Profits (π) = Revenue - Costs = P × Q - C(Q)

Profits depend on three things:

Key Insight: "From the firm's perspective, maximizing profits is the same as minimizing costs. The way to maximize profits is to minimize costs. And that's how the firm produces as efficiently as possible, by minimizing costs."

Mathematical Intuition:

π = P × Q - C(Q)

If P and Q are determined by market forces (demand and competition), then:

max π ⟺ min C(Q) for any given Q

In the real world, there are complications, but in our model—and in most of the real world—we say firms' goal is to minimize costs.

"In this course, firms are simply black boxes for much of the course. They're simply boxes where inputs go in and outputs come out. And in the middle is something called a production function. That production function is the magic that makes—is the magic of the firm."

Production Function: A mathematical description of what is technically feasible for the firm to produce given its inputs.

q = f(L, K)

where q = output, L = labor, K = capital

"Just like the utility function is a mathematical representation of how goods turn into happiness, the production function, which, in some sense, is easier to understand, is a mathematical representation of how inputs turn to outputs."

Student Question: "Why isn't the production function just q = L + K?"

Gruber's Answer:

"No. q could be L plus K. That would be a particular form of f. But just like you can't tell me what form my utility function is, I can't tell you what form your production function is. This is some rich function of how you translate L and K to q."

"So just like utility isn't just cookies plus slices, it's some function of cookies and slices, production is some function of labor and capital."

Key Idea: Think of ALL inputs as being rented.

"In other words, think of whatever you use in production as rented. You rent a machine. You rent a building. In some sense, workers, the wage is the price you pay to rent an hour of a worker's time."

Why this framing?

"This makes it easy and allows for a parallel analysis between workers and machines. They're both things you rent."

This allows us to write the cost function simply as:

C = wL + rK

"Where does the wage come from? We'll discuss that in a few lectures when we cover labor markets."

For now, we take w and r as given (exogenous) and focus on how firms choose L and K.

Important Note:

"I'm not going to tell you the short-run is six months and three days and four hours. And the long-run is past that. I can't tell you that. I can't tell you what they represent, except theoretically."

Rule of Thumb: "Think of the short-run as days and the long-run as decades. And where in between days and decades things lay depends."

"Of course, once again, in reality, this isn't so distinct. This isn't so clean."

• You can't just fire workers right away and replace them the next day, especially in tight labor markets
• You can build machines pretty quickly with modern manufacturing

"So the short-run/long-run distinction is obviously not perfect, but it allows us to essentially distinguish the concept of things you can change quickly versus things you can't."

q = f(L, K̄)

Capital is fixed by definition in the short-run

Definition:

MP_L = ∂q/∂L = ∂f(L, K̄)/∂L

"How valuable is the next worker in terms of delivering output?"

Remember: This course always works on marginal decisions!

• Not "should I hire workers?" but "should I hire one more worker?"
• Not "is labor valuable?" but "how valuable is the next unit of labor?"

Key Assumption:

∂²q/∂L² < 0

The second derivative is negative

Student Question: "Does diminishing marginal product mean MP_L is negative?"

Gruber's Answer: "No! This is diminishing, not negative!"

"This does not mean diminishing total product. Once again, like with consumers, more is generally better, more workers are better. But each worker, given fixed capital, adds less to output."

Total Product Curve (q vs L):

• Starts at origin (0 workers = 0 output)
• Increases as L increases (more workers = more output)
• Curve is concave (bends downward) due to diminishing MP_L

Marginal Product Curve (MP_L vs L):

• Positive (above x-axis)
• Downward sloping (diminishing)
• MP_L = slope of Total Product curve at each point

Setup: "You want to dig a hole and you have one piece of capital, a shovel. And that can't be changed in the short-run. All you can change is how many people you have working on digging the hole."

Key Insight: "100 workers is certainly better than 90 workers, because you can rotate more quickly. But clearly, the 100th worker is not as valuable as the second worker in terms of digging that hole."

"The idea of diminishing marginal product is: given that you only have one shovel, given your fixed capital, each additional worker does less and less good."

"It's not quite as simple an intuition as diminishing marginal utility, but I think it's sort of natural."

Exception: "And it's not always true. Like I said, you can imagine the second worker might be as good as the first worker. But in general, each additional worker will add less and less value if they're working with the same number of machines."

"The long-run is more interesting because now, you can vary both capital and labor. So now, the problem is going to look just like the consumer choice problem."

Parallel Structure:

• Consumer: Choose between cookies and pizza
• Producer: Choose between labor and capital

q = √(L × K) = L^0.5 × K^0.5

A Cobb-Douglas production function—familiar from consumer theory!

Example Calculations:

Notice: Same output can be achieved with different combinations!

Definition: An isoquant is the set of all combinations of K and L that deliver the same amount of output q.

