Final answer:
To solve the problem of finding the equilibrium price and quantity, we need to set the demand and supply equations equal to each other and solve the system algebraically or graphically.
Step-by-step explanation:
To solve the problem, we need to set the demand and supply equations equal to each other since the quantity demanded equals the quantity supplied:
x = 24 - 1/4Px
y = 42 - 1/2Py
We can then solve this system of equations algebraically or graphically to find the equilibrium price and quantity.
In Economics, particularly at the College level, understanding demand functions and market equilibrium can be achieved through graphically plotting demand and supply curves. Where these curves intersect represents the market equilibrium point, confirming the equilibrium price and quantity as derived through algebra.
The student's question is related to the subject of Economics, and specifically to demand functions and market equilibrium analysis, which is typically a College-level topic. The demand function for the commodity x is given by x = 24 − ¼Px, and for commodity y it is y = 42 − ½Py. To find the market equilibrium without algebra, we use graphs to plot the demand and supply curves.
For example, if we're dealing with the situation where the quantity supplied (Qs) equals 12 when the price (P) is $2, we would plot this point on a supply curve. Likewise, the quantity demanded (Qd) would also be 12 at that price, indicating we've found the equilibrium where Qs = Qd.
The demand curve equation would be re-arranged as P = 8 - 0.5Qd, and the supply curve as P = -0.4 + 0.2Qs. Graphically, the point where the two curves intersect represents the market equilibrium, confirming the price and quantity just as algebra does. Understanding these concepts is crucial for anyone studying solving models with graphs or any supply and demand curve analysis.