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

<p>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>

<p><span class="math-container">$ p_m = P(p_m)^{1-\varepsilon} \bigg[ \frac{I}{Y_m} \bigg]^{1/\varepsilon}$</span></p>

<p>which underlies an under-identified system. So, yes you can include the <span class="math-container">$P-1=0$</span> equation in your solver.</p>

<p>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), <span class="math-container">$\varepsilon &gt; 1$</span>, and profit maximization leads to a constant mark-up on variable cost</p>

<p><span class="math-container">$p_m = \frac{\varepsilon}{\varepsilon - 1} c_m(w_m)$</span></p>

<p>where <span class="math-container">$c_m(w_m)$</span> is the unitary labor cost of producer <span class="math-container">$m$</span>, function of wages <span class="math-container">$w_m$</span>. So there are <em>a priori</em> nothing to guess <em>freely</em> at that level, just use the relation you have between prices and wages. </p>

<p>Also, the level of capital, <span class="math-container">$K_m$</span>, 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.</p>

<p>I would not calibrate <span class="math-container">$\varepsilon$</span> 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?</p>

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

<p>Need more precisions from you to go further.</p>

CC BY-SA 4.0
2019-02-01T16:58:19.453Z
Thanks for your thoughtful comments! It's a unique price index! (1) Each variety $m$ is produced by many firms, so that there is perfect competition within each variety $m$, i.e. firms are price takers. (2) I'm not sure I understand your point on $K_m$? In my model, $K_m$ is perfectly elastically supplied at rate $r$. So surely that is determined in GE...? (3) $I$ is not indexed by $m$. It's the total income in the economy (labour and capital income $\sum_m w_m L_m + r \sum_m K_m$). At the moment, I ignore capital income (assume it's rented from abroad). Again, many thanks! @keepAlive
Anna SKH
CC BY-SA 4.0
2019-02-01T08:37:56.987Z
2019-02-01T10:14:43.103Z
2019-02-01T10:14:43.103Z
keepAlive
<p>I've built a model that I am trying to solve in Matlab. The model has Dixit-Stiglitz preferences and <span class="math-container">$m$</span> goods, so admits a price index of the following form:</p>

<p><span class="math-container">$ P = \bigg[ \sum_m p_m^{1-\varepsilon} \bigg]^{\frac{1}{1-\varepsilon}} $</span></p>

<p>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>

<p><span class="math-container">$ p_m = P^{1-\varepsilon} \bigg[ \frac{I}{Y_m} \bigg]^{1/\varepsilon} $</span></p>

<p>where <span class="math-container">$I$</span> is total income (you can think of it as <span class="math-container">$\sum_m p_m Y_m$</span>) and <span class="math-container">$Y_m$</span> is production of good <span class="math-container">$m$</span>. </p>

<p>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 <span class="math-container">$Y_m$</span> is determined, pinning down <span class="math-container">$p_m$</span>. It seems like I can't set the price index <span class="math-container">$P =1$</span>?</p>

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

<p>Thank you so much! </p>

CC BY-SA 4.0
2019-02-01T03:42:37.677Z
What do you gain from normalization? Is this a matter of convenience or does it somehow allow important insight? If it is simply a matter of convenience...well...maybe just don't worry about it?
123
CC BY-SA 4.0
2019-02-01T17:06:03.473Z
Thanks for your comment! I want to normalise as to have a "unit of account" in the model. Do you have any other suggestions on that...? @123
Anna SKH
<macroeconomics><mathematical-economics>
Can I normalise the Dixit-Stiglitz price index to 1?
Anna SKH
# Can I normalise the Dixit-Stiglitz price index to 1?

(Asked by: username_0 on 2019-01-31)

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!

**Tags:** macroeconomics, mathematical-economics
> **Comments:**
> **username_1**: What do you gain from normalization? Is this a matter of convenience or does it somehow allow important insight? If it is simply a matter of convenience...well...maybe just don't worry about it?  
> **username_0**: Thanks for your comment! I want to normalise as to have a "unit of account" in the model. Do you have any other suggestions on that...? @123  

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## 1 answers

### Answer by username_2 on 2019-02-01. Score: 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.

> **Comments:**
> **username_0**: Thanks for your thoughtful comments! It's a unique price index! (1) Each variety $m$ is produced by many firms, so that there is perfect competition within each variety $m$, i.e. firms are price takers. (2) I'm not sure I understand your point on $K_m$? In my model, $K_m$ is perfectly elastically supplied at rate $r$. So surely that is determined in GE...? (3) $I$ is not indexed by $m$. It's the total income in the economy (labour and capital income $\sum_m w_m L_m + r \sum_m K_m$). At the moment, I ignore capital income (assume it's rented from abroad). Again, many thanks! @keepAlive  

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