The demand and supply functions for a product are expressed in dollars such that

Question

The demand and supply functions for a product are expressed in dollars such that Find both the consumer surplus and producer surplus if the market for the given product is in equilibrium. Suppose that customer demand depends upon the price trend according to the formula If the supply function is q^s(p) = -24 + 20p, write down the condition for equilibrium and determine the general solution for the equilibrium price p(t). A market model in which sellers keep an inventory of a commodity is given by the equations: Create a difference equation involving Pt only. Classify this equation as first order or second order. In a certain state, it is found that the distribution of income for professional league soccer players is given by the Lorenz Curve y = Li(x) where while that for the professional golfers is given by y = I>2(x) where Which of these sports professions has the more equitable income distribution?b. In applying the Cobb-Douglas production model P = 1.01 K^0.25L where K is the units of Capital and L is the units of Labour employed, we note that each unit of Capital costs $ 175 and each unit of Labour costs $125. If there is a total of $125,000 to invest in the economy, develop a mathematical optimisation problem for finding the number of units of capital and the number of units of labour that will maximize the total production in the manufacturing sector oj the economy. ii. solve the model interpret the derived value of Lagrange Multiplier. Jingle Bells Inc. has received a rush order for as many of two types of Easter eggs as can be purchased and shipped during a 2-week period. Preliminary analysis by Jingle Bells indicates that the profit on this order is related to the number of each type of Easter egg manufactured by the function: i) Formulate a mathematical model that can be used to determine how many thousands of each Type of egg should be manufactured in the two-week period so as to maximise profit. Identify a suitable solution method and state the assumptions that underpin your choice of method. Find a solution to your mathematical model.

Explanation / Answer

(a) D(q) = 24 / (q + 2) & S(q) = - 8 + 3q

In equilibrium, D(q) = S(q)

24 / (q + 2) = - 8 + 3q

24 = - 8q + 3q2 - 16 + 6q

3q2 - 2q - 40 = 0

3q2 - 12q + 10q - 40 = 0

3q(q - 4) + 10(q - 4) = 0

(q - 4) (3q + 10) = 0

q = 4, or q = - 10/3 (inadmissible)

When q = 4, P = - 8 + (3 x 4) = - 8 + 12 = 4

Consumer surplus (CS) = Area between demand curve & price

From demand function, when q = 0, P = 24/2 = 12 (Reservation price)

So CS = (1/2) x (12 - 4) x 4 = (1/2) x 8 x 4 = 16

Producer surplus = Area between supply curve & price

PS = (1/2) x 4 x 4 = 8

Note: First question is answered.

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