Production
A firm's production is given by Q = K^0.3 L^0.7. Where the price of capital is $24/unit and the current wage is $8/unit. What is the optimal capital-labor ratio? Explain your process.
K / L = 1/7
Equate the MRTS of the Isoquant with the slope of the Isocost.
A student bakery on campus rents an oven for $300 per month. The variable cost of producing cupcakes increases with the square of output and is given by VC=3Q2VC=3Q2, where QQ is the number of dozens of cupcakes. At what output are per-unit costs minimized? If cupcakes sell for $18/dozen, how many should they produce and what is the profit?
Q = 10, at minimum costs
Q* = 3, at maximum profit, Profit = -273, Short-run, produce; Long-run, shut-down
Bakeries in a town with a thriving dining scene are in a zero-economic-profit long-run equilibrium.
Each restaurant pays $44,100 in annual rent, and its variable costs for wait staff and ingredients depend on the number of cookies served per week according to the equation
C(q)=44,100+q^2
The bakeries in the market face an aggregate demand function
Q=18,060−P
where Q is the total number of cookies served per week across the town and P is the average amount a party spends on cookies. How many cookies are served throughout the town and how many bakeries are there?
q* = 210
p* = 420
Q* = 17,640
N = 84
We read an article that noted how dock-workers are being replaced by automation, explain how this relates to optimal firm production?
(1) Previous technological improvements -- Containerization deduced the cost from $6/ton to 16 cents/ton. Which reduced the need for longshore workers
(2) Current tensions -- Dockworker unions demands a high marginal cost of labor, and so some port operators are considering deploying automation to reduce the reliance on this expensive input.
Describe prefect competition
A market state where
- Many firms exist, and all are price takers
- They all produce the same product
- No firm has any market power
SO
P = MC to maximize profit and P = min (AC) for sustainability long-term. This is the most efficient and productive market state.
A firm’s production function is Q=500L^0.8 K^0.2. The wage rate is $50,000 per worker per year, and the rental rate of capital is $150,000 per machine per year.
If total cost cannot exceed $3,000,000, find the combination of labor and capital that maximizes output.
L = 48, K = 4
Qmax = 14,601
Using ... K = L / 12 ... & ... 50,000L+150,000K - 3,000,000
A cookie company has the total cost function:
C=12,000+3,000Q−20Q^2+0.05Q^3
where C is total cost and Q is the number of cookie batches. If the market price is $999 per batch, should the firm continue production? If so, what quantity would they produce?
Yes, continue production in the short-run
min (AVC) = 1000 at Q = 200
With P = 999, the firm should shut down.
[DO 100 FIRST]
Bakeries in a town with a thriving dining scene are in a zero-economic-profit long-run equilibrium.
Each restaurant pays $44,100 in annual rent, and its variable costs for wait staff and ingredients depend on the number of cookies served per week according to the equation
C(q)=44,100+q^2
The bakeries in the market face an aggregate demand function
Q=18,060−P
where Q is the total number of cookies served per week across the town and P is the average amount a party spends on cookies. If there is a temporary cookie-scare that shifts demand to be
Q = 9030 - P
What is the new market equilibrium price and quantity? Should the restaurants shut down?q new = 105
P new = 210
Q new = 8820
Short term continue
Describe how California increasing the minimum wage impacted the fast food firms in the article we read for class.
Draw and explain the MRTS (Marginal Rate of Technical Substitution) in relation to optimal inputs.
MRTS shows how much of one input (e.g., capital K) a firm can give up if it uses one more unit of another input (e.g., labor L), keeping output constant.
Optimal inputs occur when the MRTS equals the ration of input prices (wage / price of capital)
[Write Equation]
A cookie company has the following marginal product of labor schedule:
10 hrs labor --> 15 MPL
20 hrs --> 12
30 hrs --> 9
40 hrs --> 6
50 hrs --> 3
If cookies are sold for $30 each (yikes!) what is the labor-demand schedule (e.g., what will the wage be at each amount of labor)? If minimum was is $180, how many hours will be hired? Why?
10 hrs labor --> 15 MPL --> $450 wage
20 hrs --> 12 --> $360
30 hrs --> 9 --> $270
40 hrs --> 6 --> $180
50 hrs --> 3 --> $90
At $180, they hire 40 hours of labor.
Three friends are considering running a weekend cookie truck. Their costs are as follows:
Truck Rental = $2,000
Equipment Rental = $1,500
Ingredients and Supplies = $5 per cookie
Miscellaneous Costs = $1 per cookie
Each cookie sells for $10. If the friends could instead work as delivery drivers and earn $2,400 each for the summer. They estimate that throughout the summer they might sell 1000 cookies. From the view of an accountant and an economist, should they operate the truck? What would you do?
Accounting profit = 500 --> Operate
Economic profit = -6700 --> NOT OPERATE
WE ARE ECONOMISTS
A firm in a perfectly competitive industry has
C(q)=60+8q+2q^2
The current market price is P=28
Given market demand Q=240−4P find the total market quantity demanded and the number of firms in the short run. Determine the long-run equilibrium price, output per firm, and number of firms. Explain qualitatively what happens to the market as it transitions to the long-run equilibrium.
Q short = 128
N short = 25.6
q long = 5.477
p long = $29.9
Q long = 120
N long = 21.97
Describe how the article on Walmart and Amazon relate to their anticipated returns to scale?
