Diminishing returns From Wikipedia, the free encyclopedia Jump to: navigation, search In economics, diminishing returns (also called diminishing marginal returns) refers to how the marginal production of a factor of production starts to progressively decrease as the factor is increased, in contrast to the increase that would otherwise be normally expected. According to this relationship, in a production system with fixed and variable inputs (say factory size and labor), each additional unit of the variable input (i. e. , man-hours) yields smaller and smaller increases in outputs, also reducing each worker’s mean productivity.

Conversely, producing one more unit of output will cost increasingly more (owing to the major amount of variable inputs being used, to little effect). This concept is also known as the law of diminishing marginal returns or the law of increasing relative cost. Contents[hide] * 1 Statement of the law * 2 History * 3 Examples * 4 Returns and costs * 5 Returns to scale * 6 See also * 7 References * 8 Sources| [edit] Statement of the law The law of diminishing returns has been described as one of the most famous laws in all of economics. 1] In fact, the law is central to production theory, one of the two major divisions of neoclassical microeconomic theory. The law states “that we will get less and less extra output when we add additional doses of an input while holding other inputs fixed. In other words, the marginal product of each unit of input will decline as the amount of that input increases holding all other inputs constant. “[2] Explaining exactly why this law holds true has sometimes proven problematic. Diminishing returns and diminishing marginal returns are not the same thing.

Diminishing marginal returns means that the MPL curve is falling. The output may be either negative or positive. Diminishing returns means that the extra labor causes output to fall which means that the MPL is negative. In other words the change in output per unit increase in labor is negative and total output is falling. [3] [edit] History | This section requires expansion. | The concept of diminishing returns can be traced back to the concerns of early economists such as Johann Heinrich von Thunen, Turgot, Thomas Malthus and David Ricardo.

However, classical economists such as Malthus and Ricardo attributed the successive diminishment of output to the decreasing quality of the inputs. Neoclassical economists assume that each “unit” of labor is identical = perfectly homogeneous. Diminishing returns are due to the disruption of the entire productive process as additional units of labor are added to a fixed amount of capital. Karl Marx developed a version of the law of diminishing returns in his theory of the tendency of the rate of profit to fall, described in Volume III of Capital. [edit] Examples

Suppose that one kilogram of seed applied to a plot of land of a fixed size produces one ton of crop. You might expect that an additional kilogram of seed would produce an additional ton of output. However, if there are diminishing marginal returns, that additional kilogram will produce less than one additional ton of crop (ceteris paribus). For example, the second kilogram of seed may only produce a half ton of extra output. Diminishing marginal returns also implies that a third kilogram of seed will produce an additional crop that is even less than a half ton of additional output, say, one quarter of a ton.

In economics, the term “marginal” is used to mean on the edge of productivity in a production system. The difference in the investment of seed in these three scenarios is one kilogram — “marginal investment in seed is one kilogram. ” And the difference in output, the crops, is one ton for the first kilogram of seeds, a half ton for the second kilogram, and one quarter of a ton for the third kilogram. Thus, the marginal physical product (MPP) of the seed will fall as the total amount of seed planted rises. In this example, the marginal product (or return) equals the extra amount of crop produced divided by the extra amount of seeds planted.

A consequence of diminishing marginal returns is that as total investment increases, the total return on investment as a proportion of the total investment (the average product or return) decreases. The return from investing the first kilogram is 1 t/kg. The total return when 2 kg of seed are invested is 1. 5/2 = 0. 75 t/kg, while the total return when 3 kg are invested is 1. 75/3 = 0. 58 t/kg. This particular example of Diminishing Marginal Returns in formulaic terms: Where D = Diminished Marginal Return, X = seed in kilograms, and = crop yield in tons gives us: Substituting 3 for X and expanding yields:

Another example is a factory that has a fixed stock of capital, or tools and machines, and a variable supply of labor. As the firm increases the number of workers, the total output of the firm grows but at an ever-decreasing rate. This is because after a certain point, the factory becomes overcrowded and workers begin to form lines to use the machines. The long-run solution to this problem is to increase the stock of capital, that is, to buy more machines and to build more factories. [edit] Returns and costs There is an inverse relationship between returns of inputs and the cost of production.

