Question

Q3. Use the envelop theorem to estimate the maximum value of x +x4.1y2 on the constraint set 2x +2y s1, x 20, y 20

Answer 1

In economics and econometrics. the Cobb- Douglas production function is a particular functional form of the production function, widely used to represent the technological relationship between the amounts of two or more inputs ( particularly physical capital and labor ) and the amount of output that can be produced by those inputs. The Cobb - Douglas form was developed and tested against statistical evidence by Charles Cobb and Paul Douglas during 1927-1947.

Paul Douglas explained that his first formulation of the Cobb - Douglas production function was developed in 1927 when seeking a functional form to relate estimates that he had calculated for workers and capital, he spoke with mathematician and colleague Charles Cobb. A major criticism at that time was that estimates of the production function, although seemingly accurate, were based on such sparse data that it was hard to give them much credibility. Douglas remarked " I must admit I was discouraged by this criticism and thought of giving up the effort, but there was something which told me I should hold on. The breakthrough came in using US census data, which is cross-sectional and provided a large number of observations. Douglas presented the results of these findings, along with those for other countries, at his 1947 address as president of the American Economic Association. Shortly afterward Douglas went into politics.

Following Munnell (1990), Baltagi and Pinnoi (1995) considered the Cobb Douglas production function relationship investigating the contribution of different types of public infrastructure on private production. The results based on the Cobb - Douglas production function are the same as in Baltagi. The coefficients on both labor and private capital are found to be positive and statistically significant. On the other hand, the coefficient on public capital is quite small and statistically insignificant. These results have caused some to suggest that public capital is unproductive.

Such a large difference in the returns between public and private capital is difficult to explain. The ammunition for this debate has primarily been arrived at by using the results from a Cobb -Douglas production function. However, assuming a particular production function assumes a particular form for the underlying production function, which may or may not be correct. Further, by construction, the elasticities of this model are exactly the same across all states and over all years. Thus, it seems natural to ask whether the results from the Cobb - Douglas model can be trusted. In fact, if the true model is nonlinear and one ignores it, the resulting estimates of returns to inputs are likely to be inconsistent.

Thus, it is true to say that the model is underspecified and the estimates will be biased and the current production should also be a function of past investments in production.

Current production costs (reflecting the concept of economic cost ) and replacement costs thus belong to different long -term horizons; the former reflect past investments, the latter consider the future situation, with all the margins of risk inherent in this. There might not be much difference between the two values if they did not refer to an activity which (normally) manifests declining yields, in other words, which operates at growing costs