"This illustrates a set of combinations of K and L which deliver the same amount of q. Look familiar? It should look familiar. These look like indifference curves. But in this context, we call them isoquants, because we like making up cool words."

Figure 5-1: Isoquants for q = √(K·L). Each curve represents a different output level.

Picture: Graph with K on vertical axis, L on horizontal axis

• Each curve represents a different output level (q = 1, q = 2, q = 3, ...)
• Curves further from origin = higher output
• Along each curve, output is constant

For q = √(L×K), the isoquant for q = 4:

4 = √(L×K) → 16 = L×K → K = 16/L

This is a rectangular hyperbola.

Student Answer: "Depending on how the different inputs go with each other?"

Gruber: "Exactly. It's going to depend on the substitutability of the different inputs. The more substitutable, the straighter is the isoquant; the less substitutable, the more bent is the isoquant."

Cereal Box Example (Perfect Complements):

"For every amount of cereal you got, you got to have a box. If you only have one box, it doesn't matter how much cereal you have, it's just going to sit there."

"Just like the indifference curves for left shoe, right shoe—this is cereal, cereal boxes, things which are non-substitutable."

Reality: "Nothing's ever at the extremes. We'll generally be in between these two cases."

Terminology Warning: "Now with consumers, you had marginal rate of substitution and marginal rate of transformation, MRS and MRT. Now I'm going to have MRTS, which is the marginal rate of technical substitution—just to really mess you up."

MRTS: The rate at which you can trade off labor and capital while producing the same quantity of output.

MRTS = dK/dL|_{q constant} = Slope of isoquant

Question: What happens with a marginal increase in L and marginal decrease in K, staying on the same isoquant?

Step 1: Along an isoquant, output is constant (dq = 0)

Step 2: Total differential of production function:

dq = (∂q/∂L)dL + (∂q/∂K)dK = MP_L·dL + MP_K·dK

Step 3: Set dq = 0 (staying on isoquant):

0 = MP_L·dL + MP_K·dK

Step 4: Solve for dK/dL:

MP_K·dK = -MP_L·dL

dK/dL = -MP_L/MP_K = MRTS

Compare to Consumer Theory:

For q = √(L × K) = L^0.5K^0.5:

Step 1: Find MP_L

MP_L = ∂q/∂L = 0.5 × L^-0.5 × K^0.5 = 0.5 × K/√(LK) = 0.5 × √(K/L)

Step 2: Find MP_K

MP_K = ∂q/∂K = 0.5 × L^0.5 × K^-0.5 = 0.5 × L/√(LK) = 0.5 × √(L/K)

Step 3: Calculate MRTS

MRTS = -MP_L/MP_K = -[0.5√(K/L)] / [0.5√(L/K)]

MRTS = -K/L

"For this production function, that's the marginal rate of technical substitution."

Figure 5-3: Isoquants and MRTS. Points A(1,4), B(2,2), C(4,1) show how MRTS changes along the isoquant.

Intuition:

"When you've got four shovels and one worker, you'd happily give up shovels to get more workers. So you want to move to the right."

"Likewise, at point C, you've got four workers and one shovel. That's doing you no good. You'd happily give up a worker to get a shovel, so you want to move to the left."

Why? Because marginal products are diminishing! When you have lots of something, its marginal product is low, so you're willing to trade it away.

As you move along an isoquant from upper-left to lower-right:

• L increases, K decreases
• MP_L decreases (diminishing returns to L)
• MP_K increases (less K means higher marginal product)
• |MRTS| = MP_L/MP_K decreases

This is why isoquants are convex to the origin!

Note: "I'm going to talk about two things which are not parallel to consumer theory."

"Producer theory is one step harder than consumer theory. It's all the fun stuff for consumer theory plus one additional step."

Returns to Scale: What happens when we increase all inputs proportionally?

"When you say in your mind, 'what happens when a firm gets bigger?', you're thinking, what happens when everything increases? When all inputs increase proportionally is our notion of how a firm gets bigger."

Consider multiplying all inputs by a factor t > 1:

f(tL, tK) compared to t × f(L, K)

Let's check: f(2L, 2K) = √(2L × 2K) = √(4LK) = 2√(LK) = 2f(L, K)

Result: This production function exhibits constant returns to scale!

"Think about the idea that basically, if you're still trying to dig a hole, there's one guy with one shovel, you go two guys with two shovels, they're in each other's way, they can't quite work as efficiently as one guy with one shovel because you're still trying to dig one hole."

Causes of DRS:

• Coordination problems
• Communication difficulties
• Workers getting in each other's way
• Management challenges

"That might arise through something like the fact that when the firm gets bigger, it can specialize more. So maybe as a firm gets bigger and bigger, it can get better and better at what it does, and production can become more and more efficient."

