I saw this post ‘The Equation that Governs Your Sales Team’s Effectiveness‘ by Thomasz Tunguz. It is not a very long read, but every sentence is dense with details, so it provides a very good list of factors that affect ‘Sales Velocity’, or the rate at a sales team can deliver money.

The basic argument is that the Sales Velocity depends on the amount of possible sales in progress (or, ‘Work in Progress’) and the Sales Cycle (the time it takes to close a sale). If there are 20 possible sales contracts being negotiated and it takes 30 days to close a contract, then on average, 20/30 or about 0.66 contracts will close every day. If 20% of those closings are sales (the ‘Win Rate’) and each sale brings in $15,000 (‘Average Price’), then the Sales Velocity is: Work in Progress x Win Rate / Sales Cycle x Average Price.

For these figures, we would expect the ‘Sales Velocity’ to be 20 x 0.2 / 30 x $15,000 = $2,000 per day


What happens if the Sales Cycle (the time it takes for the buyer to make a decision on whether or not to sign the sales contract) drops from 30 days down to 20?


In that case, the ‘Sales Velocity’ to be 20 x 0.2 / 20 x $15,000 = or $3,000 per day. By making the Sales Cycle shorter, the Sales Velocity goes up.

This all seems very straightforward, but as Mr. Tunguz points out, “these numbers don’t move in isolation; they are not independent”. To see how that might work, I made a more-detailed simulation of the Sales process dynamics using a piece of software called STELLA.

Conceptually, I think of the flow of money to be like the flow of water through a pipe. I have shown Mr. Tunguz’s Sales Velocity as a pipe in the diagrambelow. The ‘Sales Velocity’ depends on the ‘Average Price’ (in dollars per sales contract) and the rate of ‘signing Contracts’ (which is the remainder of his equation: ‘Work in Progress x Win Rate / Sales Cycle’)

I think of the ‘Work in Progress’ (the number of possible sales contracts that are being negotiated) as a bucket, represented by the rectangle. There are only two ways to get out of the that bucket – have the customer say No and flow out through the ‘losing Contracts’ pipe or have the customer say Yes and flow out through the ‘signing Contracts’ pipe’. The rate of ‘closing Work in Progress’ is the amount of Work in Progress divided by the Sales Cycle and the rate of signing Contracts and therefore, the rate of losing Contracts, both depend on the ‘Win Rate’.

deeper-Sales-VelocityMr. Tunguz mentions that Sales Development Representatives deliver a certain number of qualified leads per time, adding possible sales to the ‘Work in Progress’ bucket.

Now we have a more-detailed, more kneebones-connected-to-the-thighbone operational model of what is going on an how all the big pieces are connected. I ran a simulation using a value of ‘Sales Cycle’ of 30 for the first 20 days. We can see that the Sales Velocity is $2,000/day. At Day 20, the value of ‘Sales Cycle’ drops to 20 days. That is, the account executives get faster at closing (both winning and losing contracts).

We can see that the Sales Velocity jumps up to $3,000 per day… but just for one day! It slowly drops back to $2,000/day by Day 90.

Sales-Velocity-graphEven though the Sales Cycle is permanently shorter, the effect of the shortening quickly wears off. What is going on here? When we shorten the Sales Cycle, the flow rates out of ‘Work in Progress’ get faster (there are more contracts won and lost each day). But, the rate at which the Sales Development Representatives deliver leads (‘developing Work in Progress’) does no change. Higher out-flows from the bucket but the same rate of adding new work to the bucket means that the total amount of ‘Work in Progress’ must drop over time.

Shortening the Sales Cycle without adding more SDRs (or increasing their per-person productivity), means that the effect on the Sales Velocity will be short-lived.

Mr. Tunguz lays out all sorts of other scenarios (“increasing the average price”, “seasonality”, “company maturity”) that are too complex to deal with right here, but the simulation I have made could be the basis for exploring the ramifications of those in greater detail.




I saw this fascinating article from Science Magazine, ‘Slaughter at the bridge: Uncovering a colossal Bronze Age battle’, about an archaeological dig in northern Germany that seems to have been the site of a pre-historic battle.

One of the images was of some of the artifacts that the men at the battle site carried:

I thought that the bronze ring in the center looked very nice:

Warrior-ring So I have made a copy from sterling silver (5 grams):

I used hollow tubing, since I did not have any large-gauge solid round wire on hand. It was simple to make, but it did not occur to me until after I had fashioned it that I had made it backwards – If you look closely, it is actually the mirror image of the one in the article. I am left-handed and just made the ring the way that feels most natural to me, so this makes me think that the person who made the original might have been right-handed.

Kiddush Cup

Argentium (93.5%) silver and 14K (585) gold. 82g.

I have used the same design concept here as with the silver cup and reliquary pendant. This is a kiddush cup – a cup used by Jews to celebrate Shabbat. I am becoming increasingly interested in the use of silver for religious purposes. This is the first kiddush cup I know of based on a Catholic reliquary.

For decoration, I used an old gold ring that I bought at jeweler’s shop and then hammered into a large enough size to fit around the perimeter.

