<p>There exist implicit assumptions in the set up that exclude various cross-price effects on the demand side, related to business behavior as well as to consumer behavior. Accepting the set up (and assuming downward-sloping demand and upward-sloping supply), the old-fashioned verbal treatment goes as follows:  </p>

<p>An increase in $G$ will move the demand schedule $Q_1^d$ outwards, in the $\{Q_1, P_1\}$ space. This will tend to bring a new equilibrium for good 1, at a higher price $P_1$, and a higher quantity $Q_1$.</p>

<p>A higher $P_1$ increases costs for the supply of good $2$, and so it will shift upwards the supply curve for good 2 in the $\{Q_2, P_2\}$ space, while leaving the demand schedule for good $2$ unaffected. This will produce an equilibrium with higher $P_2$ and lower $Q_2$. A higher $P_2$ in turn will tend to shift upwards the supply curve for good 1 in the $\{Q_1, P_1\}$ space (again, because due to higher costs, lower quantity is supplied at any own price than previously).</p>

<p>This will tend to increase equilibrium $P_1$ even more, but it will tend to offset the increase in the new equilibrium $Q_1$.  </p>

<p><strong>So we have learned:</strong><br>
In market $2$, an increase in $G$ for good $1$ will lead to an equilibrium with higher $P_2$ and lower $Q_2$.<br>
In market $1$ an increase in $G$ will lead to an equilibrium with higher $P_1$, but ambiguous effect on the new equilibrium $Q_1$.</p>

<p>I'll leave to the OP the task of mathematizing these results and see also whether the ambiguity can be resolved.</p>

CC BY-SA 3.0
2016-10-29T15:03:47.420Z
2016-10-29T15:03:47.420Z
Alecos Papadopoulos
<p>I'm not sure if it's more economic than mathematical question but it's about analyzing a model from economics so I'd rather ask it here.</p>

<p>There's a model with two goods, 1 and 2. The first one has both consumer demand ($D_1(P_1)$) and the government demand ($G$, the variable is predefined externally). The second one has only $D_2(P_2)$.</p>

<p>The supply functions for both goods are: $Q^s_{1,2}=S(P_1,P_2)$ (some quantity of good 2 is used in good 1 production and vice versa) The supply quantity of good 1 increases when its own price $P_1$ increases but decreases when there's an increase in $P_2$ (as the costs increase). Also own effects dominate, so $\frac {\partial S_1}{\partial P_1} &gt; |\frac {\partial S_1}{\partial P_2}|$ in absolute terms. The same is true for $S_2$. Again, I'll write out all the equations below for more clarity:</p>

<p>$$Q^d_1 = D_1(P_1)+G$$
$$Q^d_2 = D_2(P_2)$$
$$Q^s_1 = S^1(P_1, P_2)$$
$$Q^s_2 = S^2(P_1, P_2)$$</p>

<p>Where I got stuck is in calculation of $\frac {\partial Q}{\partial G}$ for $Q_1, Q_2$ (the change in $Q_1$ or $Q_2$ after a change in G). The only thing that comes to my mind is:</p>

<p>$$\frac {\partial D_1}{\partial P_1}*\frac {dP_1}{dG} + 1 = \frac {\partial S^1}{\partial P_1}*\frac {dP_1}{dG} + \frac {\partial S^1}{\partial P_2}*\frac {dP_2}{dG}$$ for $Q_1$, but how to calculate the $\frac {\partial Q_1}{\partial G}$ from here?</p>

CC BY-SA 3.0
2016-10-28T15:48:40.527Z
A clarification: by the setup, it appears that for both goods a _business_ demand exists, since they are also used as inputs. Are we to assume that the demand function are in essence not only consumer demand but "consumer + business" demand?
Alecos Papadopoulos
CC BY-SA 3.0
2016-10-29T06:25:46.073Z
As far as I have understood the setup, $D_i(P_i)$ means any non-governmental demand (including business and consumer demand)
sempol
CC BY-SA 3.0
2016-10-29T06:42:30.397Z
Upd: Just checked the setup again, yes, $D(P)$ is the "private demand"
sempol
<microeconomics><mathematical-economics>
Mathematical analysis of an economic model - stuck in calculations
sempol
# Mathematical analysis of an economic model - stuck in calculations

(Asked by: username_0 on 2016-10-27)

I'm not sure if it's more economic than mathematical question but it's about analyzing a model from economics so I'd rather ask it here.

