# Equilibrium price determination in a 2 commodity framework

Following are the set of equations describing the demand and supply of two goods X and Y:

Demand functions: $$X_d = a_1 - b_1P_x + c_1P_y$$

$$Y_d = a_2 - b_2P_y +c_2P_x$$

$$a_1,~ a_2,~ b_1,~ b_2,~ c_1,~ c_2$$ are positive.

Supply functions:

$$X_s = -f_1 + g_1P_x$$

$$Y_s = -f_2 + g_2Py$$

$$f_1,~ f_2,~ g_1,~ g_2$$ are positive

Government imposed $$t\%$$ tax on $$X$$ and allows $$s\%$$ subsidy on $$Y$$ such that government budget remains balanced. Find out the required tax rate. (Clearly show How the set of equations change and the equilibrium prices ?)

I replaced $$P_x$$ by $$P_x(1+t)$$ in the two demand equations as according to my understanding consumers now face a higher effective price for $$X$$. But I'm not sure how to incorporate the subsidy and what would be the economic logic behind such a change.

• Why can't you use s the same way you use t? If Px(1+t) why not Py(1-s)?
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Commented Jul 14, 2019 at 2:39
• The reasoning behind replacing Px by Px(1+t) in the demand functions only was that consumers now have to pay a new price but the producers still recieve the same so supply functions were unchanged. Now in case of subsidy, producers will charge a lower price, Py(1-s), so I can replace it in supply equations. The problem is that consumers also face the lower price of Y so should I change the price of Y in demand functions as well? Commented Jul 14, 2019 at 4:20