Question 1.2 Chapter 5
Question : If the inverse demand function for toasters is \[p = 60-q \ \ ,\] what is the CONSUMER SUPRLUS if the price is 30?
- Answer:
- What do we know ?
- Demand Function
- \(Price = 30\)
- \(Price = 30\)
- Consumer Surplus : [Economist]: monetary difference between the max amount the consumer is willing to pay for a quantity of a good purchased and the actual price of the good in the market. [Layman] How much is the gap between what I want to pay for the good and the price that I have to pay. [Geometrically:] Area under demand curve and above price line.
- Start with a empty 2-d plot :
- Draw the demand curve , Price Line and the intercept points.
- Next, we figure out the dimensions of the triangle:
- Do you spot the Consumer Surplus :D
- For the Area of the \(\Delta ABC\) : Calculate \(\mathbf{Q^*}\) , \(\mathbf{BC}\) and \(\mathbf{AB}\)?
- To find \(\mathbf{Q^*}\) , find the point where the price line cuts the demand line. All you have to do is replace the value of \(p\) in the demand equation with the given price \(p = 30\) and calculate \(\mathbf{Q^*}\)
- \[ \begin{split} \mathbf{Q^{*}}: p &= 60-q^{*} \\ 30 &= 60-q^{*} \\ \implies q^* = 30 \end{split} \]
- This \(\implies \ BC = 30\)
- \(AB\) is simple : \(AB = 60 -30 = 30\)
- Now we have everything for the area of the triangle. We are ready to calculate the consumer surplus: \[ \mathbf{CS:} \begin{split} Area(\Delta ABC) & = \dfrac{1}{2} \times \texttt{base} \times \texttt{height} \\ & =\dfrac{1}{2} \times 30 \times 30 \\ & = 450 \end{split} \]
- So CONSUMER SURPLUS = 450
- What do we know ?
Question 1.7 Chapter 5
Question : Two linear demand curves go through the initial equilibrium, \(e_1\). One demand curve is less elastic than the other at \(e_1\). For which demand curve will a PRICE INCREASE cause the large consumer surplus loss?
- Start with a empty 2-d plot :
- Draw a demand curve that appears to be inelastic and call it A
- Draw a demand curve that appears to be elastic and call it B
- Let’s bring them together and add a price line:
- Find each person’s consumer surplus
- Add the new price increase line and label the areas
- First, what is the consumer surplus for A at original price \(P_0\):
- First, what is the consumer surplus for B at original price \(P_0\):
- PRICE INCREASES TO \(P_1\) :
- For Person A, the new CONSUMER SURPLUS and LOSS is:
- For Person B, the new CONSUMER SURPLUS and LOSS is:
- All in one figure :
So the LARGER CONSUMER LOSS IS : A - inelastic
QUESTION 7 : Previous Year
Question: Assume John’s Utility is \[U = min(2q_1, 4q_2)\]
The price of each good : \[p_1 = p_2 = \$2.00 \] and his monthly income \[m = \$ 4,000 \]
New Scenario : Firm relocates to another city where the scenario is : \[p_1 = 2.00\] \[p_2 = 8.00\] \[ m = 4,000\]
Find: COMPENSATING VARIATION AND EQUIVALENT VARIATION
COMPENSATING VARIATION:
- Find Optimal Bundle and Optimal Utility at Original City: \[U = min(2q_1, 4q_2) \] \[\texttt{subject to:} p_1 q_1 + p_2 q_2 = m \]
- At equilibrium \[ 2q_1 = 4q_2 \implies q_1 = 2q_2 \]
- Put this equality into the budget constraint: \[p_1 q_1 + p_2 q_2 = m \implies 2\underbrace{(2q_2)}_{q_1} +2q_2 = 4,000 \implies q_2^{*} = \dfrac{4000}{6} \approx 666.6667 \] And \[ q_1^{ * } = 2 * q_2 = 2*666.6667 = 1333.333\]
- Therefore, the Maximized Utility in the Original city : \[U = min(2q_1, 4q_2) = min( 2 * ( 1333.333), 4* (666.6667)) = min(2666.667,2666.667) = 2666.667\]
- Therefore, the compensating variation : \[CV: m+cv = p_1 \times q_1^{ * } + p_2^{new} \times {q_2}^{*} \] \[CV: 4000+cv = 2 \times 1333.333 + 8 \times 666.6667 \] \[CV: 4000+cv = 8000 \] \[CV = $ 4,000\]
This is the amount of money he NEEDS to achieve the SAME utilty with new prices
EQUIVALENT VARIATION
- Find Optimal Bundles and Utility in the New Location With New Prices: \[m = 2q_1 + 8q_2\] \[4,000 = 2(2q_2)+8q_2 = 12q_2\] \[q_2^* = \dfrac{4,000}{12} = 333.333\] \[q_1^* = 2q_2 = 2\times 333.333 = 666.6667 \]
- Let’s find the optimal Utility at these bundles: \[U = min(2 \times 666.6667 , 4 \times 333.333 ) = 1333.33 \]
- Thus, the EQUIVALENT VARIATION : \[EV: m - ev = p_1 \times q_1^* + p_2^{OLD} \times q_2^*\] \[EV: ev = 4,000 - [(2 * 666.6667 ) + (2 * 333.333)]\] \[EV = 2,000\]
- This EQUIVALENT VARIATION is the amount of money to TAKE AWAY from the person to hurt him as much as the price increases.