The Production Function and Cobb-Douglas production

The Production Function:

The first component of the model is a behavioral relationship called the production function. This behavioral relationship tells us how the productive resources of the economy—the labor force, the capital stock, and the level of technology that determines the efficiency of labor—can be used to produce and determine the level of output in the economy. The total volume of production of the goods and services that consumers, investing businesses, and the government wish for is limited by the available resources. The production function tells us how available resources limit production.

Tell the production function what resources the economy has available, and it will tell you how much the economy can produce. We will use the so-called Cobb-Douglas production function, a functional form that economists use because it makes many kinds of calculations relatively simple. The Cobb-Douglas production function states:

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The economy's level of output per worker (Y/L) is equal to the capital stock per worker K/L raised to the exponential power of some number α, and then multiplied by the current efficiency of labor E raised to the exponential power (1-α).

The efficiency of labor E and the number α are parameters of the model. The parameter α is always a number between zero and one. The best way to think of it as the parameter that governs how fast diminishing returns to investment set in. A level of α near zero means that the extra amount of output made possible by each additional unit of capital declines very quickly as the capital stock increases.

By contrast, a level of α near one means that the next additional unit of capital makes possible almost as large an increase in output as the last additional unit of capital. When α equals one, output is proportional to capital: double the stock of capital per worker, and you double output per worker as well. When the parameter α is near to but less than one, diminishing returns to capital accumulation do set in, but they do not set in rapidly or steeply. And as α varies from a high number near one to a low number near zero, the force of diminishing returns gets stronger.

The other parameter E tells us the current level of the efficiency of labor. A higher level of E means that more output per worker can be produced for each possible value of the capital stock per worker. A lower value of E means that the economy is very unproductive: not even huge amounts of capital per worker will boost output per worker to achieve what we would think of as prosperity. It illustrates how to use the production function once you know its form and parameters--how to calculate output per worker once you know the capital stock per worker.

The Cobb-Douglas production function is "flexible" in the sense that it can be tuned to fit any of a wide variety of different economic situations. It shows a small part of the flexibility of the Cobb-Douglas production function. Is the level of productivity high? The Cobb-Douglas function can fit with a high initial level of the efficiency of labor E. Does the economy rapidly hit a wall as capital accumulation proceeds and find that all the investment in the world is doing little to raise the level of production? Then the Cobb-Douglas function can fit with a low level--near zero--of the diminishing-retuns-to-capital parameter α. Is the speed with which diminishing-returns-to-investment sets in moderate? Then pick a moderate value of α and the Cobb-Douglas function will once again fit.

No economist believes that there is, buried somewhere in the earth, a big machine that forces the level of output per worker to behave exactly as calculated by the algebraic production function above. Instead, economists think that the Cobb-Douglas production function above is a simple and useful approximation.

The true process that does determine the level of output per worker is an immensely complicated one: everyone in the economy is part of it. And it is too complicated to work with. Writing down the Cobb-Douglas production function is a breathtakingly large leap of abstraction. Yet it is a useful leap, for this approximation is good enough that using it to analyze the economy will get us to approximately correct conclusions.

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