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From Wikipedia

Experience curve effects

Experience curve' re-directs here. For its use in video games seeExperience point.

Models of the learning curve effect and the closely related experience curve effect express the relationship between equations for experience and efficiency or between efficiency gains and investment in the effort.

Learning curve and learning curve effect

The experience of "learning curves" was first observed by the 19th Century German psychologist Hermann Ebbinghaus according to the difficulty of memorizing varying numbers of verbal stimuli. Subsequent learning about the complex processes of learning are discussed in the Learning curve article.

The experienced learning rates for exploratory discovery and development processes, for individuals and organizations, is more the focus of the main Learning curve article.

The rule used for representing the learning curve effect states that the more times a task has been performed, the less time will be required on each subsequent iteration. This relationship was probably first quantified in 1936 at Wright-Patterson Air Force Base in the United States, where it was determined that every time total aircraft production doubled, the required labour time decreased by 10 to 15 percent. Subsequent empirical studies from other industries have yielded different values ranging from only a couple of percent up to 30 percent, but in most cases it is a constant percentage: It did not vary at different scales of operation. Learning curve theory states that as the quantity of items produced doubles, costs decrease at a predictable rate. This predictable rate is described by Equations 1 and 2. The equations have the same equation form. The two equations differ only in the definition of the Y term, but this difference can make a significant difference in the outcome of an estimate.

1. This equation describes the basis for what is called the unit curve. In this equation, Y represents the cost of a specified unit in a production run. For example, If a production run has generated 200 units, the total cost can be derived by taking the equation below and applying it 200 times (for units 1 to 200) and then summing the 200 values. This is cumbersome and requires the use of a computer or published tables of predetermined values.

\ Y_x = K x^{\log_2 b}


  • \ K is the number of direct labour hours to produce the first unit
  • \ Y_x is the number of direct labour hours to produce the xth unit
  • \ x is the unit number
  • \ b is the learning percentage

2. This equation describes the basis for the cumulative average or cum average curve. In this equation, Y represents the average cost of different quantities (X) of units. The significance of the "cum" in cum average is that the average costs are computed for X cumulative units. Therefore, the total cost for X units is the product of X times the cum average cost. For example, to compute the total costs of units 1 to 200, an analyst could compute the cumulative average cost of unit 200 and multiply this value by 200. This is a much easier calculation than in the case of the unit curve.

\overline{Y_x} = K\frac{\frac{1}{1+\log_2b}x^{1+\log_2{b}}}{x}


  • \ K is the number of direct labour hours to produce the first unit
  • \ Y_x is the average number of direct labour hours to produce First xth units
  • \ x is the unit number
  • \ b is the learning percentage

The experience curve

The experience curve effect is broader in scope than the learning curve effect encompassing far more than just labor time. It states that the more often a task is performed, the lower will be the cost of doing it. The task can be the production of any good or service. Each time cumulative volume doubles, value added costs (including administration, marketing, distribution, and manufacturing) fall by a constant and predictable percentage.

In the late 1960s Bruce Henderson of the Boston Consulting Group (BCG) began to emphasize the implications of the experience curve for strategy.

Research by BCG in the 1970s observed experience curve effects for various industries that ranged from 10 to 25 percent.

These effects are often expressed graphically. The curve is plotted with cumulative units produced on the horizontal axis and unit cost on the vertical axis. A curve that depicts a 15% cost reduction for every doubling of output is called an “85% experience curve�, indicating that unit costs drop to 85% of their original level.

Mathematically the experience curve is described by a power law function sometimes referred to as Henderson's Law:

  • \ C_n = C_1 n^{-a}


  • \ C_1 is the cost of the first unit of production
  • \ C_n is the cost of the nth unit of production
  • \ n is the cumulative volume of production
  • \ a is the elasticity of cost with regard to output

Reasons for the effect


NASA quotes the following experience curves:

The primary reason for why experience and learning curve effects apply, of course, is the complex processes of learning involved. As discussed in the main article, learning generally begins with making successively larger finds and then successively smaller ones. The equations for these effects come from the usefulness of mathematical models for certain somewhat predictable aspects of those generally non-deterministic processes. They include:

  • Labour efficiency - Workers become physically more dexterous. They become mentally more confident and spend less time hesitating, learning, experimenting, or making mistakes. Over time they learn short-cuts and improvements. This applies to all employees and managers, not just those directly involved in production.
  • Standardization, specialization, and methods improvements - As processes, parts, and products become more standardized, efficiency tends to increase. When employees specialize in a lim

Learning curve

A learning curve is a graphical representation of the changing rate of learning (in the average person) for a given activity or tool. Typically, the increase in retention of information is sharpest after the initial attempts, and then gradually evens out, meaning that less and less new information is retained after each repetition.

