<p>To use a simple example, assume that consumer $i$ maximizes</p>

<p>$$U(x_i,y_i,I_i) = \alpha\ln x_i +(1-\alpha)\ln y_i + (\bar I_i-p_xx_i -p_yy_i)\\
s.t. \bar I_i \geq p_xx_i +p_yy_i$$</p>

<p>In other words, it may spend all his income on the two goods (but not more), or he may keep some for other goods, not modeled here. Due to the inequality in the constraint we need to use a Karush-Kuhn-Tucker non-negative multiplier, rather than the usual Lagrange multiplier.
Assuming a fixed income, the lagrangean function is</p>

<p>$$\Lambda = \alpha\ln x_i +(1-\alpha)\ln y_i + (\bar I_i-p_xx_i -p_yy_i) +\lambda_i(\bar I_i-p_xx_i -p_yy_i)$$</p>

<p>Assuming income is fixed, the first order conditions are</p>

<p>$$\frac {\alpha}{x_i} - (1+\lambda_i)p_x \leq 0,\;\; \frac {1-\alpha}{y_i} -  (1+\lambda_i)p_y  \leq  0$$</p>

<p>$$\lambda(\bar I_i-p_xx_i -p_yy_i) = 0 ,\; x_i\cdot \left(\frac {\alpha}{x_i} -  (1+\lambda_i)p_x\right)=0, \; y_i\cdot \left(\frac {1-\alpha}{y_i} -  (1+\lambda_i)p_y\right) =0$$</p>

<p>it follows that at the optimum both goods will be demanded in positive quantities, which in turn requires that the first-derivatives be set equal to zero. This gives us</p>

<p>$$\frac {\alpha}{p_xx_i} = \frac {1-\alpha}{p_yy_i} \implies x_i^D=\frac {\alpha}{1-\alpha}\frac {p_y}{p_x}y_i^D \tag{1}$$</p>

<p>from which we can obtain the optimal relation for the Expenditure $E_i$ on the two goods</p>

<p>$$E_i^* = \frac {1}{1+\lambda_i^*} , \;\; E_i^* \leq \bar I_i$$</p>

<p>Since $\lambda_i \geq 0 \implies  \max E_i^* \leq 1$ so if it so happens that $\bar I_i &gt;1$ then not all income will be spent on the two goods, the constraint won't be binding and so $\lambda_i^* =0 $. (Is $\max E_i^* \leq 1$ a strange result?) On the other hand if $\bar I_i \leq 1$ then we will necessarily have $E_i^* = \bar I_i$, and $\lambda_i^* = (1-\bar I_i)/\bar I_i$.</p>

<p>In either case, optimal relation $(1)$ remains valid. So, assuming identical consumers with respect to preferences (not necessarily with respect to Income), aggregating $(1)$ we obtain the market-level relation</p>

<p>$$ X^D=\frac {\alpha}{1-\alpha}\frac {p_y}{p_x}Y^D \tag {2}$$</p>

<p>At equilibrium, we have $X^D = X^S,\; Y^D = Y^S \tag{3}$.</p>

<p>Equations $(2)$ and $(3)$ hold for any supply. 
We now change the supply of $X$ but leave the supply of $Y$ unchanged. Indexing  the initial situation by $0$ and the second situation by $1$, they are described by</p>

<p>$$ X^D_0=\frac {\alpha}{1-\alpha}\frac {p_{y0}}{p_{x0}}Y^D_0,\;\;X^D_0 = X^S_0,\;\; Y^D_0 = Y^S_0 \tag {4}$$</p>

<p>$$ X^D_1=\frac {\alpha}{1-\alpha}\frac {p_{y1}}{p_{x1}}Y^D_1,\;\;X^D_1 = X^S_1&gt;X^S_0,\;\; Y^D_1 = Y^S_1= Y^S_0\tag {5}$$</p>

