# Introduction to Macroeconomics

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• Published : April 19, 2013

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Ep = - 0.36, P = 4
-0.36 = 4/(4-a)
-0.36 (4-a) = 4
0.36a = 5.4
a = 5.4/0.36 = 15
Consider the following production functions: ? Y = 10K1/2L1/2 ? Y = 2K + 3L a. Fixing labor employment (L) at 16 units, what is the marginal product of capital when capital employment is 25, 35, and 45 for each production function? Do these production functions exhibit diminishing returns to capital employment? Explain. b. Are labor and capital complements under these production functions? Explain. c. Do these production functions have the property of constant returns to scale? Explain. d. Is either production function a “Cobb-Douglas” function? Explain.  n economics, the Cobb–Douglas functional form of production functions is widely used to represent the relationship of output and two inputs. The Cobb-Douglas form was developed and tested against statistical evidence by Charles Cobb and Paul Douglas during 1900–1947.In its most standard form for production of a single good with two factors, the function is

Y=AL^{\beta}K^{\alpha}

where:

Y = total production (the monetary value of all goods produced in a year) L = labor input (the total number of person-hours worked in a year) K = capital input (the monetary worth of all machinery, equipment, and buildings) A = total factor productivity

a and ß are the output elasticities of capital and labor, respectively. These values are constants determined by available technology.

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 labor 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...