The learning curve can also represent at a glance the initial difficulty of learning something and, to an extent, how much there is to learn after initial familiarity. For example, the Windows program Notepad is extremely simple to learn, but offers little after this. On the other extreme is the UNIX terminal editor vi, which is difficult to learn, but offers a wide array of features to master after the user has figured out how to work it. It is possible for something to be easy to learn, but difficult to master or hard to learn with little beyond this.

Learning curve in psychology and economics

The first person to describe the learning curve was Hermann Ebbinghaus in 1885. He found that the time required to memorize a nonsense word increased sharply as the number of syllables increased. Psychologist Arthur Bills gave a more detailed description of learning curves in 1934. He also discussed the properties of different types of learning curves, such as negative acceleration, positive acceleration, plateaus, and ogive curves. In 1936, Theodore Paul Wright described the effect of learning on labor productivity in the aircraft industry and proposed a mathematical model of the learning curve.

The economic learning of productivity and efficiency generally follows the same kinds of experience curves and have interesting secondary effects. Efficiency and productivity improvement can be considered as whole organization or industry or economy learning processes, as well as for individuals. The general pattern is of first speeding up and then slowing down, as the practically achievable level of methodology improvement is reached. The effect of reducing local effort and resource use by learning improved methods paradoxically often has the opposite latent effect on the next larger scale system, by facilitating its expansion, or economic growth, as discussed in the Jevons paradox in the 1880s and updated in the Khazzoom-Brookes Postulate in the 1980s.

Broader interpretations of the learning curve

Initially introduced in educational and behavioral psychology, the term has acquired a broader interpretation over time, and expressions such as "experience curve", "improvement curve", "cost improvement curve", "progress curve", "progress function", "startup curve", and "efficiency curve" are often used interchangeably. In economics the subject is rates of "development", as development refers to a whole system learning process with varying rates of progression. Generally speaking all learning displays incremental change over time, but describes an "S" curvewhich has different appearances depending on the time scale of observation. It has now also become associated with the evolutionary theory ofpunctuated equilibrium and other kinds of revolutionary change in complex systems generally, relating to innovation, organizational behavior and the management of group learning, among other fields. These processes of rapidly emerging new form appear to take place by complex learning within the systems themselves, which when observable, display curves of changing rates that accelerate and decelerate.

Common terms

The familiar expression "steep learning curve" may refer to either of two aspects of a pattern in which the marginal rate of required resource investment is initially low, perhaps even decreasing at the very first stages, but eventually increases without bound.

Early uses of the metaphor focused on the pattern's positive aspect, namely the potential for quick progress in learning (as measured by, e.g., memory accuracy or the number of trials required to obtain a desired result) at the introductory or elementary stage. Over time, however, the metaphor has become more commonly used to focus on the pattern's negative aspect, namely the difficulty of learning once one gets beyond the basics of a subject.

In the former case, the "steep[ness]" metaphor is inspired by the initially high rate of increase featured by the function characterizing the overall amount learned versus total resources invested (or versus time when resource investment per unit time is held constant)—in mathematical terms, the initially high positive absolute value of the first derivative of that function. In the latter case, the metaphor is inspired by the pattern's eventual behavior, i.e., its behavior at high values of overall resources invested (or of overall time invested when resource investment per unit time is held constant), namely the high rate of increase in the resource investment required if the next item is to be learned—in other words, the eventually always-high, always-positive absolute value and the eventually never-decreasing status of the first derivative of that function. In turn, those properties of the latter function dictate that the function measuring the rate of learning per resource unit invested (or per unit time when resource investment per unit time is held constant) has a horizontal asymptote at zero, and thus that the overall amount learned, while never "plateauing" or decreasing, increases more and more slowly as more and more resources are invested.

This difference in emphasis has led to confusion and disagreements even among learned people.

The most effective solution to problems arising from a steep learning curve is to find a different method of learning that features a differently shaped (or at least less steep) curve. Such a discovery, often characterized as an aha!' moment" or "breakthrough", often results from a seemingly radical intuitive change in direction.


