Define & Explain
Marginal Utility
The change in a consumer's overall utility (happiness) produced by consuming one more unit of a good.
Explain why the price elasticity of Electric Vehicles matters to policymakers?
Price elasticity measures how sensitive consumers are to price changes -- meaning how much their behavior will change (like demanding more EVs in response to a lower price through subsidies). The research found that the demand for EVs was quite elastic: -2.1. This is good because it means more consumers will adopt EVs in response to the lower price.
If P = 2000 - 0.1Q, what is the maximum possible revenue?
Max revenue occurs when Q = 10,000
R = 10,000,000
What is the marginal rate of substitution of a Utility function U(x1, x2) =
(x1^0.4)(x2^0.7)?
MRS = 4x2 / 7 x1. This is the amount of one good you would be willing to give up in order to get more of the other.
Where does the demand curve come from?
The demand curve shows the relationship between the price of a good and the amount consumers are willing to buy. It is downward-sloping, meaning lower prices lead to higher quantities demanded.
The aggregate market demand curve comes from millions of consumers maximizing their utility subject to their budget constraints.
Indifference Curve
An indifference curve shows all combinations of two goods that give a consumer the same level of happiness (assigned a numerical value ie 60 "utils").
Each curve represents a constant utility level, and curves moving outward and upward represent higher utility. Curves can never cross, as this would mean having two different utility levels simultaneously.
Explain how getting the correct number of taxis in a city is an optimization problem.
Too few cars cause long waits and poor accessibility; too many create congestion, pollution and lower wages. RideAustin solved this by graphing the "utilization rate": ride requests per hour against unoccupied driver miles. More ride requests per hour = longer wait times. They found where unoccupied driver miles dropped pretty low without adding too much to wait time.
Note a firm's optimization problem is straightforward (maximize profit), but policymakers have to consider a much wider range of problems, and defining the optimum can be tricky.
For a company with R(Q) = -10Q^2+10Q; C(Q) = 5Q, how much Q should they make to make the most money?
We maximize profit where MR = MC or where Maginal profit = 0.
MR = R′(Q) = −20Q + 10; MC = C′(Q) = 5
−20Q + 10 = 5 → Q = 0.25
Q* = 0.25
If a product has a price elasticity of -1.6 and the initial quantity demanded was 20 units, what will happen if the firm cuts the price by 10%? Why would we expect this to happen?
Quantity increases by 16%, to 23.2 units. This good is elastic, so consumers are price sensitive, meaning demand changes by more than the change in price.
Define & Explain -- Income Elasticity.
Income elasticity measures how responsive the quantity demanded of a good is to a change in consumer income.
It shows whether a good is normal (Ey > 0) or inferior (Ey < 0), and whether a good is a necessity (0< Ey < 1) or a luxury (Ey > 1).
Substitute
A substitute is a good that can replace another in consumption (eg yogurt for cereal/ almond for oat milk). It has a positive cross-price elasticity, so when the price of one good rises, the demand for its substitute increases.
Explain why consumers were upset with Instacart and how Instacart defended itself.
Uses were upset that Instacart was charging different prices to different consumers at the same time. They claimed it was evidence of “personalized pricing.” (First degree price discrimination)
Instacart argued that they were running an experiment to test consumer responsiveness to price changes -- i.e. they were testing consumers' *elasticity* to try to help them determine the shape of their demand curve.
If Qb has a demand of Qb = 100 - 0.5Pb; and a cost of Cb = 50Qb. And Qk has a demand of Qk = 600 - 5Pk and a cost of Ck=80Qk. How would suggest we maximize profit against the two goods (b) and (k)? Why? Solve it.
This is an unconstrained maximization problem, so maximize profit for both and add! Qb*= 37.5; Qk*=100; Total profit = $4,812.50
If U(x1,x2) = (x1^0.6)(x2^0.9) and and you have a budget of $1000, with x1 costing $50 and x2 costing $25, what is the optimal bundle? Why is this the optimal bundle?
x1 =8; x2 = 24; At this ratio, the consumer will maximize their potential utility given their budget constraint.
Explain how a producer could capture the entire area under the demand curve as their own producer surplus?
Using first-degree price discrimination. (Charging each individual consumer what they are willing to pay). Auctions are a classic example of this, but personalized pricing may move ordinary firms closer to being able to do this.
Inelastic
A good is inelastic when its elasticity is between 0 and 1, meaning that demand changes little in response to the change in price (or income or cross-price).
Define pass-through and explain its connection to EV subsidies.
Pass-through is defined as how much of subsidy actually goes to consumer to change prices (and therefore demand), versus how much went to producers simply being able to raise prices.