<p>Then we have</p>

<p>$$X^D_1 - X^D_0 = \frac {\alpha}{1-\alpha}\frac {p_{y1}}{p_{x1}}Y^D_1 - \frac {\alpha}{1-\alpha}\frac {p_{y0}}{p_{x0}}Y^D_0   $$</p>

<p>and using the various relations</p>

<p>$$\implies X^S_1 - X^S_0 = \frac {\alpha}{1-\alpha}\frac {p_{y1}}{p_{x1}}Y^S_1 - \frac {\alpha}{1-\alpha}\frac {p_{y0}}{p_{x0}}Y^S_0 &gt;0   $$</p>

<p>$$\implies \frac {\alpha}{1-\alpha}Y^S_0\cdot \left[\frac {p_{y1}}{p_{x1}}-\frac {p_{y0}}{p_{x0}}\right] &gt;0$$ </p>

<p>$$\implies \frac {p_{y1}}{p_{x1}}-\frac {p_{y0}}{p_{x0}} &gt;0 \implies \frac {p_{y1}}{p_{y0}}&gt;\frac {p_{x1}}{p_{x0}} \tag{6}$$</p>

<p>$(6)$ tells us that the price of $Y$, if it falls, it will fall certainly less than the price of $X$, but <em>in proportional terms</em>.  </p>

<p>It appears that in this benchmark example, we cannot say anything about the price changes in terms of <em>levels</em>, as the OP asks about.  </p>

<p>If the levels of the two prices are very different we can easily have a smaller proportional fall for $p_y$, while at the same time a higher fall in monetary units. Set for example $p_{y0} = 100, p_{x0} = 1$, and assume that the price of $Y$ falls only by $10$% while the price of $X$ falls by $20$%.  </p>

<p>In turn the levels of prices depends on the magnitude of the supply of the two goods also, which are treated as exogenous here.</p>

CC BY-SA 3.0
2015-03-17T19:00:03.677Z
2015-03-17T19:00:03.677Z
Alecos Papadopoulos
<p>In a competitive market, there are 3 types of goods: X, Y and money (m). X and Y are <a href="https://en.wikipedia.org/wiki/Substitute_good" rel="nofollow">substitute goods</a>, and the utility functions are quasi-linear with respect to money, i.e:</p>

<p>$$U(x,y,m) = u(x,y) + m$$</p>

<p>The supply of X and Y is  determined exogeneously, and the prices $p_x$ and $p_y$ are then determined such that the system is in equilibrium (i.e. demands for X and Y equal their supply).</p>

<p>Now the supply of X is increased, and a new equilibrium is achieved with new prices $p_x'$ and $p_y'$. What can we say about the new prices?</p>

<ul>
<li>Certainly $p_x'&lt;p_x$, since we must have new consumers buying the newly available units of X.</li>
<li>Certainly $p_y'\leq p_y$, since X and Y are substitutes (when the price of X decreases, the demand for Y weakly decreases so the price of Y also weakly decreases).</li>
<li>My main question is: <strong>what happens to the price difference $p_x-p_y$?</strong> I did some anecdotal simulations, and it seems that $p_x'-p_y'\leq p_x-p_y$. I.e, even if there is a decrease in the price of Y, the decrease in the price of X is weakly larger.</li>
</ul>

<p>Is this observation correct?</p>

<p>What really happens to the price difference when the supply increases?</p>

CC BY-SA 3.0
2015-03-15T15:29:53.827Z
The question is meaningless as stated, because an economy with two goods has only one relative price, not two.  Do you mean to assume there are three goods?
Steven Landsburg
CC BY-SA 3.0
2015-03-15T18:25:29.503Z
There are two goods and money. The net utility of an agent is the utility of the good, less the amount of money paid for that good. I think this is called "quasi-linear preferences" but I am not sure.
Erel Segal-Halevi
CC BY-SA 3.0
2015-03-17T09:26:27.177Z
I created a nice simulation of a particular case of the market described in the question - the case in which all consumers are unit-demand: http://tora.us.fm/geometry/CompetitiveMarket.html
Playing with the simulation (by changing the supply of X to x-1 or x+1) shows that, indeed, the decrease in the price of X is weakly larger than that of Y. But is there an economic proof for this?
Erel Segal-Halevi
<general-equilibrium><consumer-theory><price-theory>
Price difference of substitute goods when the supply increases?
Erel Segal-Halevi
# Price difference of substitute goods when the supply increases?

