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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 $y$?

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  • $\begingroup$ Are you sure $MRS(x,y) = -x/y$ and not $-y/x$? The first one would make the utility function concave. $\endgroup$ – denesp Nov 22 '18 at 14:11
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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

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  • $\begingroup$ why would a consumer maximize over quantities if she is price maker? (cc @user617643) $\endgroup$ – Kanak Nov 22 '18 at 13:27
  • $\begingroup$ @Kanak If there is a bijective mapping between $p_y$ and $y$ why not? $\endgroup$ – denesp Nov 22 '18 at 14:07
  • $\begingroup$ The utility function $U(x,y) = Ax^ay^a$ does not have the given marginal rate of substitution though. $\endgroup$ – denesp Nov 22 '18 at 14:10
  • $\begingroup$ @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... $\endgroup$ – Kanak Nov 22 '18 at 15:52
  • $\begingroup$ @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? $\endgroup$ – Alecos Papadopoulos Nov 22 '18 at 19:49
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You are actually talking about 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.

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