In designing this cup, I learned that Jewish religious law requires a kiddush cup to be at least one revi’it in volume, though there seems to be disagreement as to how large this is. The minimum size, according to Wikipedia, seems to be 90.7 ml and I wanted to keep it as small (and, therefore, simple) as possible, so that is the size I aimed for.

To work out the size and proportion, I made a variety of paper prototypes. If you look closely, you can see the kiddush cup and the previous smaller silver cup I made in there too.

Silver Cup

Argentium silver (93.5% silver. Sterling silver is at least 92.5%, so this silver is one grade above sterling). 25 ml in volume. 35 grams in weight (1 1/8 troy oz). Polished by Christine Guibara.

It did not occur to me when I made this, but the job of fabricator and polisher used to be separate professions. I found Christine on the internet and had her polish this cup for me simply because I do not have the right equipment. It was only after I dropped it off that I appreciated the historical parallel.

This cup has a circular profile on top and hexagonal on the bottom – similar to the reliquary pendant. I am intending for it to be a test-run for a larger piece I am currently working on.

It seems easy enough to me to create the Laffer Curve from first principles that I just went ahead in Excel and did it.
We can assume that as price rises, the Demand for a good drops. Conversely, the Supply increases. In the simplest case, each of these relationships is just a straight line.


In economics literature, the Supply and Demand curves are often shown with Price on the y-axis and the Supply and Demand on the x-axis. The point where they cross is the equilibrium price. For example, at a price of $50, buyers will demand 50 units and suppliers will provide 50 units.

The Laffer Curve represents the amount of revenue the government might receive by imposing a sales tax. If the tax rate is 0, then the government will receive no money from the sales. If the tax rate is enormous, there will be no buyers, so the government will also receive no money. Somewhere in the middle, there should be a revenue-maximizing tax rate.

We can construct the Laffer Curve from the demand curve and a little bit of arithmetic. If the government imposes a $1 sales tax and raises the price to $51, then the sellers behavior should not change, since they are still getting their $50 per sale.

However, the demand from the buyers should drop a little. In this straight line case, the demand will drop to 49 units, with the government getting $1 per sale, or $49 total.

If the tax is raised to $2, demand drops to 48 units, with the government getting $2 per sale, or $96 total. As the tax rises further, the revenue-per-sale continue to rise, bu the total number of sales continues to fall. At some point, a revenue-maximizing tax level is reached and then, eventually, the market is killed off entirely.


This is the version of the diagram I have often seen in economics presentations, including at the Laffer Center’s website.

If the current tax rate is above the revenue-maximizing rate, then there is an ‘eat your cake and have it too’ situation where the government can reduce tax rates, but increase its own revenue by allowing the market to get larger.

However, real supply and demand curves are probably not straight lines. For the oil industry, I often see supply curves that look like the red line below. There are a few very low-cost producers (the Middle East), a large number of medium-cost producers (conventional sources outside the Middle East), and then a number of increasingly high-cost unconventional sources (tar sands, the Arctic, etc).


Demand is usually less elastic. There is a narrow bell curve of demand-vs-price, so a small change in price has little effect on demand. (Often, I have seen demand as just a downward-sloping straight line cutting through the supply curve).

In this case, repeating the arithmetic to find government revenue (that is, revenue-per-sale increasing with higher tax rates, while demand for units drops) gives the relationship shown below.


In principle, this is qualitatively the same as the simplified Laffer Curve above: Zero at low and high tax rates, and high in the middle.

But the thing I notice first about this new curve is the asymmetry. If the current tax rate is well above the revenue-maximizing rate, then the industry will die off very quickly.


Whatever your personal politics, you can probably find something here to support your current beliefs: Either high taxes kill markets and reducing taxes can increase market size and tax revenue. Or, current taxes levels in most markets are probably not too high, or else the market would not exist at all. Take your pick. I just thought it was interesting to see what the Laffer Curve might look like for a real product, not just the simplified form I have often seen before.

I am interested in finding out more about Poisson processes, like the arrival of cars at a toll booth. I do not have much experience with discrete event systems, so this is a learning experience for me.

There is a parking garage near an office building with an electronic sign outside it that displays how many spaces are available. I went there at 6:00 on a Friday and saw there were 635 empty spaces.


I recorded the number of space available every 5 minutes until 9:40. This does not give an exact count of the number of cars arriving, since some cars arrive, drop someone off and then leave, and similar other exceptions. But, the parking garage and building are at the end of a dead-end road, so there is little other traffic at that time of day except people going to that garage.

The graph below shows the observed number of spaces available over time, as well as a piecewise function that attempts to model the net arrival rate of cars to the parking garage.


Taking the difference in number of spaces available at each 5 minute interval gives the net arrival rate of cars. You can see this in the graph below.


I have attempted to represent the arrival rate using the Erlang distribution, which is related to the expected time between events.


The specific parameter values I used were k=79 and lambda=9. I then multiplied this entire distribution by a factor of 70 to fit it to the net total number of cars that arrived.

This does not seem to work well after 9:00 am, so after that point I have simply modeled the arrival rate with a downward sloping line with a slope of -1, since I did not have enough data to make anything meaningful from this time interval.

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