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production function discussed

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See attachment for full questions. Questions include:

(a) What is a production function?
(b) How does a short-run production function differ from a long-run production function?
(c) Explain the term "Marginal Rate of technical Substitution".
(d) Why are isoquants assumed to be downward sloping?

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Summarize the production function.

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Problem 1:
(a) What is a production function?

A production function represents how inputs are transformed into outputs by a firm. We focus on the firm with one output and aggregate all inputs or factors of production into one of several categories, such as labor, capital, and materials. It is a function summarizing the process of conversion of factors into a particular commodity.
y = f(x1, x2, ..., xm)
which relates a single output y to a series of factors of production x1, x2, ..., xm.

(b) How does a short-run production function differ from a long-run production function?

In the short run, one or more factors of production cannot be changed. As time goes by, the firm has the opportunity to change the levels of all inputs. In the long-run production function, all inputs are variable.

(c) Explain the term "Marginal Rate of technical Substitution".

The isoquant identifies all the combinations of the two inputs which can produce the same level of output. The curvature of the isoquant is measured by the slope of the isoquant at any given point. The slope of the isoquant measures the rate at which the two inputs can be exchanged and still keep output constant, and this rate is called the marginal rate of technical substitution.

(d) Why are isoquants assumed to be downward sloping?

Isoquants are downward sloping because the marginal rate of technical substitution usually diminishes as you move down along the isoquant. Note the possibility of substituting one input for another in the production process while keeping the level of output constant is shrinking.

Problem 2:
You are given production technologies that use labor and capital as inputs. For each of the following technologies, you are to calculate (i) marginal productivity of labor (MPL), (ii) ...

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