Now in the last session this week we're going to look at cost curves geometrically. And what we'll show is, there's a direct relationship between productivity and how costs behave. A direct relationship between the total product curve and the total variable cost curve. So, the law of diminishing returns that we talked about earlier in the week has a direct relationship to what we'll end up seeing in the marginal cost curve, in the average variable cost curve, in the average total cost curve, and we'll, we'll see why. Let's first look a little bit more closely behind cost relationships. Marginal cost or MC is the rate at which total variable cost changes per unit of output, per unit of Q. So we can either take the change in total cost or the change in total variable cost. They're the same as we move from a particular unit of output to the other. The change in total variable cost, if we hold one input constant in the short run, and in this case let's say we're holding capital constant, the change in total variable cost will be w, or the wage we pay per worker, multiplied by how much we scale up labor, or L, or the change in L. So, we can take the change in TVC in the numerator and it's the same as the way it's multiplied by the extra label. And, then if we divide both numerator and denominator by changing L, what we end up with is marginal cost is the same as the wage divided by the marginal product of labor. because when we divide the denominator, changing q by changing L, that's the same as the marginal product of labor. How much each additional unit of labor usage increases total output by. Average variable cost, the last equation on this page is total variable cost divided by quantity. And total variable cost is when we're holding some input constant, like capital, it's just the wage we pay times the number of workers we use. So in the numerator it's w times L over q, or output. Let's divide both numerator and denominator of this equation by L. So what we can see is average variable cost is the same thing as the variable inputs cost, w, divided by the average product of labor. And we get average product of labor, because when we divide the denominator q by L, q over L is the same as the average product per unit of labor. Now what we can see is, if the law of diminishing returns applies, so if the marginal product of labor goes down after a certain point, the marginal cost will go up. We're going to be dividing that same numerator, that same w, by a smaller and smaller marginal product of labor. Marginal cost will go up and up as we raise labor. Same thing if the law of diminishing returns applies, it's going to start dragging down at a certain point the average product of labor. So the average variable cost at a certain point will also start going up. That same wage divided by a lower and lower average product of labor. Let's see this graphically. First on figure 8.1. We've seen the total product curve before, but it also gives us the relationship between output and total variable cost. This is a case where, as we move from point A where we're using eight units of labor, we decided to go to 18 units of labor. And, let's assume labor cost 10 bucks now. So, at point A the total variable cost is 80$. We've, we've a very simplified case, just one variable input. If we use 18 units of labor. We've gone up to a 180$. And notice what happens when we've doubled the amount of total output along the way from four to eight. Total cost has more than doubled. So behind this total product curve was the law of diminishing returns. As we'll see soon, it also translates into what we see out of per unit cost curves. It also translates into what we see out of total variable cost curve. Total variable cost starts increasing at a quicker and quicker rate as we increase input usage at the same rate, if the law of diminishing returns applies. Our final figure for this session, this week's session, is figure 8.2. Two panels. In the top one, it shows the relationship between the total cost curves and output. There are three total cost curves. Total fixed cost is flat. And these number correspond to what we saw in figure, in table 8.1. So the same exact figures. It's a flat curve. $60 at any conceivable output. Because we're stuck with $60, no matter what output you choose. Total variable cost starts at zero. It first goes up at a slower and slower rate. Again, because law of diminishing returns hasn't kicked in yet, and then goes up at a quicker and quicker rate. Total cost is the sum of total variable cost and total fixed cost, so it starts at zero units of output at the same height as the total fixed cost, and then it's just parallel shifted up Horizontally by the amount of total fix costs. So the distance between C and D, between the total variable cost and the total cost curve at that particular output level, is the same height as the total fixed cost for that output level, $60. There are four per unit cost curves in the panel of, panel b of figure 8.2. Let's look at all of them. Average fixed cost is the easiest one to think about. We're dividing total fixed cost by a higher and higher quantity. So this is a curve that's going to keep going asymtotically towards zero as we increase output. We're going to keep dividing by bigger and bigger number. Average fixed cost also has the property that if you take any particular point. Let's say we looked at a height of $15. And using, and four units of output. If we multiply height times width we get $60. Which is the amount of total fixed cost. If you pick any point on that average fixed cost curve and create a rectangle with it, height times width, it will have the exact same area, $60. Marginal cost first falls and then rises. Why? Because of the law of diminishing returns. At an output level of four, the marginal product of labor is 2 3rds. As we keep increasing output, which requires more labor usage, marginal product of labor falls to a quarter. And marginal cost, as a result, ends up going from $15 per unit to $40 a unit. Average variable cost, similarly, because it's wage divided by the average product of labor, if the law of diminishing returns holds. At a certain point, this curve starts rising. It may fall for a while, where the law of diminishing returns still hasn't kicked in, or hasn't kicked in enough. But then sooner or later it'll start getting pulled up as the, as the average product to labor ends up falling. And average total cost is the sum. Vertically of average fixed cost and average variable cost. So it'll have that same u shape as average variable cost. But asympotically it'll get closer and closer to average variable costs, as we increase output further and further. Last thing to note about these curves. There's that same relationship between marginal and average. In particular between marginal cost and average variable cost; and marginal and average total cost. When marginal cost is below the average variable cost curve. So when the incremental unit of output adds less to cost than the prevailing average, average variable cost is still falling. When marginal cost is above average variable cost, when the incremental unit costs you more to produce than the existing average, it'll start raising up the average variable cost. And when the marginal unit costs the exact same amount as the prevailing average variable cost of production, average variable cost is flat. It's at the bottom of its u. The same applies, and test out the reasoning, with the average total cost curve. In closing for this week, let me give you one example of productivity and cost curves at work. And that involves Walmart. A McKinsey study looking at between 1995 and 2005 at what was the single most important factor to improving productivity in the United States, points to Walmart. Walmart now accounts for over 10% of overall retail sales, if you exclude automobiles. Over, well over 20% of grocery store sales, well over 23% of toy sales. Walmart has been extremely adept at figuring out, how do we pivot total product curves to the left? So from the origin of any given output, at any given input usage we get more total output. By keeping track of inventory, by keeping track of where trucks are so that they don't return from dropping off shipments empty. When you can increase total productivity, when you can increase marginal product per unit of input, you also end up lowering cost curves. And that result in lowering cost curves has allowed them to establish their preeminent position. At the same time drive down the cost of providing retail sales and providing important benefit in terms of the prices consumers end up paying for the goods they, they buy. And the estimates are that nowadays 5% lower retail prices in urban areas, 8% lower in, in, rural areas, due to the Walmart effect.