Marginal product of labor

In economics, the marginal product of labor also known as MPL or MPN is the change in output from hiring one additional unit of labor. It is the increase in output added by the last unit of labor.^{[1]} Ceteris paribus that no other inputs to production change, the marginal product of a given input (X) can be expressed as:
 MP = ΔY/ΔX = (the change of Y)/(the change of X).^{[2]}
Marginal Product of Labor is not simply the additional output produced by the last worker hired
In the short run the only way a firm can vary production is by changing the quantity of variable input.^{[3]} The variable input is conventionally assumed to be labor. Output is referred to as the product of labor.^{[4]} The total product of labor is the amount of output produced.^{[4]} The total product curve is the short run relationship between the amount of output produced and the quantity of labor used. Capital which is fixed serves as a capacity constraint.^{[5]} Marginal product of labor is the slope of the total product curve. Marginal product is the increase in output resulting from an incremental increase in the amount of labor employed. Increasing the amount of an input while holding other factors of production constant affects the efficiency of the entire production process. The effects can be either positive or negative. Typically, at low levels of production, increases in the variable input have positive effects due to specialization of labor. In these early stages each additional unit of labor produces more output than did the previous unit of labor. Thus the marginal product of labor is increasing. Graphically the total product curve is increasing and getting steeper as units of labor are added. At some point the additional units of labor begin to interfere with the productivity of all other units of labor being employed  people are getting into each other’s way.^{[6]} The result is that the average product of labor falls. This disruption to productivity gets worse as more units of labor are added. During this range of production, total product is still increasing but each successive addition is smaller (the marginal product of labor is falling). The curve is still rising but it is becoming flatter. The point at which this change in the shape of the total product curve occurs (from getting steeper to getting flatter) is called the point of diminishing marginal returns. Beyond this point additions to output due to a unit increase in labor are successively smaller.
The point that the marginal product of labor is not the addition to output directly attributable to the marginal unit is well stated in the following paragraph:
[I]t is important to underline that the marginal product is not, properly speaking, the contribution of the marginal unit by itself….The second man may very well produce nine or ten or eleven and still the total output increases only to eighteen because the first man reduces his output to nine, or eight or seven in the presence of the second. In our example, output increases from ten bushels to eighteen bushels when one adds the second man not because the second man only adds eight, but rather because his presence on the field makes the situation such that the total output of both men is eighteen. Notice that the contribution per man is reduced: the average product is actually nine. This may very well be how much each of the two laborers contributes. But this is not what interests us: what we wish to note is that by adding the second man, output was increased by eight. Thus, the marginal product of the second man is eight. But his actual contribution may be very different than this.
^{[7]}
Examples
 There is a factory which produces toys. When there are no workers in the factory, no toys are produced. When there is one worker in the factory, six toys are produced per hour. When there are two workers in the factory, eleven toys are produced per hour. There is a marginal product of labor of 5 when there are two workers in the factory compared to one. When the marginal product of labor is positive, this is called increasing marginal returns. However, as the number of workers increases, the marginal product of labor may not increase indefinitely. When not scaled properly, the marginal product of labor may go down when the number of employees goes up, creating a situation known as diminishing marginal returns.When the marginal product of labor becomes negative, it is known as negative marginal returns.
Deriving the marginal product of a factor of production from the production function
 Assume that the production process involves three inputs x, y and z.
 The production function is Q = 2x^{2}+3xy + 6.45y^{2}+1.75yz +0.35z^{3}.^{[8]} The marginal product of a factor of production is the partial derivative of that function with respect to the specific input.

 MP_{x}= 4x +3y

 MP_{y}= 3x + 12.90y +1.75z

 MP_{z}= 1.75y + 1.05z^{2}
Definitions
The marginal product of labor is the change in output per unit change in labor. In discrete terms the marginal product of labor is ∆Q/∆L. In marginal terms the MPL is the first derivative of the total product curve or short run production function or ∂Q/∂L.^{[9]} The precise mathematical formula is MP_{L} = (dAP_{L}/dL)q + AP_{L}. Graphically the MP_{L} is the slope of the total product curve/short run production function.
MP_{L} and MC
The marginal product of labor is directly related to costs of production. Cost are divided between fixed and variable costs. Fixed costs are costs that relate to the fixed input, capital, or rK where r is the rate of return and K is the quantity of capital. Variable costs are the costs of the variable input, labor, or wL where w is the wage rate and L is the amount of labor employed. Thus VC = wL . MC is the change in total cost per unit change in output or ∆C/∆Q. In the short run, production can be varied only by changing the variable input. Thus only variable costs change as output increases ∆C = ∆VC = ∆Lw. Marginal costs is ∆Lw/∆Q. Now, ∆L/∆Q is the reciprocal of the marginal product of labor (∆Q/∆L). Therefore, marginal cost is simply the wage rate w divided by the marginal product of labor
 MC = ∆VC∕∆q;
 ∆VC = w∆L;
 ∆L∕∆q the change in quantity of labor to affect a one unit change in output = 1∕MPL.