Causes of IRS:

• Specialization and division of labor
• Bulk purchasing discounts
• Fixed costs spread over more output
• Network effects

"This is an important concept because basically, this is how we think about economies growing. This is the core of any model of economic growth."

Key Insight:

"Typically, economists do not believe that there is everywhere and always increasing returns to scale. We typically believe returns to scale may increase initially as you specialize as a firm, but eventually you have to decrease."

"Now it turns out, for search engines, maybe that's not such a bad description. We'll see in the trial that's going on now where Google may get broken up."

"So it may be that increasing returns to scale may last longer than we thought as economists."

The pattern economists expect:

• Initially: Increasing returns (specialization benefits)
• Eventually: Decreasing returns (coordination problems dominate)

"But typically, we think of firms as maybe initially having increasing returns to scale, and then eventually as they mature, having decreasing returns to scale."

Important Distinction:

"One of the most exciting topics in economics is thinking about productivity. One of the most famous—and the idea of productivity comes actually from a very famous application from 1798 of the idea of decreasing returns to scale or diminishing marginal product."

Thomas Malthus's Argument:

1. "Think about the production of agriculture. There's essentially two inputs: people and land."
2. "The difference between capital and land is land is always fixed. Even in the long-run. There's only so much land. We got the Earth. Unless we go colonize another planet."
3. "There's a fixed amount of land, so that means that the marginal product of labor must be everywhere diminishing."
4. "That means that eventually, we're going to starve."

"Malthus actually said that there would be cycles—the world would be marked by cycles of starvation."

"Really not a very happy picture of the future."

(Gruber describes Malthus as "a very, I think, probably pretty depressed philosopher")

Since Malthus wrote that book in 1798:

"Not that there's not starvation. And let's be clear, there's horrible starvation around the world."

Amartya Sen's Nobel Prize-winning observation:

"So we think about famines and starvation all over the world, remember, that's a political failure, not technological failure. There's never been a famine in a democracy. Famines only happen when corrupt governments don't spread the food they need to the people who need to get it."

Technology! Productivity!

q = A × f(L, K)

where A = Productivity factor (Total Factor Productivity)

Standard of Living = How much stuff we get to consume per person

Standard of Living ∝ q/Population = A × f(L, K)/Population

"What's great about productivity is these two are expensive. Working harder is costly. Capital, you got to build a machine. Productivity, boom, you got more. A brilliant idea, and suddenly, boom, you got more stuff for everyone."

"So productivity is the magic that makes economic growth happen almost costlessly."

"And as a result, productivity is the central determinant of our standard of living."

"The IT revolution started in the late '80s, but productivity didn't increase. People were saying, well, where's the IT revolution?"

Famous Quote (Robert Solow):

"You guys have no idea how different life is now than it was 25 years ago in terms of the internet and everything you have, but somehow, we're not that much more productive as an economy."

"Why did the IT revolution cause this brief burst and then eventually kind of wear off?"

"Is it because basically all our innovation is watching TikTok and not in making stuff? I don't know."

The puzzle: Life is radically different from 25 years ago, but we're not that much more economically productive. Why not?

Possible explanations:

• Measurement problems (we're not capturing value correctly)
• Innovation is in consumption, not production
• Benefits going to leisure, not measured output
• Takes time for technology to diffuse

"I said it was free, it's not free, it's not magical. Productivity is through real things."

"So basically, there are things we can do as a society to increase our productivity—we don't just sit back and let it happen. Having more educated population, more investments in R&D. Those can really make a difference in terms of how fast we grow."

"Imagine you have a cool new idea that can make things more efficient. Take Henry Ford's idea. He could do two things with that:"

"If A goes up, you can reduce L, and with the same amount of output. There's no reason an increase in A has to go into more q. An increase in A could go into less L, which people like. They like time off."

"One thing we learn in this class, the hardest thing about teaching a class at MIT, the single hardest thing is in the real world, people actually don't like work. I find this hard to believe, guys, but in the real world, people would rather not be at work than at work."

"What's the better thing to have? Good question. When I was young, I probably thought stuff. Now that I'm older, I might think time off."

"The pandemic may have shook this up in the US. The pandemic was the first chance that people had to actually be at home and say, wait a second, maybe I don't want to work that hard."

Post-pandemic changes:

• Huge reduction in labor supply
• Many people retired early
• People aren't working as hard
• People want better jobs
• People want to work from home

"So the pandemic really shook things up in terms of people maybe starting to value their leisure more than maybe they did before."

"Which group in society benefits from a more productive economy?"

The Numbers

The Bottom Line:

"A doesn't magically go to everyone. It's not like a more productive economy, everyone benefits. That's determined by a set of decisions made by individual actors in the economy and by the government. And that's what we'll talk about later in the semester, is how does the government think about those decisions?"

"And that comes to the topic of economic fairness, which we won't spend nearly enough time on, but we will come back to towards the end of the semester."