(Asked by: username_0 on 2015-03-15)

In a competitive market, there are 3 types of goods: X, Y and money (m). X and Y are [substitute goods](https://en.wikipedia.org/wiki/Substitute_good), and the utility functions are quasi-linear with respect to money, i.e:

$$U(x,y,m) = u(x,y) + m$$

The supply of X and Y is determined exogeneously, and the prices $p\_x$ and $p\_y$ are then determined such that the system is in equilibrium (i.e. demands for X and Y equal their supply).

Now the supply of X is increased, and a new equilibrium is achieved with new prices $p\_x'$ and $p\_y'$. What can we say about the new prices?

* Certainly $p\_x'* Certainly $p\_y'\leq p\_y$, since X and Y are substitutes (when the price of X decreases, the demand for Y weakly decreases so the price of Y also weakly decreases).
  * My main question is: **what happens to the price difference $p\_x-p\_y$?** I did some anecdotal simulations, and it seems that $p\_x'-p\_y'\leq p\_x-p\_y$. I.e, even if there is a decrease in the price of Y, the decrease in the price of X is weakly larger.

Is this observation correct?

What really happens to the price difference when the supply increases?

**Tags:** general-equilibrium, consumer-theory, price-theory
> **Comments:**
> **username_1**: The question is meaningless as stated, because an economy with two goods has only one relative price, not two.  Do you mean to assume there are three goods?  
> **username_0**: There are two goods and money. The net utility of an agent is the utility of the good, less the amount of money paid for that good. I think this is called "quasi-linear preferences" but I am not sure.  
> **username_0**: I created a nice simulation of a particular case of the market described in the question - the case in which all consumers are unit-demand: http://tora.us.fm/geometry/CompetitiveMarket.html Playing with the simulation (by changing the supply of X to x-1 or x+1) shows that, indeed, the decrease in the price of X is weakly larger than that of Y. But is there an economic proof for this?  

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## 1 answers

### Answer by username_2 on 2015-03-17. Score: 2 | Accepted answer

To use a simple example, assume that consumer $i$ maximizes

$$U(x\_i,y\_i,I\_i) = \alpha\ln x\_i +(1-\alpha)\ln y\_i + (\bar I\_i-p\_xx\_i -p\_yy\_i)\\
s.t. \bar I\_i \geq p\_xx\_i +p\_yy\_i$$

In other words, it may spend all his income on the two goods (but not more), or he may keep some for other goods, not modeled here. Due to the inequality in the constraint we need to use a Karush-Kuhn-Tucker non-negative multiplier, rather than the usual Lagrange multiplier.
Assuming a fixed income, the lagrangean function is

$$\Lambda = \alpha\ln x\_i +(1-\alpha)\ln y\_i + (\bar I\_i-p\_xx\_i -p\_yy\_i) +\lambda\_i(\bar I\_i-p\_xx\_i -p\_yy\_i)$$

Assuming income is fixed, the first order conditions are

$$\frac {\alpha}{x\_i} - (1+\lambda\_i)p\_x \leq 0,\;\; \frac {1-\alpha}{y\_i} - (1+\lambda\_i)p\_y \leq 0$$

$$\lambda(\bar I\_i-p\_xx\_i -p\_yy\_i) = 0 ,\; x\_i\cdot \left(\frac {\alpha}{x\_i} - (1+\lambda\_i)p\_x\right)=0, \; y\_i\cdot \left(\frac {1-\alpha}{y\_i} - (1+\lambda\_i)p\_y\right) =0$$

it follows that at the optimum both goods will be demanded in positive quantities, which in turn requires that the first-derivatives be set equal to zero. This gives us

