Hence, the relationship between /3g and equilibrium inflation is non-monotonic. Inflation rises as (3g increases from a low level (while holding j3p constant), but falls eventually. The appendix proves that:

The intuition for the non-monotonicity result is simple. A bit of conservatism increases inflation while lowering output, and hence welfare falls as central bank conservatism rises from an initially low level. A larger doses of conservatism is necessary for the benefits of lower inflation to outweigh those of lower output, and hence raise welfare.

Fig. 2: CBC and welfare 4.3. Increasing the number of unions

What is the effect of the number of unions on employment, output, inflation and welfare? Here we tackle these questions holding constant the degree of central bank conservatism.

Note, first, from equations 3.4-3.7, that all the variables of interest depend on n exclusively via the coefficient ф. When this coefficient is equal to one, employment and output are at their first-best levels, inflation is zero, and welfare is maximized. Since, for finite {3p and f3g, ф is always below its optimal level of one, employment, output, and welfare increase with 0, and inflation decreases with ф.

The key relationship, then, is that between the number n of unions and the coefficient ф. The former affects the latter in two ways:

a) First, holding (^?r) constant, ф is decreasing in n. Intuitively, the larger the number of unions, the less each internalizes the inflationary consequences of its actions. Henceforth we term this the “internalization effect.”

b) Second, and as we saw earlier, ф is increasing in and in turn the elasticity of labor demand ф depends on n. Above we termed this the “competition effect.” Recall that this effect has an ambiguous sign, which depends on parameter values. If a (1 — a) < 1, so that < 0, then a larger number of unions reduces competition; the partial effect of this is to reduce ф.