2
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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? $\endgroup$ – Alecos Papadopoulos Oct 28 '16 at 15:48
  • $\begingroup$ As far as I have understood the setup, $D_i(P_i)$ means any non-governmental demand (including business and consumer demand) $\endgroup$ – sempol Oct 29 '16 at 6:25
  • $\begingroup$ Upd: Just checked the setup again, yes, $D(P)$ is the "private demand" $\endgroup$ – sempol Oct 29 '16 at 6:42
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.