 Therefore MC = w ∕ MP_{L}
Thus if the marginal product of labor is rising then marginal costs will be falling and if the marginal product of labor is falling marginal costs will be rising.(assuming a constant wage rate).^{[10]}
Relation between MP_{L} and AP_{L}
The average product of labor is the total product of labor divided by the number of units of labor employed or Q/L.^{[9]} The average product of labor is a common measure of labor productivity.^{[11]}^{[12]} The AP_{L} curve is shaped like an inverted “u”.^{[6]} At low production levels the AP_{L} tends to increases as additional labor is added. The primary reason for the increase is specialization and division of labor.^{[6]} At the point the AP_{L} reaches its maximum value AP_{L} equals the MP_{L}.^{[13]} Beyond this point the AP_{L} falls.
During the early stages of production MP_{L} is greater than AP_{L}. When the MP_{L} is above the AP_{L} the AP_{L} will increase. Eventually the MP_{L} reaches it maximum value at the point of diminishing returns. Beyond this point MP_{L} will decrease. However, at the point of diminishing returns the MP_{L} is still above the AP_{L} and AP_{L} will continue to increase until MP_{L} equals AP_{L}. When MP_{L} is below AP_{L}, AP_{L} will decrease.
Graphically the AP_{L} curve can be derived from the total product curve by drawing secants from the origin that intersect (cut) the total product curve. The slope of the secant line equals the average product of labor (Slope = dQ/dL).^{[6]} The slope of the curve at each intersection marks a point on the average product curve. The slope increases until the line reaches a point of tangency with the total product curve. This point marks the maximum average product of labor. It also marks the point where MP_{L} (which is the slope of the total product curve)^{[14]} equals the AP_{L} (the slope of the secant).^{[15]} Beyond this point the slope of the secants become progressively smaller as AP_{L} declines. The MP_{L} curve intersects the AP_{L} curve from above at the maximum point of the AP_{L} curve. Thereafter, the MP_{L} curve is below the AP_{L} curve.
The Law of Diminishing Marginal Returns
The falling MP_{L} is due to the law of diminishing marginal returns. The law states,”as units of one input are added (with all other inputs held constant) a point will be reached where the resulting additions to output will begin to decrease; that is marginal product will decline.”^{[16]} The law of diminishing marginal returns applies regardless of whether the production function exhibits increasing, decreasing or constant returns to scale.^{[17]} The key factor is that the variable input is being changed while all other factors of production are being held constant. Under such circumstances diminishing marginal returns are inevitable at some level of production.^{[17]}
Diminishing marginal returns differs from diminishing returns.^{[18]} Diminishing marginal returns means that the marginal product of the variable input is falling.^{[18]} Diminishing returns occur when the marginal product of the variable input is negative.^{[18]} That is when a unit increase in the variable input causes total product to fall.^{[18]} At the point that diminishing returns begin the MP_{L} is zero.
MP_{L}, MRP_{L} and Profit Maximization
The general rule is that firm maximizes profit by producing that quantity of output where marginal revenue equals marginal costs. The profit maximization issue can also be approached from the input side. That is, what is the profit maximizing usage of the variable input?^{[19]} To maximize profits the firm should increase usage "up to the point where the input’s marginal revenue product equals its marginal costs".^{[19]} So mathematically the profit maximizing rule is MRP_{L} = MC_{L}^{[19]} The marginal profit per unit of labor equals the marginal revenue product of labor minus the marginal cost of labor or Mπ_{L} = MRP_{L}  MC_{L}A firm maximizes profits where Mπ_{L} = 0.
The marginal revenue product is the change in total revenue per unit change in the variable input assume labor.^{[19]} That is MRP_{L} = ∆TR/∆L. MRP_{L} is the product of marginal revenue and the marginal product of labor or MRP_{L} = MR x MP_{L}.
 Derivation:
 MR = ∆TR/∆Q
 MP_{L} = ∆Q/∆L
 MRP_{L} = MR x MP_{L} = (∆TR/∆Q) x (∆Q/∆L) = ∆TR/∆L
Example
 Assume that production function is Q = 90L  L^{2}^{[16]}
• MC_{L} = 30
• Output price is $40 per unit.
 MP_{L} = 90  2L
 MRP_{L} = 40(90  2L)
 MRP_{L} = 3600  80L
 MRP_{L} = MCL (Profit Max Rule)
 3600  80L = 30
"3570 = 80L
 L = 44.625
 44.625 is the profit maximizing number of workers.