$$\frac {\alpha}{p\_xx\_i} = \frac {1-\alpha}{p\_yy\_i} \implies x\_i^D=\frac {\alpha}{1-\alpha}\frac {p\_y}{p\_x}y\_i^D \tag{1}$$

from which we can obtain the optimal relation for the Expenditure $E\_i$ on the two goods

$$E\_i^\* = \frac {1}{1+\lambda\_i^\*} , \;\; E\_i^\* \leq \bar I\_i$$

Since $\lambda\_i \geq 0 \implies \max E\_i^\* \leq 1$ so if it so happens that $\bar I\_i >1$ then not all income will be spent on the two goods, the constraint won't be binding and so $\lambda\_i^\* =0 $. (Is $\max E\_i^\* \leq 1$ a strange result?) On the other hand if $\bar I\_i \leq 1$ then we will necessarily have $E\_i^\* = \bar I\_i$, and $\lambda\_i^\* = (1-\bar I\_i)/\bar I\_i$.

In either case, optimal relation $(1)$ remains valid. So, assuming identical consumers with respect to preferences (not necessarily with respect to Income), aggregating $(1)$ we obtain the market-level relation

$$ X^D=\frac {\alpha}{1-\alpha}\frac {p\_y}{p\_x}Y^D \tag {2}$$

At equilibrium, we have $X^D = X^S,\; Y^D = Y^S \tag{3}$.

Equations $(2)$ and $(3)$ hold for any supply.
We now change the supply of $X$ but leave the supply of $Y$ unchanged. Indexing the initial situation by $0$ and the second situation by $1$, they are described by

$$ X^D\_0=\frac {\alpha}{1-\alpha}\frac {p\_{y0}}{p\_{x0}}Y^D\_0,\;\;X^D\_0 = X^S\_0,\;\; Y^D\_0 = Y^S\_0 \tag {4}$$

$$ X^D\_1=\frac {\alpha}{1-\alpha}\frac {p\_{y1}}{p\_{x1}}Y^D\_1,\;\;X^D\_1 = X^S\_1>X^S\_0,\;\; Y^D\_1 = Y^S\_1= Y^S\_0\tag {5}$$

Then we have

$$X^D\_1 - X^D\_0 = \frac {\alpha}{1-\alpha}\frac {p\_{y1}}{p\_{x1}}Y^D\_1 - \frac {\alpha}{1-\alpha}\frac {p\_{y0}}{p\_{x0}}Y^D\_0 $$

and using the various relations

$$\implies X^S\_1 - X^S\_0 = \frac {\alpha}{1-\alpha}\frac {p\_{y1}}{p\_{x1}}Y^S\_1 - \frac {\alpha}{1-\alpha}\frac {p\_{y0}}{p\_{x0}}Y^S\_0 >0 $$

$$\implies \frac {\alpha}{1-\alpha}Y^S\_0\cdot \left[\frac {p\_{y1}}{p\_{x1}}-\frac {p\_{y0}}{p\_{x0}}\right] >0$$

$$\implies \frac {p\_{y1}}{p\_{x1}}-\frac {p\_{y0}}{p\_{x0}} >0 \implies \frac {p\_{y1}}{p\_{y0}}>\frac {p\_{x1}}{p\_{x0}} \tag{6}$$

$(6)$ tells us that the price of $Y$, if it falls, it will fall certainly less than the price of $X$, but *in proportional terms*.

It appears that in this benchmark example, we cannot say anything about the price changes in terms of *levels*, as the OP asks about.

If the levels of the two prices are very different we can easily have a smaller proportional fall for $p\_y$, while at the same time a higher fall in monetary units. Set for example $p\_{y0} = 100, p\_{x0} = 1$, and assume that the price of $Y$ falls only by $10$% while the price of $X$ falls by $20$%.

In turn the levels of prices depends on the magnitude of the supply of the two goods also, which are treated as exogenous here.


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