Question

Consider the following general linear demand and supply functions that represent a market: Qd =Z −GP (3) Qs = D + EP+ CS (4) where P is the price, S is a variable denoting the average amount of production shipping costs, and Qd and Qs are the quantity demanded and the quantity supplied. Assume D, E, G, and Z all have values greater than zero.

(a) What equation (in addition to equations 3 & 4) completes our mathematical model of market equilibrium?
(b) Identify the parameters, endogenous variables, and exogenous variables in the above system of equations.
(c) Derive expressions for the equilibrium market price (P ∗ ) and quantity (Q ∗ ) and illustrate your answers with a graph. Be sure to specify the symbolic values of the demand and supply curves where they intersect with the P-axis and Q-axis in the positive quadrant.
(d) Given your results from part (a), use calculus to determine the effect of a small increase in shipping on the equilibrium price (P*). What is the sign of the expression you find? Be sure to briefly describe the logic you use to determine whether C is positive or negative.

Answer 1

To complete the market equilibrium model, the equation Qd = Qs is required. Endogenous variables are P, Qd, and Qs, while exogenous variables include S (shipping costs). Equilibrium price and quantity can be derived algebraically or graphically, and calculus is used to find the effect of shipping costs on equilibrium price.

Step-by-step explanation:

To complete the mathematical model of market equilibrium, we need an equation that sets the quantity demanded (Qd) equal to the quantity supplied (Qs), as they must be equal at equilibrium. Therefore, the additional equation is Qd = Qs. In the given demand and supply functions Qd = Z − GP and Qs = D + EP + CS, the parameters are Z, G, D, E, and C.

The endogenous variables are those determined within the system, namely the price (P) and quantities (Qd and Qs), and exogenous variables are those determined outside the system, such as S (production shipping costs). To find the equilibrium price (P*) and quantity (Q*), we set Qd = Qs and solve for P, then substitute back to find Q. Graphically, we would solve each equation for P and graph the demand curve P = (Z/Qd) - (G), and the supply curve P = (D/Qs) + (E) + (CS). The equilibrium is found where these curves intersect on a graph, and P* and Q* are the coordinates of this point.

Using calculus to determine the effect of a small increase in shipping costs on the equilibrium price, we take the derivative of the price with respect to S, holding other factors constant. The sign of the derivative will indicate whether the equilibrium price will rise or fall with an increase in S, and this sign is determined by the positive or negative influence of C on the supply function.