# Tag Archive for 'Dixit-Stiglitz'

### Approximating a demand function with shocks to the elasticity of demand

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.

### Be careful interpreting Lagrangian multipliers

Last year I wrote up a derivation of the New Keynesian Phillips Curve using Calvo pricing.  At the start of it, I provided the standard pathway from the Dixit-Stiglitz aggregator for consumption to the constant own-price elasticity individual demand function.  Let me reproduce it here:

There is a constant and common elasticity of substitution between each good: $$\varepsilon>1$$.  We aggregate across the different consumptions goods:

$$!C=\left(\int_{0}^{1}C\left(i\right)^{\frac{\varepsilon-1}{\varepsilon}}di\right)^{\frac{\varepsilon}{\varepsilon-1}}$$

$$P\left(i\right)$$ is the price of good i, so the total expenditure on consumption is $$\int_{0}^{1}P\left(i\right)C\left(i\right)di$$

A representative consumer seeks to minimise their expenditure subject to achieving at least $$C$$ units of aggregate consumption. Using the Lagrange multiplier method:

$$!L=\int_{0}^{1}P\left(i\right)C\left(i\right)di-\lambda\left(\left(\int_{0}^{1}C\left(i\right)^{\frac{\varepsilon-1}{\varepsilon}}di\right)^{\frac{\varepsilon}{\varepsilon-1}}-C\right)$$

The first-order conditions are that, for every intermediate good, the first derivative of $$L$$ with respect to $$C\left(i\right)$$ must equal zero. This implies that:

$$!P\left(i\right)=\lambda C\left(i\right)^{\frac{-1}{\varepsilon}}\left(\int_{0}^{1}C\left(j\right)^{\frac{\varepsilon-1}{\varepsilon}}dj\right)^{\frac{1}{\varepsilon-1}}$$

Substituting back in our definition of aggregate consumption, replacing $$\lambda$$ with $$P$$ (since $$\lambda$$ represents the cost of buying an extra unit of the aggregate good $$C$$) and rearranging, we end up with the demand curve for each intermediate good:

$$!C\left(i\right)=\left(\frac{P\left(i\right)}{P}\right)^{-\varepsilon}C$$

If that Lagrangian looks odd to you, or if you’re asking where the utility function’s gone, you’re not alone.  It’s obviously just the dual problems of consumer theory – the fact that it doesn’t matter if you maximise utility subject to a budget constraint or minimise expenditure subject to a minimum level of utility – but what I want to focus on is the resulting interpretation of the lagrangian multipliers.

Let’s rephrase the problem as maximising utility, with utility a generic function of aggregate consumption, $$U\left(C\right)$$.  The Lagrangian is then:

$$!L=U\left(\left(\int_{0}^{1}C\left(i\right)^{\frac{\varepsilon-1}{\varepsilon}}di\right)^{\frac{\varepsilon}{\varepsilon-1}}\right)+\mu\left(M-\int_{0}^{1}P\left(i\right)C\left(i\right)di\right)$$

The first-order conditions are:

$$!U’\left(C\right)\left(\int_{0}^{1}C\left(j\right)^{\frac{\varepsilon-1}{\varepsilon}}dj\right)^{\frac{1}{\varepsilon-1}}C\left(i\right)^{\frac{-1}{\varepsilon}}=\mu P\left(i\right)$$

Rearranging and substituting back in the definition for $$C$$ then gives us:

$$!C\left(i\right)=\left(P\left(i\right)\frac{\mu}{U’\left(C\right)}\right)^{-\varepsilon}C$$

In the first approach, $$\lambda$$ represents the cost of buying an extra unit of the aggregate good $$C$$, which is the definition of the aggregate price level.  In the second approach, $$\mu$$ represents the cost of buying an extra unit of income, which is not the same thing.  Comparing the two results, we can see that:

$$!\lambda=P=\frac{U’\left(C\right)}{\mu}$$

Which should cause you to raise an eyebrow.  Why aren’t the two multipliers just the inverses of each other?  Aren’t they meant to be?  Yes, they are, but only when the two problems are equivalent.  These two problems are slightly different.

In the first one, to be equivalent, the term in the lagrangian would need to be $$V – U\left(\left(\int_{0}^{1}C\left(i\right)^{\frac{\varepsilon-1}{\varepsilon}}di\right)^{\frac{\varepsilon}{\varepsilon-1}}\right)$$, which would give us Hicksian demands as a function of utility level ($$V$$).  But since we assumed that utility is only a function of aggregate consumption, then in order to pin down a level of utility, it’s sufficient to pin down a level of aggregate consumption; and that is useful to us because a) a level of utility doesn’t mean much to us as macroeconomists but a level of aggregate consumption does and b) it means that we can recognise the lagrange multiplier as the aggregate price level.

Which, when you think about it, makes perfect sense.  Extra income must be adjusted by the marginal value of the extra consumption it affords in order to arrive at the price that the (representative) consumer would be willing to pay for that consumption.

In other words:  be careful when interpreting your Lagrangian multipliers.

### Deriving the New Keynesian Phillips Curve (NKPC) with Calvo pricing

The Phillips Curve is an empirical observation that inflation and unemployment seem to be inversely related; when one is high, the other tends to be low.  It was identified by William Phillips in a 1958 paper and very rapidly entered into economic theory, where it was thought of as a basic law of macroeconomics.  The 1970s produced two significant blows to the idea.  Theoretically, the Lucas critique convinced pretty much everyone that you could not make policy decisions based purely on historical data (i.e. without considering that people would adjust their expectations of the future when your policy was announced).  Empirically, the emergence of stagflation demonstrated that you could have both high inflation and high unemployment at the same time.

Modern Keynesian thought – on which the assumed efficacy of monetary policy rests – still proposes a short-run Phillips curve based on the idea that prices (or at least aggregate prices) are “sticky.”  The New Keynesian Phillips Curve (NKPC) generally looks like this:

$\pi_{t}=\beta E_{t}\left[\pi_{t+1}\right]+\kappa y_{t}$

Where $y_{t}$ is the (natural) log deviation – that is, the percentage deviation – of output from its long-run, full-employment trend and $\beta$ and $\kappa$ are parameters.  Notice that (unlike the original Phillips curve), it is forward looking.  There are criticisms of the NKPC, but they are mostly about how it is derived rather than its existence.

What follows is a derivation of the standard New Keynesian Phillips Curve using Calvo pricing, based on notes from Kevin Sheedy‘s EC522 at LSE.  I’m putting it after this vile “more” tag because it’s quite long and of no interest to 99% of the planet.