Approximating a demand function with shocks to the elasticity of demand

Published on 16 November 2009 in Academia and Economics. Tags: , , , , . 0 Comments

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.

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