Download Solutions 10 - Emilio Cuilty

Econ 101 Discussion Section - Handout #10, Nov. 7th
Exercise 1: Perfect Complements
Consider the economy in Mars. There are two goods in Mars, right shoe and left shoe.
The price of right shoe is 10 mars dollars and the price of left shoe is 5 mars dollars.
Martian, a mars alien, earns 30 mars dollars per day.
(a) Find Martian’s (daily) budget constraint (equation that characterize his choice set)
and then draw his budget line.
Budget constraint is given by Total expenditure = Total income. In this example, total
expenditure is 10*R + 5*L and total income is 30. Thus, budget equation is
30 = 10*R + 5*L
As a normal mars alien, Martian has one right leg and two left legs so he gains utility
from the number of combinations of one right shoe and two left shoes.
(b) Draw two indifferent curves each represents different levels of utility.
The slope of the grey line is 2.
(c) Find Martian’s optimal consumption bundle.
(3/2R, 3L). Solve R and L from 10R + 5L = 30 when L = 2 R.
A huge meteorite hits a right shoe plant in mars and the price of right shoes suddenly
increases to 20 mars dollars.
(d) Draw a new budget line and find Martian’s optimal consumption bundle under this
new budget line.
The new budget line has a slope of -4, L-intercept of 6, and R-intercept of 3/2.
(R, 2L). Solve R and L from 20R + 5L =30 when L = 2R.
(e) Illustrate income and positive effect in your graph.
The black line is the initial budget line BL1. The blue line is the new budget line BL2.
The yellow line is the imaginary budget line BL3. Point C is the optimal bundle when
Martian is compensated to stay at this original utility level and faces the new prices
(BL3//BL2). Point C coincides with point A.
Shift from A to C represents substitution effect while C to B represents income effect.
Quantity demanded for right shoes does not change from A to C. Quantity demanded for
right shoes decreases from C to B. Thus there is no substitution effect and the income
effect implies the real income and quantity demanded for right shoes both decrease
when the price of right shoes increase.
(f) Is cross-price elasticity of left shoe demand with respect to right shoe price positive?
Are these two goods complements or substitutes?
No, the cross-price elasticity is negative because right shoe’s price increase causes left
shoe’s demand decrease. They are complements.
Exercise 2: Cost function and supply curve
Charlie is an owner of the chocolate factory. His factory uses two inputs, a single
machine and labor. The rental cost of the machine is $100 per month and is not
refundable. The labor cost per month associated with each level of output is given in the
following table:
Quantity
Wage (Total
variable cost)
Total
cost
Average
variable cost
Average total
cost
Marginal
cost
1
$95
$195.00
$95.00
$195.00
$95.00
2
$180
$280.00
$90.00
$140.00
$85.00
3
$270
$370.00
$90.00
$123.33
$90.00
4
$370
$470.00
$92.50
$117.50
$100.00
5
$490
$590.00
$98.00
$118.00
$120.00
6
$620
$720.00
$103.33
$120.00
$130.00
(a) Calculate the total cost, the average variable cost, the average total cost and the
marginal cost for each quantity of output.
See the filled table above.
(b) What is the shutdown price?
MC = AVC = 90 at Q = 3. Thus, 90 is the shutdown price.
(c) At the beginning of November, Charlie rents a machine without knowing the market
price in the perfectly competitive chocolate market. After he pays rental cost of machine,
he realizes the market price of chocolate is $100. How many units does he produce?
What is his revenue?
P = MC holds at Q = 4. At this point the revenue is P*Q = 4*100 = 400, which is greater
than total variable cost at Q = 4. Thus, he produce Q = 4 and receives profits P*Q-TC =
P*Q - VC - FC = 400 - 370 - 100 = -70. Notice that the rental cost is not refundable so
his profit when Q = 0 is just -100, which is smaller than -70. So, producing 4 is better
than producing nothing even though the profit is negative.
(d) On December, he observes the market price of chocolate is $100 before he pays the
rental costs. How many units does he produce? What is his revenue?
In this case, he is able to decide not to rent the machine and produce nothing. This gives
him zero revenue and zero profit. From (c), we know his profit is at most -70 if he
produce positive amount. Thus, he optimally chose not to produce. Thus, Q = 0 and
revenue = 0.
Exercise 3: More on cost function and supply curve
A gummy bear plant has a total production cost function given by the following
equation:
TC = 𝑄 2 + 2𝑄 + 100
(a) What is the total cost when the plant produces 10 units of gummy bears? What is the
total variable cost?
At Q = 10, total cost is 102 + 2 × 10 + 100 = 220. Variable cost is 𝑄 2 + 2𝑄. Thus,
total variable cost is 102 + 2 × 10 = 120.
(b) What is the average total cost when the plant produces 10 units of gummy bears?
What is the average variable cost?
Average total cost is a total cost divided by quantity. Thus at Q=10, ATC = 220/10=22
Similarly, average variable cost is 120/10=12
(c) Derive average variable cost function and average total cost function. Draw these
two functions.
Average variable cost is variable cost divided by Q, Therefore,
𝐴𝑉𝐶 =
𝑄 2 + 2𝑄
= 𝑄+2
𝑄
Similarly, average total cost is
𝑄 2 + 2𝑄 + 100
100
𝐴𝑇𝐶 =
=𝑄+2+
𝑄
𝑄
The marginal cost function is given by the following equation:
MC = 2Q + 2
(d) Draw the marginal cost function and find the point where average total cost and
marginal cost cross.
Two lines cross at Q = 10, P = 22.
(e) Derive short run and long run supply curves.
In the short run, the supply curve can be described as

when the price is below the shutdown price, the firm should produce nothing.
Q = 0 for P < Shutdown price.

when the price is at the shutdown price, the firm is indifferent between
producing and not P = Shutdown price.

when the price is above the shutdown price, the firm would produce when P
= MC = 2Q + 2 for P > Shutdown price.
In the long run, the supply curve can be described as

when the price is below the breakeven price, the firm should produce
nothing. Q = 0 for P < Breakeven price.

when the price is at the breakeven price, the firm is indifferent between
producing and not P = Breakeven price.

when the price is above the breakeven price, the firm would produce when P
= MC = 2Q + 2 for P > breakeven price.
(f) Find the shutdown point and breakeven point in your graph. Explain what would
happen if the price is below the breakeven price but above the shutdown point.
Shutdown point is when MC = AVC (2Q + 2 = Q + 2) which gives Q=0, P=2 and
breakeven point is when MC = ATC (2Q + 2 = Q + 2 + 100/Q) which gives Q=10, P=22.
In the short run, the plant continues to operate when the market price is below
breakeven price of 22. However, in a long run, they stop producing goods.