Yield curve

Not to be confused with Yield-curve spread - seeZ-spread

In finance, the yield curve is the relation between the interest rate (or cost of borrowing) and the time to maturity of the debt for a given borrower in a given currency. For example, the U.S. dollar interest rates paid on U.S. Treasury securities for various maturities are closely watched by many traders, and are commonly plotted on a graph such as the one on the right which is informally called "the yield curve." More formal mathematical descriptions of this relation are often called the term structure of interest rates.

The yield of a debt instrument is the overall rate of return available on the investment. For instance, a bank account that pays an interest rate of 4% per year has a 4% yield. In general the percentage per year that can be earned is dependent on the length of time that the money is invested. For example, a bank may offer a "savings rate" higher than the normal checking account rate if the customer is prepared to leave money untouched for five years. Investing for a period of time t gives a yield Y(t).

This functionY is called the yield curve, and it is often, but not always, an increasing function of t. Yield curves are used by fixed income analysts, who analyze bonds and related securities, to understand conditions in financial markets and to seek trading opportunities. Economists use the curves to understand economic conditions.

The yield curve function Y is actually only known with certainty for a few specific maturity dates, while the other maturities are calculated by interpolation (seeConstruction of the full yield curve from market data below).

The typical shape of the yield curve

Yield curves are usually upward sloping asymptotically: the longer the maturity, the higher the yield, with diminishing marginal increases (that is, as one moves to the right, the curve flattens out). There are two common explanations for upward sloping yield curves. First, it may be that the market is anticipating a rise in the risk-free rate. If investors hold off investing now, they may receive a better rate in the future. Therefore, under the arbitrage pricing theory, investors who are willing to lock their money in now need to be compensated for the anticipated rise in rates—thus the higher interest rate on long-term investments.

However, interest rates can fall just as they can rise. Another explanation is that longer maturities entail greater risks for the investor (i.e. the lender). A risk premium is needed by the market, since at longer durations there is more uncertainty and a greater chance of catastrophic events that impact the investment. This explanation depends on the notion that the economy faces more uncertainties in the distant future than in the near term. This effect is referred to as the liquidity spread. If the market expects more volatility in the future, even if interest rates are anticipated to decline, the increase in the risk premium can influence the spread and cause an increasing yield.

The opposite position (short-term interest rates higher than long-term) can also occur. For instance, in November 2004, the yield curve for UK Government bonds was partially inverted. The yield for the 10 year bond stood at 4.68%, but was only 4.45% for the 30 year bond. The market's anticipation of falling interest rates causes such incidents. Negative liquidity premiums can exist if long-term investors dominate the market, but the prevailing view is that a positive liquidity premium dominates, so only the anticipation of falling interest rates will cause an inverted yield curve. Strongly inverted yield curves have historically preceded economic depressions.

The shape of the yield curve is influenced by supply and demand: for instance, if there is a large demand for long bonds, for instance from pension funds to match their fixed liabilities to pensioners, and not enough bonds in existence to meet this demand, then the yields on long bonds can be expected to be low, irrespective of market participants' views about future events.

The yield curve may also be flat or hump-shaped, due to anticipated interest rates being steady, or short-term volatility outweighing long-term volatility.

Yield curves continually move all the time that the markets are open, reflecting the market's reaction to news. A further "stylized fact" is that yield curves tend to move in parallel (i.e., the yield curve shifts up and down as interest rate levels rise and fall).

Types of yield curve

There is no single yield curve describing the cost of money for everybody. The most important factor in determining a yield curve is the currency in which the securities are denominated. The economic position of the countries and companies using each currency is a primary factor in determining the yield curve. Different institutions borrow money at different rates, depending on their creditworthiness. The yield curves corresponding to the bonds issued by governments in their own currency are called the government bond yield curve (government curve). Banks with high credit ratings (Aa/AA or above) borrow money from each other at the LIBOR rates. These yield curves are typically a little higher than government curves. They are the most important and widely used in the financial markets, and are known variously as the LIBOR curve or the swap curve. The construction of the swap curve is described below.

Besides the government curve and the LIBOR curve, there are corporate (company) curves. These are construc

Horizontal plane

In geometry, physics, astronomy, geography, and related sciences and contexts, a planeis said to be horizontal at a given point if it is locally perpendicular to thegradient of the gravityfield, i.e., with the direction of the gravitational force (per unit mass) at that point.

In radio science, horizontal plane is used to plot an antenna's relative field strength in relation to the ground (which directly affects a station's coverage area) on a polar graph. Normally the maximum of 1.000 or 0 dB is at the top, which is labeled 0o, running clockwise back around to the top at 360°. Other field strengths are expressed as a decimal less than 1.000, a percentage less than 100%, or decibels less than 0 dB. If the graph is of an actual or proposed installation, rotation is applied so that the top is 0otrue north. See also the perpendicular vertical plane.