RETURNS TO SCALE: Retail, like Walmart and Amazon, exhibit positive returns to scale. So the bigger a firm gets, the more it can lower is long-run average cost.
FIXED COSTS: Both Walmart and Amazon have already invested heavily in their fixed costs, while the average per unit cost is low, so adding units/options doesn't drive up the cost.
Define and draw minimum efficient scale
The Minimum Efficient Scale (MES) is the smallest level of output at which a firm can produce at the lowest possible long-run average cost (LAC) — that is, the bottom of the long-run average cost curve.
It represents the scale of production where economies of scale are fully exploited and average cost stops decreasing.
d(LAC)/d(Q) = 0
A cookie company operates to factories (home-bakers) in different states, X & Y with production functions:
Qx = 40Lx - 0.5Lx^2
Qy = 30Ly - 0.25Ly^2
The wage is state X is $20, while the wage in state Y is only $10. The Cookie CFO (CCFO or "grand cookie") has allocated $70 for labor. What is the optimal allocation across the two plants?
Lx = 16.666, Ly = 36.666
Note:
2Lx + Ly = 70
Company Z produces camping tents, with a winter-season demand of P=500−0.5QP=500−0.5Q.
Its cost function is C=80,000+100Q. What is the optimal output, price and associated profit.
Q* = 400
P* = 300
Profit = 0
Market supply and demand are:
QS=2P−6 & QD=30−P
The government imposes a unit tax of t=3 on producers. What was the pre-tax equilibrium and quantity? What is the post-tax equilibrium and quantity? What is the consumer surplus, producer surplus, gov't revenue and deadweight loss? Who is loosing the most?
Po = 12
Qo = 18
Pre-Tax Consumer Surplus = 162
Pre-Tax Producer Surplus = 81
Post-Tax Consumer Surplus = 128
Post-Tax Producer Surplus = 64
Gov't Revenue = 48
Deadweight loss = 3
Explain how the article on food banks demonstrated perfect competition?
The food banks employed a market mechanisms with fake-money to allow different food banks to 'bid' on goods. This allowed for marginal valuation of goods -- some banks really wanted one good rather than another -- and enabled the whole system to more efficiently distribute food to where it would be most valued. This is the premise of the 'free market' allocating goods to where they are most valued (maximizing total surplus)
Explain short run and long run shut down price, using graphs
Short run: the minimum price to avoid shut down is P=min (AVC) because this is where your'e price just allows you to cover variable costs
Long run: the minimum price to avoid shut down is P = min(AC) to cover TOTAL COSTS
[THIS IS A MEAN PROBLEM] A company called EcoCharge produces portable batteries using two plants: North and South.
The production functions are:
QN=60KN^0.3 LN^0.7,QS=40KS^0.5 LS^0.5
The price of capital is the same for both plants, r=$20,000 per unit per year),
but the wage differs:
wN=$10,000 per worker per year
wS=$5,000 per worker per year.
The company has a total annual production budget of $1,000,000, which must cover all labor and capital at both plants. Note, North can only produce up to 2,500 units and South can only produce up to 5,000 units. How should the firm allocate this budget across the two plants, and across labor and capital?
Kn/Ln = 3/14; Ks/Ls = 1/4
Cost N = $377.97; Cost S = $500
because N < S, max that capacity first.
N = 2,500 units, Cn = $944,935
South has $55,075 left over, so
S = 110 units
Q total = ~2610 units
Ln = 66.1; Kn = 14.2
Ls = 5.5; Ks = 1.375
A firm manufactures electric fans on a single day shift. Each labor-hour produces 2 fans.
Fixed cost is $400,000 per month. Day-shift labor costs $25/hour, and other variable costs are $12 per fan. Market demand is
P=200−0.02Q
where PP is price (dollars) and QQ is monthly quantity.
At most 1,800 labor-hours can be scheduled per month on the day shift. The firm is deciding if they should add a $30/hr night shift to increase their profits. Should they?
MCday = 24.5
Q max = 4,387.5 (constraint binds)
Profit day only = -27,400 (short run operation only)
MCnight = 27
Q* = 4325, P* = 113.5
Profit (day & night) = -16,887.5
YES! NIGHT SHIFT
The market has linear demand and a cost-based supply:
Demand: P=80−2Q
Marginal Cost (industry): MC=20+Q.
What is the equilibrium price and quantity under perfect competition? What about it is was under a monopoly instead? How do consumer and producer surplus change between these scenarios?
Perfect Comp: Q=20, P=40
Monopoly: Q=12, P=56
Consumer Surplus: 400 --> 144
Producer Surplus: 200 --> 360
DWL: 96
Explain why Oasis fans should be in favor of high-ticket prices?
Because the high-prices reflect that the Oasis tickets are being allocated to the users that value the band the most (e.g., genuine devotees). It also sends a signal to the bank to increase the number of shows (the supply) to serve the fans that are still willing to pay -- In the article, Oasis added more shows.
Describe how the Cobbs-Douglas Function tells you something about returns to scale
A production function where combining capital and labor interact through the form
Q = A K^(alpha) L^(beta)
Where alpha and beta are the output elasticities and A is the total factor productivity.
Importantly when (alpha) + (beta) > 1 you get increasing returns to scale!