as the one we would get using calculus methods.

The fraction representing how W changes with N is what we called \(η\) in the main part of this section. We have now derived the expression for marginal cost that we stated there:

\[\text{MC} = (1+\eta)\frac{W}{\lambda}\]

\(η\) is a positive number that tells us how much the firm has to raise the wage to increase employment. It is ‘the employment elasticity of the wage’: the percentage change in the wage for a 1% change in employment.

The higher the value of \(η\), the more the firm can control its own wage by varying employment: the no-shirking wage curve will be steep. When \(η\) is high, the firm has an incentive to keep employment and wages low—because the cost of increasing output is high.

We noted before that we can call \(η\) the markdown: it tells us how far the wage is marked down so that the average cost \(W/ λ\) is below MC. We can think of \(η\) as a measure of the firm’s labour market power. It depends on how much competition the firm faces in the labour market. If the firm faces a lot of competition for workers from other employers, it will have to pay a wage similar to other firms irrespective of the number of employees; its labour market power will be low. When competition is low and its employees have few other options, the firm has the power to mark down the wage: that is, \(η\) is high.

Remember that in our macroeconomic model, we assume the intensity of competition in both the product and labour market is constant: it doesn’t change with employment and output. For the product market, this means that PED is a constant; for the labour market, it means that labour market power \(η\) is a constant. So we can conclude from the equation above that the firm’s marginal cost is proportional to W, the nominal wage.

Step 3: The price-setting real wage

We now combine the equations from the first two steps:

\[P = \frac{1}{1 - \mu} \text{MC}\]

and

\[\text{MC} = (1+\eta)\frac{W}{\lambda}\]

to get:

\[P = \frac{1+\eta}{\lambda(1 - \mu)}W\]

The price is proportional to the nominal wage, so we can rearrange this equation to get the price-setting real wage, w:

\[w = \frac{W}{P} = \frac{1 - \mu}{1 + \eta}\lambda\]

The wage share and the profit share

The price-setting real wage is a constant: whatever the firm’s level of output and employment, the way it sets the price means that the real wage is a constant fraction of output per worker, \(λ\). (Since \(\mu\) is between 0 and 1, and \(η\) is positive, \((1 - \mu)/(1 + η)\) lies between 0 and 1.) The fraction that goes to the worker is the wage share, and the remaining fraction for the firm is the profit share, \(\sigma\):

\[w = (1 - \sigma) \lambda\] \[\text{where the wage share is } 1 - \sigma = \frac{1 - \mu}{1 + \eta} \text{ and the profit share is } \sigma = \frac{\mu + \eta}{1 + \eta}\]

The profit share, \(\sigma\), depends on competition in the product and labour markets, because it depends on \(\mu\) and \(η\):

  • High product market competition means a low value of the markup, \(\mu\). From the expression for the wage share, this means the wage share will be high, and hence that the profit share will be low.
  • High labour market competition means a low value of labour market power, markdown \(η\). Again from the expression for the wage share, this means the wage share will be high, and so profit share will be low.

In summary, more competition in either or both markets benefits workers and reduces profit.

  1. David Card. 2022. ‘Who set your wage?’ American Economic Review 112 (4): pp. 1075–1090.