The demand curve for Widgets is given by QD 5800 200p 30p

The demand curve for Widgets is given by QD = 5800 - 200p + 30pG where QD is the quantity of widgets demanded, y is the per capital income and pG is the price of Gizmos. The supply of Widgets is given by: QS = 250p - 1250 a. Solve for the equilibrium price and quantity of widgets in terms of the price of Gizmos. b. Compute the comparative static derivatives for the changes in the equilibrium price and quantity of Widgets with respect to a change in the price of Gizmos. please show all steps in math and equation solving thank you.

Solution

a.   Set quantity demand equal to quantity supply to find price:

            5800 - 200p + 30pG = 250p - 1250

      Therefore, p = (6050 + 30pG)/450

                        = 6050/450 + 30pG /450

                        = 13.44 + 0.067pG

      Substituting the value of p into demand or supply to find quantity:

            Q = 250(6050 + 30pG)/450 - 1250

            Q = (1512500 + 7500pG)/450 - 1250

               = 1512500/450 + 7500pG/450 -1250

                = 3361.11 + 16.67pG - 1250

                = 2111.11 + 16.67pG

b.   The first order derivative of Quantity demanded of Widgets with respect to price of Gizmos

dQ/dpG = d( 2111.11 + 16.67pG) / dpG

            =d( 2111.11) / dpG + d(16.67pG) / dpG

            = 0 + 16.67

   = 16.67      ( Note: 150/9 = 16.67)

and, The first order derivative of Price of Widgets with respect to price of Gizmos

dp/dpG = d(13.44 + 0.067pG) / dpG

           = d(13.44) / dpG + d(0.067pG) / dpG

            = 0 + 0.067

            = 0.067 ( Note: 1/15 = 16.67)

Note: The derivative of a constant = 0