 Q = 90L  L2
 Q = 90(44.625)  (44.625)2
 Q= 4016.25  89.25
 Q = 3927
 The profit maximizing output is 3927 units
• And the profit is
 TR  TC = π
 π = 40(3927)  30(3927)
 Π = 157,080  117,810
 π = 39,270
Marginal productivity ethics
In the aftermath of the marginal revolution in economics, a number of economists including John Bates Clark and T.N. Carver sought to derive an ethical theory of income distribution based on the idea that workers were morally entitled to receive a wage exactly equal to their marginal product. In the 20th century, marginal productivity ethics found few supporters among economists, being criticised not only by egalitarians but by economists associated with the Chicago school such as Frank Knight (in The Ethics of Competition) and the Austrian School, such as Leland Yeager [1]. However, marginal productivity ethics were defended by George Stigler.
See also
 Marginal product
 Marginal revenue
 Diminishing marginal returns
 Increasing marginal returns
 Average marginal return
Footnotes
 ^ Sullivan, arthur; Steven M. Sheffrin (2003). Economics: Principles in action. Upper Saddle River, New Jersey 07458: Pearson Prentice Hall. pp. 108. ISBN 0130630853. http://www.pearsonschool.com/index.cfm?locator=PSZ3R9&PMDbSiteId=2781&PMDbSolutionId=6724&PMDbCategoryId=&PMDbProgramId=12881&level=4.
 ^ R. Baye, Michael (1999). "Marginal Product of Labor". Managerial Economics and Business Strategy:Economic Department at Iowa State University. McGraw Hill. pp. Slide 4 of 28. http://www.econ.iastate.edu/classes/econ532/hennessy/lectures/chap05/sld004.htm. Retrieved 20090217.
 ^ Perloff, J: Microeconomics Theory & Applications with Calculus page 209. Pearson 2008.
 ^ ^{a} ^{b} Ayers, R & Collinge, R: 2003 page 178.
 ^ Silberberg, E & Suen, W: 2001. p. 181.
 ^ ^{a} ^{b} ^{c} ^{d} Perloff, J: Microeconomics Theory & Applications with Calculus page 176. Pearson 2008.
 ^ http://homepage.newschool.edu/het//essays/product/prodfunc.htm#marginal
 ^ Dowling 1980 p. 119.
 ^ ^{a} ^{b} Perloff, J: Microeconomics Theory & Applications with Calculus page 173. Pearson 2008.
 ^ Pindyck, R & Rubinfeld, D: Microeconomics 5th ed. PrenticeHall 2001.
 ^ Nicholson, W. & Snyder, C.: Intermediate Microeconomics page 215. Thomson 2007
 ^ Nicholson: Microeconomic Theory 9th ed. Page 185. Thomson 2005
 ^ Binger, B & Hoffman, E.: Microeconomics with Calculus, 2nd ed. page 253. AddisonWesley 1998.
 ^ Krugman & Wells: Microeconomics 2d ed. page 306. Worth 2009
 ^ Perloff, J: Microeconomics Theory & Applications with Calculus page 177. Pearson 2008.
 ^ ^{a} ^{b} Samuelson, W & Marks, S. Managerial Economics 4th ed. page 227. Wiley 2003.
 ^ ^{a} ^{b} Varian, H: Microeconomic Analysis 3rd ed. Norton 1992.
 ^ ^{a} ^{b} ^{c} ^{d} Perloff, J: Microeconomics Theory & Applications with Calculus page 178. Pearson 2008.
 ^ ^{a} ^{b} ^{c} ^{d} Samuelson, W & Marks, S. Managerial Economics 4th ed. Wiley 2003.
References
 1. Ayers, R & R Collinge: 2003 Microeconomics Pearson ISBN 0131463926
 2. Baye, Michael 1999 Managerial Economics and Business Strategy:Economic . McGraw Hill.
 3. Binger, B & E Hoffman, 1998. Microeconomics with Calculus, 2nd ed. AddisonWesley ISBN 0321012259
 4. Krugman & Wells: 2009 Microeconomics 2d ed. Worth
 5. Nicholson: 2005 Microeconomic Theory 9th ed. Thomson
 6. Nicholson, W. & C Snyder: 2007 Intermediate Microeconomics Thomson ISBN 0324319681
 7. Perloff, J: 2008 Microeconomics Theory & Applications with Calculus page Pearson. ISBN 9780321277947
 8. Pindyck, R & D Rubinfeld: 2001 Microeconomics 5th ed. Page PrenticeHall. ISBN 0130196738
 9. Silberberg, E & W Suen: 2001 The Structure of Economics, A Mathematical Analysis 3rd ed. McGrawHill, ISBN 97800723433526.
 10. Samuelson, W & S Marks, 2003 Managerial Economics 4th ed. Wiley
 11. Sullivan, A & Steven M. Sheffrin (2003). Economics: Principles in action. Pearson Prentice Hall.ISBN 0130630853
 12. Varian, H: Microeconomic Analysis 3rd ed. Norton 199
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