Suppose that a kilogram of seed costs one dollar, and this price does not change; although there are other costs, assume they do not vary with the amount of output and are therefore fixed costs. One kilogram of seeds yields one ton of crop, so the first ton of the crop costs one extra dollar to produce. That is, for the first ton of output, the marginal cost (MC) of the output is $1 per ton. If there are no other changes, then if the second kilogram of seeds applied to land produces only half the output of the first, the MC equals $1 per half ton of output, or $2 per ton.

Similarly, if the third kilogram produces only ? ton, then the MC equals $1 per quarter ton, or $4 per ton. Thus, diminishing marginal returns imply increasing marginal costs. This also implies rising average costs. In this numerical example, average cost rises from $1 for 1 ton to $2 for 1. 5 tons to $3 for 1. 75 tons, or approximately from 1 to 1. 3 to 1. 7 dollars per ton. In this example, the marginal cost equals the extra amount of money spent on seed divided by the extra amount of crop produced, while average cost is the total amount of money spent on seeds divided by the total amount of crop produced.

Cost can also be measured in terms of opportunity cost. In this case the law also applies to societies; the opportunity cost of producing a single unit of a good generally increases as a society attempts to produce more of that good. This explains the bowed-out shape of the production possibilities frontier. [edit] Returns to scale The marginal returns discussed refer to cases when only one of many inputs is increased (for example, the quantity of seed increases, but the amount of land remains constant). If all inputs are increased in proportion, the result is generally constant or increased output.

As a firm in the long-run increases the quantities of all factors employed, all other things being equal, initially the rate of increase in output may be more rapid than the rate of increase in inputs, later output might increase in the same proportion as input, then ultimately, output will increase less proportionately than input. See also: economies of scale The Real Numbers Sets of Numbers Natural Numbers {1, 2, 3, 4, . . . } Whole Numbers {0, 1, 2, 3, 4, . . . } Integers {. . . , -3, -2, -1, 0, 1, 2, 3, . . . } Rational Numbers {| p and q are integers and q ? 0 }

The set of rational numbers contains all numbers that can be written as fractions, or quotients of integers. Integers are also rational numbers since they can be represented as fractions. All decimals that repeat or terminate belong to the set of rational numbers. The following are all rational numbers: , -, 1, -5 =, 0 =, 0. 125 =, 0. 6666 . . . = Irrational Numbers {x | x is real but not rational } The irrational numbers are nonrepeating, nonterminating decimals. They cannot be represented as the quotient of two integers. The following are all irrational numbers: ? , , –

Real Numbers {x | x corresponds to a point on the number line } The set of real numbers consists of all the rational numbers together with all the irrational numbers. Example Given set A = {, -, 0, 2. 9, -5, 4, -, , -7, ? }, list all the elements of A that belong to the set of : a) natural numbers, b) whole numbers, c) integers, d) rational numbers, e) irrational numbers, and f) real numbers. a) 4 b) 0, 4 c) 0, -5, 4 d), 0, 2. 9, -5, 4, -, -7 e) -, , ? f) all elements of A are real numbers Order of Operations 1. Perform operations in grouping symbols (parentheses, brackets, braces, or fraction bars).

