Entirely for my own reference …
A demand function commonly used in macroeconomics is the following, derived from a Dixit-Stiglitz aggregator and exhibiting a constant own-price elasticity of demand (\gamma):
!Q_{it}=\left(\frac{P_{it}}{P_{t}}\right)^{-\gamma}Q_{t}
A demand-side shock can then be modelled as a change in the elasticity of demand:
!Q_{it}=\left(\frac{P_{it}}{P_{t}}\right)^{-\gamma D_{t}}Q_{t}
Where \ln\left(D_{t}\right) is, say, Normally distributed and plausibly autocorrelated. We can rewrite this as a function of (natural) log deviations from long-run trends:
!Q_{it}=\overline{Q_{t}}e^{q_{t}-\gamma e^{d_{t}}\left(p_{it}-p_{t}\right)}
Where:
- Variables with a bar above them are long-run trends: \overline{X_{it}}
- Lower-case variables are natural log deviations from their long run trends (so that for small deviations, they may be thought of as the percentage difference from trend): x_{it}=\ln\left(X_{it}\right)-\ln\left(\overline{X_{it}}\right)
- The long-run trend of all prices is to equal the aggregate price: \overline{P_{it}}=\overline{P_{t}}
- The long-run trend of D_{t} is unity
We’ll construct a quadratic approximation around q_{t}=p_{it}=p_{t}=d_{t}=0 but, first, a table of partial derivatives for a more general function:
Function | Value at x=y=z=0 |
f\left(x,y,z\right)=ae^{x+bye^{z}} | a |
f_{x}\left(x,y,z\right)=ae^{x+bye^{z}} | a |
f_{y}\left(x,y,z\right)=abe^{x+bye^{z}+z} | ab |
f_{z}\left(x,y,z\right)=abye^{x+bye^{z}+z} | 0 |
f_{xx}\left(x,y,z\right)=ae^{x+bye^{z}} | a |
f_{yy}\left(x,y,z\right)=ab^{2}e^{x+bye^{z}+2z} | ab^{2} |
f_{zz}\left(x,y,z\right)=abye^{x+bye^{z}+z}+ab^{2}y^{2}e^{x+bye^{z}+2z} | 0 |
f_{xy}\left(x,y,z\right)=abe^{x+bye^{z}+z} | ab |
f_{xz}\left(x,y,z\right)=abye^{x+bye^{z}+z} | 0 |
f_{yz}\left(x,y,z\right)=abe^{x+bye^{z}+z}+ab^{2}ye^{x+bye^{z}+2z} | ab |
So that in the vicinity of x=y=z=0, the function f\left(x,y,z\right) is approximated by:
!f\left(x,y,z\right)\simeq a + a\left(x+by\right) + a\left[\frac{1}{2}\left(x+by\right)^{2}+byz\right]
From which we can infer that:
!Q_{it}\simeq \overline{Q_{t}}\left[1+\left(q_{t}-\gamma\left(p_{it}-p_{t}\right)\right) + \frac{1}{2}\left(q_{t}-\gamma\left(p_{it}-p_{t}\right)\right)^{2}-\gamma\left(p_{it}-p_{t}\right)d_{t}\right]
If introduced to a profit function, the first-order components (q_{t}-\gamma\left(p_{it}-p_{t}\right)) would vanish as individual prices will be optimal in the long run.
Update (20 Jan 2010): Added the half in each of the last equations.