<p>You are actually talking about <a href="https://course-notes.org/economics/micro_economics/monopsony" rel="nofollow noreferrer">monopsony</a> -- or in a weaker form: oligopsony. </p>

<p>There are <em>a priori</em> no conventional way to deal directly with hybrid situations like the one you describe: you would rather have to deal with a (multi-level) <em>nested</em> (CES-type or whatever) <em>utility function</em>, recursively optimized, taking the goods' characteristics into account at each solving step. </p>

<p>For <em>an example</em> of how such a utility "tree" looks like, see <a href="https://scholar.google.co.jp/scholar?q=nested+ces+type&amp;hl=en&amp;as_sdt=0&amp;as_vis=1&amp;oi=scholart#d=gs_qabs&amp;p=&amp;u=%23p%3D4M_lBFThHfIJ" rel="nofollow noreferrer">Keller 1976</a>.</p>

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2018-11-22T01:56:26.927Z
2018-11-22T02:17:32.850Z
2018-11-22T02:17:32.850Z
keepAlive
<p>We have the utility function  <span class="math-container">$U(x,y) = Ax^ay^a$</span> and we want to solve</p>

<p><span class="math-container">$$\max U = Ax^ay^a,\;\;\; s.t. xp_x + yp_y(y) = I$$</span></p>

<p>The Lagrangean is </p>

<p><span class="math-container">$$\Lambda = Ax^ay^a + \lambda [I-xp_x - yp_y(y)]$$</span></p>

<p>and the first order conditions are</p>

<p><span class="math-container">$$\frac {aU}{x} = \lambda p_x$$</span>
<span class="math-container">$$\frac {aU}{y} =\lambda p_y + \lambda y\frac{\partial p_y}{\partial y}$$</span></p>

<p>It follows that at the optimum</p>

<p><span class="math-container">$$xp_x = yp_y + y^2\frac{\partial p_y}{\partial y}$$</span></p>

<p>or that</p>

<p><span class="math-container">$$\frac{x}{y} = \frac{p_y}{p_x} + y \frac{\partial p_y/\partial y}{p_x}$$</span></p>

<p>But the previous expression is more interesting because it can be viewed as a quadratic polynomial in <span class="math-container">$y$</span>, and be solved for its roots</p>

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2018-11-22T13:27:25.403Z
why would a consumer maximize over quantities if she is price maker? (cc @user617643)
keepAlive
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2018-11-22T14:07:46.817Z
@Kanak If there is a bijective mapping between $p_y$ and $y$ why not?
Giskard
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2018-11-22T14:10:17.433Z
The utility function $U(x,y) = Ax^ay^a$ does not have the given marginal rate of substitution though.
Giskard
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2018-11-22T15:52:12.670Z
@denesp this is at least worth mentioning and/or rather restrictive. Bijections between prices and quantities are far from being an invariable rule. Moreover, we are not even talking about demands here: Such a relation may not be, say, continuous...
keepAlive
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2018-11-22T19:49:09.047Z
@densep $MRS = -U_x/U_y$. $U_x = (a/x)U, U_y = (a/y)U$. So, it appears, $MRS = -(a/x)/(a/y) = -x/y$. What am I missing?
Alecos Papadopoulos
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2018-11-22T19:52:12.560Z
@Kanak 1) Quantities, not prices, are what create utility for the consumer, so it is quantities that the consumer must choose, in order to maximize its utility. 2) The consumer may "make"(influence)  the price, through his demand of the good, but he does not _set_ the price. So from both angles, the consumer is to choose quantities.
Alecos Papadopoulos
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2018-11-22T03:02:31.163Z
2018-11-22T03:02:31.163Z
Alecos Papadopoulos
<p>Consider some consumer who consumes good <span class="math-container">$x$</span> and <span class="math-container">$y$</span>. The price of <span class="math-container">$x$</span> is fixed to be <span class="math-container">$k$</span>. The price of <span class="math-container">$y$</span> is <span class="math-container">$c(y)$</span>. The consumer's marginal rate of substitution is given by <span class="math-container">$\frac{-x}{y}$</span>. If the consumer optimises and is a price taker, I know she should set the absolute value of this equal to the price ratio. However, her quantity demanded affects the price ratio, as her demand influences the price of <span class="math-container">$y$</span>. So, does it still hold that she would set</p>