Start with the innermost and work outward. 2. Calculate powers and roots, working from left to right. 3. Perform multiplication and division in order from left to right. 4. Perform addition and subtraction in order from left to right. Example Use order of operations to evaluate: a) 6(-5) – (-3)(2) b) c) -9 – {6 – 2[12 – (8 – 15)] – 4} Solution: a) 6(-5) – (-3)(2) = 6(-5) – (-3)(16) No grouping symbols; power calculated first = -30 – (-48) Multiplication performed = -30 + 48 Subtraction changed to addition = 18 Addition performed b) Begin by simplifying the numerator and denominator of fraction. Calculate powers first = Perform multiplications = Perform additions and subtractions = Simplify c) -9 – {6 – 2[12 – (8 – 15)] – 4} = -9 – {6 – 2[12 – (-7)] – 4} Start with innermost grouping symbol, parentheses, and subtract = -9 – {6 – 2[19] – 4} Working outward, perform subtraction in brackets = -9 – {6 – 38 – 4} Within braces, multiply = -9 – {-36} Within braces, subtract = -9 + 36 Change subtraction to addition = 27 Add Properties of the Real Numbers For all real numbers a, b, and c: 1. Commutative Property for Addition: a + b = b + a 2.

Commutative Property for Multiplication: ab = ba The commutative properties state that two numbers may be added or multiplied in any order. 3. Associative Property for Addition: a + (b + c) = (a + b) + c 4. Association Property for Multiplication: a(bc) = (ab)c For the associative properties, the order of the terms or factors remains the same; only the grouping is changed. 5. Identity Property for Addition: There is a unique real number, 0, such that a + 0 = a and 0 + a = a The identity property for addition tells us that adding 0 to any number will not change the number. 6.

Identity Property for Multiplication: There is a unique real number, 1, such that a·1 = a and 1·a = a The identity property for multiplication tells us that multiplying any number by 1 will not change the number. 7. Inverse Property for Addition: Each nonzero real number a has a unique additive inverse, represented by –a, such that a + (-a) = 0 and –a + a = 0 Additive inverses are called opposites. 8. Inverse Property for Multiplication: Each nonzero real number a has unique multiplicative inverse, represented by , such that and Multiplicative inverses are called reciprocals. 9. Distributive Property: a(b + c) = ab + ac

Example Identify the property illustrated in each statement: a) (x + 7) + 8 = x + (7 + 8) b) 4x + 0 = 4x c) 10 · (x) = (10 ·)x d) (x+ 1) · = 1 e) 4(x + 5) = 4x + 20 f) 3 · (5 · a) = 3 · (a · 5) g) -6x + 6x = 0 h) (2 + y) + 5 = 5 + (2 + y) i) (y + 5)(y – 3) = (y – 3)(y + 5) j) 5 · 1 = 5 Solution: a) Associative Property for Addition. Order of terms remains the same. Only the grouping changes. b) Identity Property for Addition. Adding zero to something does not change it. c) Associative Property for Multiplication. Order of factors is the same. Only the grouping changes. d) Inverse Property for Multiplication.

The product of reciprocals is 1. e) Distributive Property. f) Commutative Property for Multiplication. Order of the factors is changed. g) Inverse Property for Addition. The sum of opposites is 0. h) Commutative Property for Addition. The order of the terms is changed. i) Commutative Property for Multiplication. The order of the factors is changed. j) Identity Property for Multiplication. Multiplying a number by 1 does not change it. real number| | | | | – A real number is any element of the set R, which is the union of the set of rational numbers and the set of irrational numbers.

In mathematical expressions, unknown or unspecified real numbers are usually represented by lowercase italic letters u through z. The set R gives rise to other sets such as the set of imaginary numbers and the set of complex numbers. The idea of a real number (and what makes it “real”) is primarily of interest to theoreticians. Abstract mathematics has potentially far-reaching applications in communications and computer science, especially in data encryption and security. If x and z are real numbers such that x ; z, then there always exists a real number y such that x ; y ; z.

The set of reals is “dense” in the same sense as the set of irrationals. Both sets are nondenumerable. There are more real numbers than is possible to list, even by implication. The set R is sometimes called the continuum because it is intuitive to think of the elements of R as corresponding one-to-one with the points on a geometric line. This notion, first proposed by Georg Cantor who also noted the difference between the cardinalities (sizes) of the sets of rational and irrational numbers, is called the Continuum Hypothesis. This hypothesis can be either affirmed or denied without causing contradictions in theoretical mathematics. | |