Question

The supply function for oil is given (in dollars) by S(q), and the demand function is given (in dollars) by:

D(q)â‹…S(q)=q^2+13q
D(q)=992−17q−q^2

Required:
a. Graph the supply and demand curves on the same axes.
b. Find the point at which supply and demand are in equilibrium.
c. Find the consumers' surplus.
d. Find the producers' surplus.

Answer 1

Answer:

E ( 16 , $464 )

$4907

$4395

Step-by-step explanation:

Solution:-

- The supply curve is an upward sloping curve that denotes the relation between the quantity of goods ( oil ) supplied ( q ) and the market price ( p ). Such that the market price ( p ) increases with each additional unit of oil sold.

- The supply function S ( q ) for the commodity ( oil ) is expressed as:

                             $S ( q ) = q^2 + 13q$

- The demand curve is a downward sloping curve that is derived from the concept of diminishing marginal utility i.e with each additional unit of oil consumed people are willing to pay less for the additional unit. The demand curve D ( q ) for the commodity ( oil ) is expressed as:

                            $D ( q ) = 992 - 17q - q^2$

- The two graphs are plotted and given in the attachment with quantity of oil ( q ) on the x-axis and market price ( p ) on the y-axis.

- The equilibrium point is the ordered pair of ( quantity of oil , market price ) where the demand curve and supply curve intersect. This is the point where manufacturing sell and the customers buy the commodity.

- To determine the equilibrium point ( E ) we will equate the two curves S ( q ) and D ( q ) as follows:

                            $S ( q ) = D ( q )\\\\q^2 + 13q = 992 - 17q - q^2\\\\2q^2 + 30q - 992 = 0\\\\q = -31 , q = 16$

- We will ignore the negative value of quantity of goods and accept q = 16 units. Plug the value in either of the two equations of curve:

                           $S ( 16 ) = ( 16 )^2 + 13*( 16 )\\\\S ( 16 ) = 464 . dollars$

- The ordered pair for the equilibrium point is:

Answer:            E ( 16 , $464 )

- Consumer surplus (CS) is the region bounded by the market equilibrium price ( pe = $464 ) and the demand curve D ( q ). We will employ the use of calculus and evaluate the area of the region using integrals as follows:

                           $\int\limits^1^6_0 [ {D(q) - p_e} ] \, dq$

- Evaluate the quantity ( q ) over to the equilibrium quantity ( qe = 16 ):

                        $CS = \int [ 992-17q-q^2 - 464} ] \limits^1^6_0 . dq = \int [ 528-17q-q^2 } ].dq \limits^1^6_0\\\\CS = [ 528q - \frac{17}{2}*q^2 - \frac{1}{3}*q^3 ] \limits^1^6_0\\\\CS = [ 528(16) - \frac{17}{2}*(16)^2 - \frac{1}{3}*(16)^3 ]\\\\CS = 4907 . dollars$

Answer: Their is a consumer surplus of ( $ 4907 ) for this commodity.

- Producer's surplus ( PS ) is the region bounded by the market equilibrium price ( pe = $464 ) and the supply curve S ( q ). We will employ the use of calculus and evaluate the area of the region using integrals as follows:

                           $\int\limits^1^6_0 [ {p_e - S(q)} ] \, dq$

- Evaluate the quantity ( q ) over to the equilibrium quantity ( qe = 16 ):

                        $PS = \int [ 464 - q^2 - 13q ] \limits^1^6_0 . dq \\\\PS = [ 464q - \frac{13}{2}*q^2 - \frac{1}{3}*q^3 ] \limits^1^6_0\\\\PS = [ 464(16) - \frac{13}{2}*(16)^2 - \frac{1}{3}*(16)^3 ]\\\\PS = 4395 . dollars$

Answer: Their is a producer's surplus of ( $ 4395 ) for this commodity.