Question

1. (15 points) suppose the market demand for a good consists of two consumers, 1 and 2, where their respective individual demands are: \\(d_1(p) = 200 - 4p\\) and \\(d_2(p) = 100 - p\\). note that demand cannot be negative, so what we really mean is \\(d_1(p) = \max\\{200 - 4p, 0\\}\\) and \\(d_2(p) = \max\\{100 - p, 0\\}\\).

(a) (5 points) on one diagram graph the individual demand curves as well as the resulting market demand curve. mark the intercepts correctly.

(b) (5 points) suppose supply is given by \\(p = 40\\), that is, the supply is perfectly elastic. find the amount purchased by each consumer. illustrate your answer with a graph.

(c) (5 points) now, suppose instead the supply is given by \\(q = 20\\). find the equilibrium market price. (be very careful here.) illustrate your answer with a graph.

Explanation:

Aggregate individual demands

Using the Market Equilibrium knowledge point, we sum individual demands horizontally.
Individual demands are:

$$ D_1(p) = \max\{200 - 4p, 0\} \implies D_1(p) > 0 \text{ for } p < 50 $$

$$ D_2(p) = \max\{100 - p, 0\} \implies D_2(p) > 0 \text{ for } p < 100 $$

For \(p \ge 100\):

$$ D(p) = 0 $$

For \(50 \le p < 100\):

$$ D(p) = 100 - p $$

For \(p < 50\):

$$ D(p) = (200 - 4p) + (100 - p) = 300 - 5p $$

Find quantities at price \(p = 40\)

Substitute \(p = 40\) into individual demand functions:

$$ q_1 = D_1(40) = 200 - 4(40) = 40 $$

$$ q_2 = D_2(40) = 100 - 40 = 60 $$

Total market quantity purchased:

$$ Q = q_1 + q_2 = 100 $$

Find equilibrium price for supply \(q = 20\)

We set the market demand equal to the fixed supply \(Q = 20\).
Since \(20 < 50\) (the quantity at the kink point where \(p = 50\) is \(D(50) = 50\)), the equilibrium must occur on the upper segment of the market demand curve where only consumer 2 purchases the good.
Using the segment for \(50 \le p < 100\):

$$ D(p) = 100 - p = 20 \implies p = 80 $$

At \(p = 80\):

$$ q_1 = D_1(80) = 0 $$

$$ q_2 = D_2(80) = 20 $$

Answer:

Question 2

(a) The individual demand curves and the resulting market demand curve are defined as:

• Consumer 1 Demand: \(D_1(p) = 200 - 4p\) for \(p \in [0, 50]\), with intercepts at \((q=200, p=0)\) and \((q=0, p=50)\).
• Consumer 2 Demand: \(D_2(p) = 100 - p\) for \(p \in [0, 100]\), with intercepts at \((q=100, p=0)\) and \((q=0, p=100)\).
• Market Demand:

$$ D(p) =
LATEXBLOCK0
$$

The market demand curve has a kink at \((q=50, p=50)\), with intercepts at \((q=300, p=0)\) and \((q=0, p=100)\).

(b) When supply is perfectly elastic at \(p = 40\):

• Amount purchased by Consumer 1: \(q_1 = 40\) units.
• Amount purchased by Consumer 2: \(q_2 = 60\) units.
• Total market quantity: \(Q = 100\) units.

(c) When supply is fixed at \(q = 20\):

• The equilibrium market price is \(p = 80\).
• At this price, Consumer 1 purchases \(q_1 = 0\) units, and Consumer 2 purchases \(q_2 = 20\) units.