It is mathematically impossible for fiscal redistribution to not trigger a wage-price spiral— turns the case because real wages decrease, burying the working class in an even deeper hole

We now go back to the full model and trace back price and wage inflation to the general equilibrium effect of underlying shocks.

We focus on two shocks. First an expansionary demand shock, driven by easy monetary policy (easy fiscal policy would have similar implications).

A common view is that excessive demand would work its way from a tight labor market, to higher wages, to higher prices. Following this intuition a pure demand shock should manifest itself in increasing real wages. We show that in our model general equilibrium forces are at work on both sides of the labor market and that the direction of adjustment of the real wage is in general ambiguous. This is especially true when the scarce, inelastic input X plays an important role.

Consider a monetary shock that leads to a temporary increase in employment n0 > 0 on impact, the shock decays exponentially at rate δ, so

The responses of mplt and mrst are easily derived from (9) and (14):

Substituting in the conditions of Proposition 3 shows that price and wage inflation are both positive following the shock. What happens to the real wage, though, is in general ambiguous. The following is an immediate corollary of Proposition 3.

Proposition 4. In response to a monetary shock that leads to a transitory increase in employment, real wages fall on impact if and only if

The left-hand side of the inequality captures the direct effect on price inflation. This term depends on the effect of higher employment on marginal costs and on stickiness in price setting, captured by Λp. The effect of employment on marginal costs is larger when the scarce input X is more important in the production of the final good (higher share sX) and when the elasticity of substitution between labor and X is lower. The term on the righthand side captures direct effects on wage inflation. This term depends on the effect on the marginal rate of substitution and on stickiness in wage setting, captured by Λw. The effect on the marginal rate of substitution, in turn, depends on an income effect, captured by the term σsL, since sL is the elasticity of output to the labor input, and on the inverse Frisch elasticity η.

Overall, if the effect on firms’ marginal costs is relatively stronger than the effects on workers’ marginal rate of substitution and if prices are relatively more flexible than wages, we get a reduction in real wages.

In Figure 6 we plot the response to a temporary expansionary shock that increases n above its potential level by 2%, with a decay δ = 1 in a simple numerical example.7 The parameters used are in the Table 1.

The first panel shows the shock to n. The remaining panels show the responses of different prices.

The input price is flexible, so it jumps on impact and then gradually goes back to its initial level, as the shock goes away. This is shown in the second panel of the figure. Notice that this panel shows the level of the input price, not its rate of inflation. Due to perfect flexibility PX jumps by 20% at t = 0. This large increase is due to our assumption of a low elasticity of substitution between labor and the input X (e = 0.1), so when the employment is growing too fast relative to the supply of X, the price of X reacts strongly.

The effect of the increase in the input price is to increase firm’s marginal costs. The impact effect on the nominal marginal cost w0 − mpl0 is 2%, as the input represents 10% of the cost in steady state (sX = 0.1). This impulse translates into fast inflation on impact, due to our assumption of relatively flexible prices (λp = 4, i.e, prices reset every quarter). This is plotted in the third panel.

Wages respond because high employment translates into high real wage demands. In our simple model with η = 0, this is only due only to an income effect: as consumption grows, workers need higher wages to be induced to work. For illustration we have chosen parameters such that the impact effect on the nominal marginal cost of labor p0 + mrs0 is identical to the effect on the marginal cost of goods, both are 2%. However, wages are more sticky (λw = 1), so the effect on wage inflation is weaker. Wage inflation is also plotted in the third panel. The conditions for Proposition 4 are satisfied and the real wage falls on impact, as shown in the fourth panel.

To be clear, this is just a numerical example with numbers chosen for clarity of illustration. Nonetheless, there is clear qualitative feature that we want to highlight: the adjustment happens in three phases.

1. First, there is a bout of very fast inflation in the sector where the supply constraints are binding, here the market for input X.

2. Second, there is a phase in which price inflation is faster than wage inflation, as price setters react relatively quickly to the increase in input costs.

3. At some point (near t = 0.5 in our example) wage inflation crosses price inflation and we enter the third phase in which real wages recover. The input scarcity is going away, so the pressure on firms’ marginal costs is weaker, while workers are still trying to catch up to the higher cost of living, given their real wage aspirations.