Can I normalise the Dixit-Stiglitz price index to 1?

Question

I've built a model that I am trying to solve in Matlab. The model has Dixit-Stiglitz preferences and $m$ goods, so admits a price index of the following form:

$ P = \bigg[ \sum_m p_m^{1-\varepsilon} \bigg]^{\frac{1}{1-\varepsilon}} $

I would like to normalise this to 1. That is, everything in my model would be in terms of the final goods basket. However, the goods prices are already determined by the following well-known equation

$ p_m = P^{1-\varepsilon} \bigg[ \frac{I}{Y_m} \bigg]^{1/\varepsilon} $

where $I$ is total income (you can think of it as $\sum_m p_m Y_m$) and $Y_m$ is production of good $m$.

Furthermore, given a set of wages and rental rates (supplied as a "guess" for Matlab's fsolve), the FOCs already (over)identify the capital and labour levels, and so $Y_m$ is determined, pinning down $p_m$. It seems like I can't set the price index $P =1$?

What shall I do? Shall I include the $P-1=0$ equation in my Matlab solver (adding to the overidentification issues!)? Shall I calibrate $\varepsilon$ to ensure $P=1$? Or do you have any other advice?

Thank you so much!

Answer 1

To me it depends. If you are dealing with price indexes that are somehow distributed spatially, i.e. you have $P_i \ \text{for} \ i=\text{NY},...,\text{CH}$ it would make no sense to normalize only one of them. I say "only" because it would be a priori impossible to normalize them all since, say, transport costs would ineluctably force them to be different.

If you are dealing with a unique price-index, I see no reason of why you could not do so. Indeed, note that you have

$ p_m = P(p_m)^{1-\varepsilon} \bigg[ \frac{I}{Y_m} \bigg]^{1/\varepsilon}$

which underlies an under-identified system. So, yes you can include the $P-1=0$ equation in your solver.

Regarding the contradiction it creates with your (over-identifying) wages and rental rates, to me it looks like you should take things upside-down. Under Dixit-Stiglitz monopolistic market, firms set their price by assuming a constant elasticity of substitution (CES), $\varepsilon > 1$, and profit maximization leads to a constant mark-up on variable cost

$p_m = \frac{\varepsilon}{\varepsilon - 1} c_m(w_m)$

where $c_m(w_m)$ is the unitary labor cost of producer $m$, function of wages $w_m$. So there are a priori nothing to guess freely at that level, just use the relation you have between prices and wages.

Also, the level of capital, $K_m$, should be set before any king of GE-resolution. Indeed this is a "calibrating" characteristic and I do not see why -- at least given the content of your question -- you want its obtention to be melt with that of the general-equilibrium (GE) vector of prices.

I would not calibrate $\varepsilon$ if I were you, since there is a big narrative behind the choice of its value: this is somehow representative of the "market depth", see Rivera-Batiz (1988) for an intuition about the level of elasticity that might be chosen. Do you want your narrative to be chosen by your solver?

Finally, you say that we can think of $I_m$ as $\sum_m p_m Y_m$, which is (only) true at equilibrium... But how is your $I_m$ formed ? It looks like each consumer also possesses her own firm (given that $I$ is indexed by $m$)? Is it $I_m = w_m + K_mr_m$, where $r_m$ is the rate of return of the firm $m$ ?

Need more precisions from you to go further.