dfbfdf

  • Upload
    jim-kat

  • View
    217

  • Download
    0

Embed Size (px)

Citation preview

  • 8/11/2019 dfbfdf

    1/5

    B

    usine

    ssCalculus

    WSUF

    all20

    10

    14.2. CONSUMER AND PRODUCER SURPLUS 1

    14.2 Consumer and Producer Surplus

    Consumer and Producer Surplus

    Let us direct our attention towards applying some of the concepts that we learned in regards to the area of

    region bounded by two curves. In this section we will use the following notation.

    D(q) will denote the price-demand equation of product.

    S(q) will denote the price-surplus equation of a product.

    The variableq will denote the quantity of the product that is in question. In a free-market economy, the

    price of an product is determined by the relationship between its supply and demand. A positive surplus

    occurs when the price is above a certain price-level; whereas, a negative surplus (i.e. shortage) occurs the

    price is below this price-level. This price-level is known as the equilibrium price. It is the price in which the

    effects of demand is in balance with a products supply.

    Definition 14.1: Equilibrium-Point for Supply and Demand

    The equilibrium-point between forD(q)and S(q)is the point(q0,p0)such thatp0 =D(q0) = S(q0).

    Thep0 is referred to as the equilibrium price, andq0 is known as the equilibrium quantity.

    The equilibrium-pricep0is the price per unit that the consumers are willing to purchase, and the price that

    producers are willing to sell.

  • 8/11/2019 dfbfdf

    2/5

    B

    usine

    ssCalculus

    WSUF

    all20

    10

    2

    Example1.2 Equilibrium-point

    Find the equilibrium-point if the D(q) = 25 + 0.2q and S(q) = 1 + 0.01q2

    Solution

    The equilibrium-point between D(q)and S(q)can be found by solving the equation D(q) = S(q).

    D(q) = S(q)

    25 0.2q = 1 + 0.01q2

    (q 60)(q + 40) = 0

    Since it is impractical for q = 40, it must be the case that q = 60; in other words, the equilibrium-quantity is q0 =60,

    which in turns yields the equilibrium-price as S(60) = D(60) = 37. Therefore, the equilibrium-point point is(60, 37).

    Sometimes it necessary to determine the difference between the maximum price and the actual price that a

    consumer is willing to pay for a product. This quantity is know as the consumer surplus.

    Definition 14.2: Consumer Surplus

    The consumer surplus at the price-level of p = D(q)is defined as

    CS =

    q

    0

    (D(q) p)dq.

  • 8/11/2019 dfbfdf

    3/5

    B

    usine

    ssCalculus

    WSUF

    all20

    10

    14.2. CONSUMER AND PRODUCER SURPLUS 3

    Example2.2 Consumers Surplus

    Determine the consumer surplus at a price-level of of$5for the price-demand equation D(q) = 41 0.01q2.

    Solution

    The question that first needs to be addressed is what is the value ofq when p = 5.

    p = D(q)

    5 = 41 0.01q2

    3, 600 = q2

    Since quantity is a nonnegative measure, then it follows that q = 6; therefore, the consumer surplus, at the given

    price-level, is found by determining the area bounded by the functions D(q)and p = 5on the interval[0, 6].

    60

    (41 0.01q2 5)dq = 36q q3

    3000

    60=215.28

    This value$215.28represents the total savings to consumers who are willing to pay a higher price than $5in order to

    obtain the product.

    The producer surplus measures the difference between the price in which the producers are willing to

    supply to the consumers and the actual price that they receive from the consumers.

    Definition 14.3: Producer Surplus

    The producer surplus at the price-level of p = S(q)is defined as

    CS =

    q0

    ( p S(q))dq.

  • 8/11/2019 dfbfdf

    4/5

    B

    usine

    ssCalculus

    WSUF

    all20

    10

    4

    Example3.2 Producers Surplus

    Find the producers surplus at a price level of$30if the price-supply equation is S(q) = 5 + 0.01q2.

    Solution

    In this situation, we are given p = 30and we need to determine q. We can do this solving the equation p0 =S(q0).

    30 = 5 + 0.01q2

    2500 = q2

    Since quantity is a nonnegative measure, then it follows that q =

    50; therefore, the producer surplus, at the givenprice-level, is found by determining the area bounded by the functions S(q)and p = 30on the interval[0, 50].

    500

    (30 (5 0.01q2))dq =q3

    3000+ 25q

    500=

    5000

    3

    This value$ 50003 $1666.67represents the total amount gained by the producers who are willing to sell the product

    to the consumers at a lower price than$8.

  • 8/11/2019 dfbfdf

    5/5

    B

    usine

    ssCalculus

    WSUF

    all20

    10

    14.2. CONSUMER AND PRODUCER SURPLUS 5

    Consumer Surplus Producer Surplus

    The total savings to the consumers who

    are willing to pay more than the given price

    ofD(q) = pfor a product.

    The total gain to the producers who are

    willing to supply a product at a price lower

    than a given priceS(q) = p.

    Ifpis the equilibrium-price forD(q) andS(q), then there is stability between supply and demand.