Unit 2 Unemployment, wages, and inequality: Supply-side policies and institutions

2.7 Taxes and the WS–PS model

tax: A tax is a compulsory payment to the government levied, for example, on workers’ incomes (income taxes) or firms’ profits (profit taxes) or included in the price paid for goods and services (value added or sales taxes).

Until now, we have ignored the role of taxes⁠tax A tax is a compulsory payment to the government levied, for example, on workers’ incomes (income taxes) or firms’ profits (profit taxes) or included in the price paid for goods and services (value added or sales taxes). in the economy. That wouldn’t matter if tax rates on income and consumption never changed; we could just assume that workers and firms allowed for the taxes in their decisions about wages and prices.

But taxes are an important policy tool: governments can change economic outcomes by raising or lowering taxes on income, and consumption taxes such as sales taxes or value added tax (VAT). To incorporate taxes into the model explicitly, we need to distinguish between wages and prices before and after tax.

Consumption taxes mean that the price received by the firm, which we call P, is lower than the price paid by the consumer, which includes the tax. Writing $P_c$ for the consumer price level:

$P_c= P(1+t_v)$

where $t_v$ is the percentage rate of the consumption tax.

Similarly, the worker’s take-home nominal wage, which we call W, is the wage they receive after all labour taxes have been paid. We write $W^{\text{gross}}$ for the wage cost to the employer, which is higher—it is the wage before labour taxes have been paid. The relationship between them is:

$W^{\text{gross}}= W(1+t_d)$

where $t_d$ is the percentage rate of direct taxation, including both income taxes and social security contributions.

Modifying the model

We will introduce taxes into the WS–PS model by writing both the wage- and price-setting curves in terms of the real post-tax consumption wage, $w = W/P_c$ .

This is the real wage that matters for recruiting and retaining workers. It is the value of the wage in terms of the goods and services they can buy with it. So if we draw the WS curve with the real post-tax consumption wage, w, on the vertical axis, the curve will be exactly the same as before.

However, the PS curve will be different. In our model of price setting (Section 1.7), the firm sets the price that it receives (P) as a markup on the wage that it pays ( $W^{\text{gross}}$ ). So the PS curve is $W^{\text{gross}}/P = (1 – \sigma)\lambda$ . If we use the expressions above to rewrite this in terms of W and $P_c$ instead, we get the PS curve in terms of the post-tax real wage $w=W/P_c$ :

$w = \frac{(1 - \sigma) \lambda}{(1 + t_d)(1 + t_v)}$

To derive this equation:

Start from the PS curve $\frac{W^{\text{gross}}}{P} = (1 - \sigma) \lambda$
Substitute $W^{\text{gross}}=W(1+t_d)$ to get $\frac{W (1 + t_d)}{P} = (1 - \sigma) \lambda$
And then replace $P = \frac{P_c}{(1+t_v)}$ to get $\frac{W (1 + t_d) (1 + t_v)}{P_c} = (1 - \sigma) \lambda$
Rearranging gives $\frac{W}{P_c} = \frac{(1 - \sigma) \lambda}{(1 + t_d) (1 + t_v)}$

The overall effect of taxes is to reduce the real output available to be shared between the firm and the worker from $λ$ to $\frac{\lambda}{(1+t_d)(1+t_v)}$ . The firm and worker get shares $σ$ and $(1 – σ)$ of this, and the rest of $λ$ goes to the government in the form of tax.

A numerical example

Suppose the direct tax rate, $t_d$ , is 25%, and the consumption tax rate, $t_v$ , is 20%. Then output net of taxes is:

$\frac{\lambda}{(1 + t_d) (1 + t_v)} = \frac{\lambda}{1.25 \times 1.2} = \frac{2}{3} \lambda$

and the government receives one-third of $λ$ . If the profit share, $σ$ , is 40%, the wage share is 60%, so the price-setting real wage is:

$w = \frac{(1 - \sigma)\lambda}{(1 + t_d) (1 + t_v)} = 0.6 \times \frac{2}{3} \lambda = 0.4 \lambda$

Figure 2.19 shows the modified WS–PS model, with the tax rates and profit share in this example.

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https://www.core-econ.org/macroeconomics/02-unemployment-wages-inequality-07-taxes-ws-ps-model.html#figure-2-19

Figure 2.19 The WS–PS model with taxation.

Exercise 2.6 Education and training financed by taxation

Suppose the direct tax rate ( $t_d$ ) is 22%, the consumption tax rate ( $t_v$ ) is 8%, and the profit share is 34%.

Draw a WS–PS diagram like Figure 2.19 to illustrate how the average product of labour ( $\lambda$ ) is split between profits, taxes, and wages. (The horizontal axis will not have specific numbers.)
Modify your diagram to illustrate the effect of spending on education and training that is financed by taxation. Compare this equilibrium outcome with the original equilibrium (WS–PS with no taxation or spending on training).

The effect of a tax increase

What is the effect on real wages and structural unemployment of a higher rate of taxation? A larger tax taken by the government—either on labour income or consumption—leaves less output per worker for real wages and profits to be shared between firms and workers. Since nothing has changed in the conditions of competition in the market for goods and services, the shares $σ$ and $(1 – σ)$ are unaffected. From the equation of the PS curve, higher taxation reduces the price-setting real wage, so the PS curve shifts down.

Figure 2.20 shows the effect of a tax increase: in equilibrium, real wages are lower and unemployment is higher.

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https://www.core-econ.org/macroeconomics/02-unemployment-wages-inequality-07-taxes-ws-ps-model.html#figure-2-20

Figure 2.20 The effect of a rise in tax rates on wages and employment.

We can now return to the questions raised earlier in the unit about how spending on education and training that raises productivity affects real wages and structural unemployment.

Section 2.4 showed that if training raises productivity, $λ$ , the PS curve shifts upward, leading to higher wages and lower unemployment. Profits and tax revenue will also be higher. But what happens if the spending on training is financed by raising tax rates? We know from Figure 2.19 that tax increases shift the PS curve in the opposite direction, having the opposite effect.

So the need to finance training through taxation will moderate the effects of the rise in productivity. The overall rises in wages and employment will be smaller. (We can expect that they will both still rise; a training programme for which the costs outweigh the benefits should not be undertaken.)

Question 2.8 Choose the correct answer(s)

Suppose the direct tax rate ( $t_d$ ) is 30%, the consumption tax rate ( $t_v$ ) is 15%, and the profit share is 35% (original scenario). The government then raises the direct tax rate to 35% and the consumption tax rate to 20% to finance spending on education, which raises worker productivity by 20% (after-tax scenario). Based on this information, read the following statements and choose the correct option(s).

1. In the original scenario, the government receives 30% of the output.
2. In the original scenario, the price-setting real wage is 0.44 $\lambda$ (to two decimal places).
3. In the after-tax scenario, the government receives 38% of the output.
4. In the after-tax scenario, the price-setting real wage is 0.40 $\lambda$ .

The government receives $1 - \frac{1}{(1.3 \ \times \ 1.15)} = 33\%$ of the output.
The price-setting real wage is $0.65 \ \times \ 0.67 \lambda = 0.44 \lambda$ .
The government receives $1 - \frac{1}{(1.35 \ \times \ 1.2)} = 38\%$ of the output.
The price-setting real wage, accounting for the increase in productivity, is $0.65 \ \times \ 0.62 \lambda \ \times \ 1.2 = 0.48 \lambda$ .