﻿ 经济代写 | 4SSMN901 Mathematics For Economists

### 經濟代寫 | 4SSMN901 Mathematics For Economists

Q1. A sole producer has a marginal revenue function given by:
dT R
dQ = 100 ? 3Q
2
Is demand elastic or inelastic if they produce 5 units?
Q2. The Gini coefficient is a number between 0 and 1 and is used
for measuring inequality. A number closer to 1 indicates higher
inequality. Graphically, the Gini coefficient is given by the area
between a diagonal straight line with slope equal to 1 and the
Lorenz curve and this figure divided by the area below the diagonal. If The Lorenz curve is given by y =
11
12x
2 +
1
12x, estimate the
inequality (up to 4 decimals). Compute the Gini coefficient. Is
this an equal or unequal society? Figure 1 represents the diagonal
straight line in red and the Lorenz curve in gray.
Q3. Demand is given by (P + 10)(Q + 20) = 1000 and supply by
Q ? 4P + 10 = 0. Determine the equilibrium, show it graphically as accurately as possible and determine the consumer surplus (round up to the nearest integer). How will this surplus be
affected if price increases by 2 and the equilibrium is not realized?
Q4. A firm produces three goods (G1, G2, G3) and sells them in a
perfectly competitive market. Table 1 shows the number of units
(in thousands) sold of each good along with the total revenue (in
1,000\$) the firm raised by these sales in each calendar month of
2019. Find the equilibrium prices of these goods using matrices.
Q5. The demand functions for product A and B are each a function
of the prices of A and B and are given by:
QA = 50
√3

PB
PA
and QB = 75 PA
√3 P B
2
Determine whether A and B are competitive products, complementary products, or neither.
Q6. Suppose A and B are the only two firms in the market selling the
same product. The industry demand for the product is
p = 92 ? qA ? qB
Where qA and qB denote the output produced and sold by A and
B, respectively. For A, the cost function is cA = 10qA; for B, it is
cB = 0.5q
2
B
. Suppose the firms decide to enter into an agreement
on output and price control by jointly acting as a monopoly. In
this case, we say they enter into collusion. Show that the profit
function for the monopoly is given by
Π = pqA ? cA + pqB ? cB
Express Π as a function of qA and qB, and determine how output
should be allocated so as to maximise the profit of the monopoly.
Q7. Pankaj is a junior web developer working for a major company.
He divides his income between the consumption of goods x1, x2,
and x3 and paying his rent. He can afford to buy one unit of x1
by working for one hour, 1 unit of x2 by working for two hours,
and one unit of x3 by working for three hours. His rent costs him
40% of his income.
His utility function is given by:
u = x
1/2
1 + x
1/2
2 + x
1/2
3
Assuming that Pankaj works for 40 hours a week:
i. Write down Pankaj’s weekly budget constraint and briefly
describe his maximisation problem;
ii. Set up the Lagrangian and find how many of x1, x2, and x3
he would consume per week to maximise his well-being;
iii. What is the interpretation of the Lagrange multiplier? What
does it tell us in this particular example?

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