Friday 12 June 2009

How should a government invest to maximise national output?

Suppose a government has a budget and wants to invest in the economy to maximise economic output. It can choose between educating its population, buying more of its current productive goods such as roads or factories for state owned companies, or researching for new technologies and adopting foreign technologies. Which should it do?

Here is a quick answer that illustrates some of the major principles involved. Let us say that the economy's output is given by a function

Output = Y = A^a * K^b * H^c.

This says that output depends on the technology used in the economy (A), the invested money in physical inputs (K), and the education in the economy (H). The exponential terms a, b, and c describe how output may rise at a different speed to the inputs.

Suppose that the costs of A, K, and H are constant relative to each other, which may be the case if the economy is small relative to the input sources (such as the rest of the world) or if the factors used to make A, K, and H (such as finance and labour) are interchangeable. Then the country has a budget

A+d.K+e.L=f

for some total amount of money f, where d and e are the prices of K and L relative to A. There are several ways for the government to work out how to maximise output when its budget is constrained. One way is to substitute the budget into the output function and then use differential calculus to find a maximum. Another way is first to take logarithms of the output function and then substitute the budget, which is less algebraically complicated, and equivalent since the logarithm function moves up and down at the same places as the original function so will have the same maximum. Another way is to use linear programming, subtracting an undefined multiple of the budget from the output function and then using the Lagrange multiplier approach (described here. This approach is very common in academic work since the algebra tends to be by far the least complicated for big problems).

The solutions are

K/A = b/(a*d) for all f and L/A = c/(a*e) for all f.

So the ratios of the inputs at the maximum output are constant. Investment should occur to keep these ratios constant. The constants a, b, c, d, and e can be estimated from past data, although the estimation might be difficult since: 1) the economy may have changed since the data was generated, 2) the economy may not have a maximal allocation in the past, 3) the data should be for government expenditure, not for economy-wide expenditure and the data may not be available, 4) the assumptions of parameter stability are only approximate, 5) the function for output is an approximation, and 6) there are other reasons too.

The many cautions on estimation just presented show why the answer is rough, but it does have a place when there is an extreme imbalance between the amounts of each input (far more capital in the economy than technology, for example), since then the approximations are less likely to make the broad recommendations wrong. Large imbalances are not uncommon internationally; communist countries often have far higher levels of education than technology for example, so increasing expenditures on adopting new technologies is more likely to promote growth than further educational expenditures. Another example might be in closed economies that tried to import-substitute for advanced Western goods in the past; they built up large technological bases (even if the technology was difficult to observe because of limited capital stocks) and so capital investment is more likely to promote rapid growth than further domestic technological innovation. Such economies are like coiled springs, where political decisions caused deviation from growth maximisation in the past, but were able to correct rapidly to their long-run position on adoption of a growth-maximising approach to the economy.

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