<p><span class="math-container">$$\frac{x}{y}=\frac{k}{c(y)}$$</span></p>

<p>Or does this result fail since her quantity demanded has an affect on the price of <span class="math-container">$y$</span>?</p>

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2018-11-22T14:11:15.683Z
Are you sure $MRS(x,y) = -x/y$ and not $-y/x$? The first one would make the utility function concave.
Giskard
<microeconomics><mathematical-economics><consumer-theory>
If the consumer is not a price taker, does she still set marginal rate of substitution equal to the price ratio?
user617643
# If the consumer is not a price taker, does she still set marginal rate of substitution equal to the price ratio?

(Asked by: username_0 on 2018-11-21)

Consider some consumer who consumes good $x$ and $y$. The price of $x$ is fixed to be $k$. The price of $y$ is $c(y)$. The consumer's marginal rate of substitution is given by $\frac{-x}{y}$. If the consumer optimises and is a price taker, I know she should set the absolute value of this equal to the price ratio. However, her quantity demanded affects the price ratio, as her demand influences the price of $y$. So, does it still hold that she would set

$$\frac{x}{y}=\frac{k}{c(y)}$$

Or does this result fail since her quantity demanded has an affect on the price of $y$?

**Tags:** microeconomics, mathematical-economics, consumer-theory

> **Comments:**
> **username_1**: Are you sure $MRS(x,y) = -x/y$ and not $-y/x$? The first one would make the utility function concave.  

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## 2 answers

### Answer by username_3 on 2018-11-22. Score: 3 | Accepted answer

We have the utility function $U(x,y) = Ax^ay^a$ and we want to solve

$$\max U = Ax^ay^a,\;\;\; s.t. xp\_x + yp\_y(y) = I$$

The Lagrangean is

$$\Lambda = Ax^ay^a + \lambda [I-xp\_x - yp\_y(y)]$$

and the first order conditions are

$$\frac {aU}{x} = \lambda p\_x$$
$$\frac {aU}{y} =\lambda p\_y + \lambda y\frac{\partial p\_y}{\partial y}$$

It follows that at the optimum

$$xp\_x = yp\_y + y^2\frac{\partial p\_y}{\partial y}$$

or that

$$\frac{x}{y} = \frac{p\_y}{p\_x} + y \frac{\partial p\_y/\partial y}{p\_x}$$

But the previous expression is more interesting because it can be viewed as a quadratic polynomial in $y$, and be solved for its roots

> **Comments:**
> **username_2**: why would a consumer maximize over quantities if she is price maker? (cc @username_0)  
> **username_1**: @Kanak If there is a bijective mapping between $p_y$ and $y$ why not?  
> **username_1**: The utility function $U(x,y) = Ax^ay^a$ does not have the given marginal rate of substitution though.  
> **username_2**: @denesp this is at least worth mentioning and/or rather restrictive. Bijections between prices and quantities are far from being an invariable rule. Moreover, we are not even talking about demands here: Such a relation may not be, say, continuous...  
> **username_3**: @densep $MRS = -U_x/U_y$. $U_x = (a/x)U, U_y = (a/y)U$. So, it appears, $MRS = -(a/x)/(a/y) = -x/y$. What am I missing?  
> **username_3**: @Kanak 1) Quantities, not prices, are what create utility for the consumer, so it is quantities that the consumer must choose, in order to maximize its utility. 2) The consumer may "make"(influence)  the price, through his demand of the good, but he does not _set_ the price. So from both angles, the consumer is to choose quantities.  

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### Answer by username_2 on 2018-11-22. Score: 0

You are actually talking about [monopsony](https://course-notes.org/economics/micro_economics/monopsony) -- or in a weaker form: oligopsony.

There are *a priori* no conventional way to deal directly with hybrid situations like the one you describe: you would rather have to deal with a (multi-level) *nested* (CES-type or whatever) *utility function*, recursively optimized, taking the goods' characteristics into account at each solving step.

For *an example* of how such a utility "tree" looks like, see [Keller 1976](https://scholar.google.co.jp/scholar?q=nested+ces+type&hl=en&as_sdt=0&as_vis=1&oi=scholart#d=gs_qabs&p=&u=%23p%3D4M_lBFThHfIJ).


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