Mathematical analysis of an economic model - stuck in calculations

Question

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?

Answer 1

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.