Assume that a country\'s production function is Y = AK^(1/2)L^(1/2) a. What is t

Question

Assume that a country's production function is Y = AK^(1/2)L^(1/2)

a. What is the per-worker production function y = f(k) if A = 1?
b. Assume that the country possesses 40,000 units of capital and 10,000 units of labor.
What is Y? What is labor productivity computed from the per-worker production
function? Is this value the same as labor productivity computed from the original
production function?
c. Assume that 10 percent of capital depreciates each year. What gross saving rate is
necessary to make the given capital

Explanation / Answer

production function, in economics, equation that expresses the relationship between the quantities of productive factors (such as labour and capital) used and the amount of product obtained. It states the amount of product that can be obtained from every combination of factors, assuming that the most efficient available methods of production are used. The production function can thus answer a variety of questions. It can, for example, measure the marginal productivity of a particular factor of production (i.e., the change in output from one additional unit of that factor). It can also be used to determine the cheapest combination of productive factors that can be used to produce a given output. In economics, the Cobb–Douglas functional form of production functions is widely used to represent the relationship of output and two inputs. Similar functions were originally used by Knut Wicksell (1851–1926), while the Cobb-Douglas form was developed and tested against statistical evidence by Charles Cobb and Paul Douglas during 1900–1947. Output elasticity measures the responsiveness of output to a change in levels of either labor or capital used in production, ceteris paribus. For example if a = 0.15, a 1% increase in labor would lead to approximately a 0.15% increase in output. Further, if: a + ß = 1, the production function has constant returns to scale: Doubling capital K and labour L will also double output Y. If a + ß < 1, returns to scale are decreasing, and if a + ß > 1 returns to scale are increasing. Assuming perfect competition and a + ß = 1, a and ß can be shown to be labor and capital's share of output. Cobb and Douglas were influenced by statistical evidence that appeared to show that labor and capital shares of total output were constant over time in developed countries; they explained this by statistical fitting least-squares regression of their production function. There is now doubt over whether constancy over time exists.

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