"Remember, the way firms maximize profits is by minimizing costs, by producing goods as efficiently as possible at minimum costs."

Definition: The cost function C(q) tells us the total cost of producing quantity q.

C(q) = r·K̄ + w·L(q)

where r = rental rate of capital, w = wage, K̄ = fixed capital, L(q) = labor needed for q

Given:

• Production function: q = √(L × K)
• Wage: w = 5
• Rental rate: r = 10
• Fixed capital: K̄ = 1

Step 1: Solve production function for L

q = √(L × K̄) → q² = L × K̄ → L = q²/K̄

Step 2: Substitute into cost function

C(q) = r·K̄ + w·L = 10(1) + 5(q²/1) = 10 + 5q²

"Where do w and r come from? I'll teach you that in a few lectures. Remember, we peel the onion in this class. We're revealing the mystery slowly. Right now, just take those as given."

Marginal Cost (MC): The cost of producing one more unit

MC = dC/dq = dVC/dq

(In short run, dTC/dq = dVC/dq because fixed costs don't change)

In our example:

MC = d(10 + 5q²)/dq = 10q

Figure 6-1: Cost curves for C(q) = 10 + 5q². Shows MC, AC, AVC, and AFC.

"Average costs first fall, then rise. Why is that? Well, that's because at first, you have these big fixed costs you have to pay off. But over time, the fixed costs get smaller relative to the variable costs."

Key Rule: Marginal Cost crosses Average Cost at the minimum of Average Cost.

Intuition:

• If MC < AC: Producing one more unit brings down the average → AC is falling
• If MC > AC: Producing one more unit brings up the average → AC is rising
• At MC = AC: Average is neither rising nor falling → minimum!

"Wherever the marginal cost is above the average cost, the average cost is rising. Wherever the marginal cost is below the average cost, it's falling. Therefore, marginal cost intersects average cost at the minimum."

Starting from: C = r·K̄ + w·L

Differentiate with respect to q:

MC = dC/dq = w · (dL/dq)

But recall: MP_L = dq/dL, so dL/dq = 1/MP_L

MC = w / MP_L

"The marginal cost of producing the next unit of the good is higher the more you have to pay each worker and lower the more productive each worker is."

"Last lecture, I said in the long run, all inputs are variable. I'm going to already contradict that. We have some costs which in the long run are not variable. Those are called sunk costs."

"A sunk cost is something you can't ever get rid of or sell. The best example would be med school. If you want to be a doctor, you have to go to med school. Having gone to med school, you can't ever un-go to med school. You have paid that cost of time and money, and you're done. It's sunk."

"Sunk costs are irrelevant to decision making once they're paid."

"Or as the economists say, sunk costs are sunk. Doesn't matter. It's irrelevant."

Gruber's Story:

• Bought Journey concert tickets for $125 each ($250 total)
• Looked at setlist, realized he only likes 3 songs
• Decided to sell tickets on StubHub

Question: How should the $250 he paid affect the price he sets on StubHub?

The opportunity cost of going!

• If you literally don't want to go → willing to sell for $0
• If you'd pay $50 to go → set minimum price at $50
• If someone pays more than your value → sell
• If someone pays less than your value → keep tickets and go

"The amount I should be willing to get should be determined by the opportunity cost of going, not by what I paid for them."

Student: "I feel like I would think about trying to recover those sunk costs."

Gruber: "Yeah, you would because that's the way humans think. But it's wrong."

"Let me put it this way. Let's say I could go online and sell them for $500 each. Should I only sell them for $250 because that's what I paid for them? No, sell them for $500. So it's irrelevant that I paid $250."

Long-run costs ≤ Short-run costs (always!)

"Long-run costs will always be at the lower bound of short-run costs. The cost of production in the long run are always the same or lower than the cost of production in the short run."

Why? "The more variables you have to toy with, the more you can optimize."

Figure 6-2: Long-Run Average Cost (LRAC) as the envelope of Short-Run Average Cost curves.

Think of this as three different plant sizes (different K̄ choices):

• SRAC¹: Small plant (low K̄) - efficient at low output q₁
• SRAC²: Medium plant - efficient at medium output q₂
• SRAC³: Large plant (high K̄) - efficient at high output q₃

"If the firm wants to produce q₁, the most efficient K̄ is the one that yields SRAC¹. If the firm wants to produce q₃, the most efficient K̄ is the one that yields SRAC³."

The Tesla Battery Factory Story:

"Elon Musk did not produce most efficiently in the short run, so he built a bigger plant. Will he now produce most efficiently? It depends on whether demand is up at q₃ or somewhere else."

Isocost Line: All combinations of L and K that cost the same total amount.

C = w·L + r·K

Rearranging: K = C/r - (w/r)·L

Figure 6-3: Isocost lines for w=$5, r=$10. Each line shows combinations of L and K that cost the same.

Goal: For any given output q, find the combination of L and K that minimizes cost.

Solution: Find the tangency between the isoquant (from production function) and the isocost line.

Figure 6-4: Cost minimization at tangency. To produce √12.5 units at minimum cost ($50), use L=5 and K=2.5.

At the optimum:

Slope of Isoquant = Slope of Isocost

MRTS = -MP_L/MP_K = -w/r

"This is why producer theory is harder. Because I didn't tell you the right isocost. With consumer theory, I gave you a budget constraint. I'm not giving you an isocost here. I'm just saying there are multiple possible isocosts."

Long-Run Expansion Path: The set of cost-minimizing combinations of L and K for every possible output level q.

Figure 6-5a: Linear expansion path for q = √(L×K). Always use K = L/2.

"This is where I get to tell you that economics is just cooler than accounting. And let me explain why. Accounting only considers the cash-flow costs that you have when you start a firm. Economics more appropriately considers opportunity costs, as well."

Scenario: You (MIT graduate) start a web design firm after graduation.

• You work full time (no salary to yourself)
• Hire one programmer for $40,000/year
• Give programmer your extra computer (1 year left before it's useless)
• After 1 year, revenue = $60,000

"Opportunity costs are real costs. Opportunity costs are the value of the next best foregone alternative."

"Remember, that's why we call economics the dismal science. We point out nothing is free. Your time wasn't free. The time you dedicate to this firm meant you could have been doing something else. And that's not free."

"Next lecture, I will introduce the one extra step that lets you choose q."

Coming up: How firms decide how much to produce (choosing q) → Supply curve derivation