A subsidy with low-pass-through means little of the benefit makes it to the consumer (ie producers raise prices by 99% of the subsidy). A subsidy with high pass-through means the consumer catches most of the benefit.
The pass-through on EVs was found to be ~85%. So this was a successful pass-through, and because consumer demand was also price-elastic (-2.1), the subsidy should work.
(SEE 300 FIRST) If Qb has a demand of Qb = 100 - 0.5Pb; and a cost of Cb = 50Qb. And Qk has a demand of Qk = 600 - 5Pk and a cost of Ck=80Qk. BUT! Say the total of Qb and Qk can only be 200 -- What does this change? What if it can only be 100?
200 is not a constraint because the max profit for both individually summed together is below that.
For 100, now we have a constrained optimization problem. We should set Mπb = Mπk (or use the Lagrangian!)
Mπb = 150 − 4Qb; Mπk = 40 − 0.4Qk.
Set them equal and substitute Qk = 100 − Qb: 150 − 4Qb = 40 − 0.4(100 − Qb)
150 = 4.4Qb
Qb ≈ 34.09, so Qk ≈ 65.91
If a good has a cross-price elasticity of -0.8, explain what this means. What would happen to the original good if the cross-price good had a price increase of 30%?
A negative cross-price elasticity means the other good is a complement (as the price of the first good goes down, demand for the other good goes up -- or as the price of the other good goes up, demand for the first good goes down). In this case (-0.8) * (0.3) = -0.24. The original good's demand will decrease by 24%.
What is the optimal markup equation and how would you use it?
The optimal mark-up is the price increase over cost that maximizes a firm’s profit. At profit-maximizing price:
(P - MC)/ P = -1/Ep
If this equation does not hold and markup > -1/Ep, then you could earn more by selling more at a lower price. If markup < -1/Ep, you could earn more by selling fewer at a higher price.
Income effect
The income effect shows the change in consumption resulting from a change in income (or purchasing power).
When income rises, consumers usually buy more of normal goods and fewer of inferior goods.
Define stakeholder capitalism. What are the arguments for and against its inclusion in firms' optimization considerations?
Stakeholder capitalism refers to a system in which firms should seek to maximize more than shareholder value (pure profit), and instead take a broader view of society and include workers, communities and the environment.
Proponents argue that firms benefit from a functioning society (they could not operate without laws, an educated population, good infrastructure), and they should therefore have a stake in maintaining it. Also, taking care of the environment and workers benefits the firm in the long run.
Opponents argue that firms that are untrained in societal issues outside their area of expertise are not the best suited to solve societal's problems, and they have a primary duty to maximize shareholder value.
If I have a revenue equations R(q) = Aq^2 + Bq + C.... What must be true about A / B / C ? Why?
A must be negative, because the demand curve P(q) must have a downward slope. B must be positive, because it represents the maximum price anyone would be willing to pay. C is 0, because selling 0 units should produce 0 revenue.
A cereal company estimated the following function for the demand of its cereal: Qₓ = 3.0 − 2.0Pₓ + 1.5I + 0.8Pᵧ − 3.0P_M + 1.0A
Where Qx is the sales of the cereal in millions of boxes, Px is the price of the cereal in dollars, I is national income in trillions of dollars, Py is the price of the most similar competitor cereal, PM is the price of milk, and A is the advertising expenditures of the company, in hundreds of thousands of dollars.
This year, Px= $2, I = $4, Py= $2.50, PM= $1, and A = $2. If PM increases by 30% how will this change sales?
Initial quantity demanded (Qx) with these inputs is 6 million boxes. Qₓ = 3.0 − 2.0(2) + 1.5(4) + 0.8(2.50) − 3.0(1) + 1.0(2)
= 3 − 4 + 6 + 2 − 3 + 2 = 6 million boxes
Cross-price elasticity = EPm = (ΔQₓ/ΔPM) × (PM / Qₓ) = −3.0 × (1 / 6) = −0.5
Effect of a 30% milk price increase: %ΔQₓ = −0.5 × 30% = −15%
(1-0.15) * 6 = 5.1. Sales will drop by 0.9 million, from 6 million to 5.1 million boxes.
Define & Explain - Shadow Price
The shadow price is the marginal value of relaxing the constraint by one unit.
In optimization problems, it shows how much the objective (like profit or utility) would increase if one more unit of a constrained resource were available.
Example: You only have enough storage for 900 goods, when the profit-maximizing quantity is 1100. A shadow price of "20" would tell you that profit would increase by $20 if you suddenly had room for 901 goods.