Editing Nominal rigidity (section)

===Mathematical example: a little price stickiness can go a long way===
To see how a small sector with a fixed price can affect the way rest of the flexible prices behave, suppose that there are two sectors in the economy: a proportion ''a'' with flexible prices ''P<sub>f</sub>'' and a proportion 1&nbsp;−&nbsp;''a'' that are affected by menu costs with sticky prices ''P<sub>m</sub>''. Suppose that the flexible price sector price ''P<sub>f</sub>'' has the market clearing condition of the following form:

:<math> \frac{P_f}{P}=\theta  </math>

where <math>P=P_f^{a}P_m^{1-a} </math> is the aggregate [[price index]] (which would result if consumers had Cobb-Douglas preferences over the two goods). The equilibrium condition says that the real flexible price equals some constant (for example <math>{\theta}</math> could be real marginal cost). Now we have a remarkable result: no matter how small the menu cost sector, so long as ''a''&nbsp;<&nbsp;1, the flexible prices get "pegged" to the fixed price.<ref name="Dixon1992" />  Using the aggregate price index the equilibrium condition becomes
:<math> \frac{P_f}{P_f^a P_m^{1-a}}=\theta</math>
which implies that
:<math> P_f^{1-a}=P_m^{1-a}\theta,</math>
so that
:<math> P_f=P_m\theta^{1/(1-a)}.</math>
What this result says is that no matter how small the sector affected by menu-costs, it will tie down the flexible price. In macroeconomic terms all nominal prices will be sticky, even those in the potentially flexible price sector, so that changes in nominal demand will feed through into changes in output in both the menu-cost sector and the flexible price sector.

Now, this is of course an extreme result resulting from the [[real rigidity]] taking the form of a constant real marginal cost. For example, if we allowed for the real marginal cost to vary with aggregate output ''Y'', then we would have

:<math>P_f=P_m\theta Y^{1/(1-a)}</math>

so that the flexible prices would vary with output ''Y''. However, the presence of the fixed prices in the menu-cost sector would still act to dampen the responsiveness of the flexible prices, although this would now depend upon the size of the menu-cost sector ''a'', the sensitivity of <math>\theta</math> to ''Y'' and so on.