Sunday, 3 June 2012

An Economic Approach to Ski Training

An economic approach to ski training. What does that even mean?  What would an economist know about ski training?

Expenses and budgets and eating brown rice with eggs for dinner may come to mind but my intention is not to discuss the financial side of being an athlete. Rather, my intention is to spend some time discussing an approach to ski training, specifically to analyzing all the elements and details to take care of in becoming a good skier, from the perspective of an academic economist. This may be the first in a new alanaskis blog series or it may be a one-off, the economist in me knows that it's too soon to tell for sure.

We will use a few basic themes and principles of economics to guide the discussion:

Economists and athletes have one major thing in common: an obsession with optimization. An economist defines optimization as: "The process by which economic agents (firms and consumers) do the best they can given the constraints they face" (Williamson, p.29).

In general, to an economist this means things like firms will hire the efficient amount of labour and capital so as to produce a set output at the lowest possible cost, households will allocate income between different goods and services to attain the highest amount of happiness given a budget restriction and governments will implement taxes that raise real tax revenue without overly distorting the household's or firm's decisions or honesty in reporting.

To an athlete, optimization is the idea of achieving excellent performance in their sport as a result of training and preparation. If we extend or relax the terms in an economist's definition of optimization we have a very succinct and likely accurate definition for what an athlete's optimization quest comes down to: "The process by which athletes do the best they can given the constraints they face." (Thomas). The best they can refers to the best results they can achieve while the constraints they face refers to any number of challenges an athlete may face during the process. In some practical sense these constraints may be financial, they may be injuries, they may be balancing school and training, they may be adverse life events but in our model we will consider them more basically as being the recovery time needed from actual physical training.

Profit (Performance) Maximization
We will chose to analyze the approach to ski training in parallel with how a firm will chose to maximize profits. Economists assume every firm is rational and has the goal of maximizing profits and we will assume the same of athletes, athletes are rational and have the goal of maximizing sport performance, an athlete's "profit". A firm's profit is the difference between it's total revenue and its total costs. Total revenue is the selling price of output multiplied by units of output sold while total cost is as it sounds, the total cost of producing all that output.

For an athlete's profit we must define the term "profit" somewhat more creatively since we are not interested in the dollars and cents of being an athlete right now. We will think of an athlete's profit as being excellent race performances come wintertime.

The Production Function (Training)
To maximize profit a firm considers it's production function:
                                                                      Y= zf(K,L,)

Y represents output (whatever good or service the firm is producing), f(K,L) is some function of a firm's inputs, capital, K and labour, L which describes how Y is created and z is some vector of semi-permanent factors which represent the total factor productivity of the function. If z increases then output will increase without having to increase the use of capital or labour, something has changed which leads to these inputs increasing in productivity.

We relate this to training by assuming an athlete has a similar function where Y is the total training effect, f(K,L) is the function which describes how different forms of training (for simplicity let K be strength training and L be intensity training) interact to produce the training effect Y and z represents a set of less tangible things that contribute to the total training effect other than the literal training itself. Think of z as something like mental focus during workouts or having teammates to help you get great training done. Let's assume Y, total training effect, is purely a good thing to have lots of and that it is positively correlated with performing well.

Total Revenue (Using the Training in Races)
For a firm total revenue is output, Y multiplied by the market price, P for which they can sell this output. We assume the market is competitive and so the price is given by the market rather than set by the firm.

For an athlete, total revenue is a somewhat trickier concept since we are thinking of profit as excellent physical performances rather than money in the bank. We have already established that an athlete's output is their total training effect so perhaps their total revenue is training effect multiplied by race opportunities. Instead of facing a market price they face a race calendar which tells them how many events they can use their training effect at. In reality, an athlete can choose how many races to enter so we will consider "P" to be the maximum number of races offered during the NorAM or World Cup season. To summarize, an athlete's "revenue" is getting to put training effect to use in races.

For a firm, costs are straightforward. A firm's inputs, capital and labour, each cost money and if you want to increase either one, it will cost you more money.

For athletes, costs are less straightforward as we aren't looking at monetary costs. We defined an athlete's inputs as being different kinds of training, such as strength training and intensity training.  The cost of these inputs to the athlete would be the physical fatigue created by them, the breakdown of muscles and the stress and strain on joints and tendons.

We assumed earlier that a bigger total training effect is always a good thing, basically saying that training is good, more training is better and even more training again is best. The introduction
of the physical costs of training add realism to our model. It incorporates the logic that while training is necessary it is only profitable if the physical gains outweigh the physical costs over the long run just as a bakery selling lots and lots of bread is only profitable if the selling price is higher than the cost per loaf.

Statistics and Calculus (Coaches and Experience)
To actually solve the firm's puzzle of profit maximization one must collect historical data to statistically estimate representative equations for the market price, production function and costs and then use multivariable calculus to solve for approximately how much capital, labour and output is optimal for the firm.

This sounds messy but in practice but is probably more straightforward than trying to do the same to create a training plan! Coaches and athletes can't exactly consult a database at Statistics Canada to look at every skier's training history and race performance and so instead must rely on past training logs, race results and self-reported data on the perceived success of the past season's training. Tricky business!

To summarize the discussion, we have likened a firm's decision making process for the balance of inputs, output and total cost in order maximize profits to an athlete's balance of training, racing and recovery to maximize race performance. In academic economics, this is the framework for modelling and studying how a firm in a competitive market behaves and we can use the same logic to model how a smart athlete could approach training.

If you made it this far, WOW! Thank you for reading!

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