There's a model with two goods, 1 and 2. The first one has both consumer demand ($D\_1(P\_1)$) and the government demand ($G$, the variable is predefined externally). The second one has only $D\_2(P\_2)$.

The supply functions for both goods are: $Q^s\_{1,2}=S(P\_1,P\_2)$ (some quantity of good 2 is used in good 1 production and vice versa) The supply quantity of good 1 increases when its own price $P\_1$ increases but decreases when there's an increase in $P\_2$ (as the costs increase). Also own effects dominate, so $\frac {\partial S\_1}{\partial P\_1} > |\frac {\partial S\_1}{\partial P\_2}|$ in absolute terms. The same is true for $S\_2$. Again, I'll write out all the equations below for more clarity:

$$Q^d\_1 = D\_1(P\_1)+G$$
$$Q^d\_2 = D\_2(P\_2)$$
$$Q^s\_1 = S^1(P\_1, P\_2)$$
$$Q^s\_2 = S^2(P\_1, P\_2)$$

Where I got stuck is in calculation of $\frac {\partial Q}{\partial G}$ for $Q\_1, Q\_2$ (the change in $Q\_1$ or $Q\_2$ after a change in G). The only thing that comes to my mind is:

$$\frac {\partial D\_1}{\partial P\_1}\*\frac {dP\_1}{dG} + 1 = \frac {\partial S^1}{\partial P\_1}\*\frac {dP\_1}{dG} + \frac {\partial S^1}{\partial P\_2}\*\frac {dP\_2}{dG}$$ for $Q\_1$, but how to calculate the $\frac {\partial Q\_1}{\partial G}$ from here?

**Tags:** microeconomics, mathematical-economics
> **Comments:**
> **username_1**: A clarification: by the setup, it appears that for both goods a _business_ demand exists, since they are also used as inputs. Are we to assume that the demand function are in essence not only consumer demand but "consumer + business" demand?  
> **username_0**: As far as I have understood the setup, $D_i(P_i)$ means any non-governmental demand (including business and consumer demand)  
> **username_0**: Upd: Just checked the setup again, yes, $D(P)$ is the "private demand"  

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## 1 answers

### Answer by username_1 on 2016-10-29. Score: 1 | Accepted answer

There exist implicit assumptions in the set up that exclude various cross-price effects on the demand side, related to business behavior as well as to consumer behavior. Accepting the set up (and assuming downward-sloping demand and upward-sloping supply), the old-fashioned verbal treatment goes as follows:

An increase in $G$ will move the demand schedule $Q\_1^d$ outwards, in the $\{Q\_1, P\_1\}$ space. This will tend to bring a new equilibrium for good 1, at a higher price $P\_1$, and a higher quantity $Q\_1$.

A higher $P\_1$ increases costs for the supply of good $2$, and so it will shift upwards the supply curve for good 2 in the $\{Q\_2, P\_2\}$ space, while leaving the demand schedule for good $2$ unaffected. This will produce an equilibrium with higher $P\_2$ and lower $Q\_2$. A higher $P\_2$ in turn will tend to shift upwards the supply curve for good 1 in the $\{Q\_1, P\_1\}$ space (again, because due to higher costs, lower quantity is supplied at any own price than previously).

This will tend to increase equilibrium $P\_1$ even more, but it will tend to offset the increase in the new equilibrium $Q\_1$.

**So we have learned:**  
In market $2$, an increase in $G$ for good $1$ will lead to an equilibrium with higher $P\_2$ and lower $Q\_2$.  
In market $1$ an increase in $G$ will lead to an equilibrium with higher $P\_1$, but ambiguous effect on the new equilibrium $Q\_1$.

I'll leave to the OP the task of mathematizing these results and see also whether the ambiguity can be resolved.


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