In general, something that is horizontal can be drawn from left to right (or right to left), such as the x-axis in the Cartesian coordinate system.


Although the word horizontal is common in daily life and language (see below), it is subject to many misconceptions. The precise definition above and the following discussion points will hopefully clarify these issues.

  • The concept of horizontality only makes sense in the context of a clearly measurable gravity field, i.e., in the 'neighborhood' of a planet, star, etc. When the gravity field becomes very weak (the masses are too small or too distant from the point of interest), the notion of being horizontal loses its meaning.
  • In the presence of a simple, time-invariant, rotationally symmetric gravity field, a plane is horizontal only at the reference point. The horizontal planes with respect to two separate points are not parallel, they intersect.
  • In general, a horizontal plane will only be perpendicular to a vertical direction if both are specifically defined with respect to the same point: a direction is only vertical at the point of reference. Thus both horizontality and verticality are strictly speaking local concepts, and it is always necessary to state to which location the direction or the plane refers to. Note that (1) the same restriction applies to the straight lines contained within the plane: they are horizontal only at the point of reference, and (2) those straight lines contained in the plane but not passing by the reference point are not horizontal anywhere.
  • In reality, the gravity field of a heterogeneous planet such as Earth is deformed due to the inhomogeneous spatial distribution of materials with different densities. Actual horizontal planes are thus not even parallel even if their reference points are along the same vertical direction.
  • At any given location, the total gravitational force is a function of time, because the objects that generate the reference gravity field move relative to each other. For instance, on Earth, the local horizontal plane at a given point (as materialized by a pair of spirit levels) changes with the relative position of the Moon (air, sea and land tides).
  • Furthermore, on a rotating planet such as Earth, there is a difference between the strictly gravitational pull of the planet (and possibly other celestial objects such as the Moon, the Sun, etc.), and the apparent net force applied (e.g., on a free-falling object) that can be measured in the laboratory or in the field. This difference is due to the centrifugal force associated with the planet's rotation. This is a fictitious force: it only arises when calculations or experiments are conducted in non-inertial frames of reference.

Practical use in daily life

The concept of a horizontal plane is thus anything but simple, although, in practice, most of these effects and variations are rather small: they are measurable and can be predicted with great accuracy, but they may not greatly affect our daily life.

This dichotomy between the apparent simplicity of a concept and an actual complexity of defining (and measuring) it in scientific terms arises from the fact that the typical linear scales and dimensions of relevance in daily life are 3 orders of magnitude (or more) smaller than the size of the Earth. Hence, the world appears to be flat locally, and horizontal planes in nearby locations appear to be parallel. Such statements are nevertheless approximations; whether they are acceptable in any particular context or application depends on the applicable requirements, in particular in terms of accuracy.

In graphical contexts, such as drawing and drafting on rectangular paper, it is very common to associate one of the dimensions of the paper with a horizontal, even though the entire sheet of paper is standing on a flat horizontal (or slanted) table. In this case, the horizontal direction is typically from the left side of the paper to the right side. This is purely conventional (although it is somehow 'natural' when drawing a natural scene as it is seen in reality), and may lead to misunderstandings or misconceptions, especially in an educational context.

From Yahoo Answers

Question:Here is my question that I have NO IDEA how to answer :P It would be great if you could help me out, thank you! A ball is thrown out of a window at a horizontal speed of 20 m/s. The window is 45 metres from the ground. The ball has a mass of 6.2 kg. Assume that gravitational acceleration of 9.8 metres per second squared. How much horizontal distance, in centimetres, will the ball travel from the house before hitting the ground?

Answers:Treat x and y separately (break it down to components) First solve for t Yf=Yo + vt + 1/2*a*t^2 0=45m + 0*t + 1/2(-9.8)t^2 -45/-4.9 = t^2 t = 3.03 s to hit the ground Then solve for x Xf=Xo + vt + 1/2*a*t^2 Xf=0 + 20(3.03) + 0 Xf = 60.6 m ---> convert to cm **mass has nothing to do with it

Question:How do i find the intersection of two lines (one following y=ax^2+bx+c and the other y=ax+b) with a calculator or other technology. I know how to do this by calculating, but is there any way i can just insert these into a calculator or something and get the answers...?

Answers:You can use matlab or excel. You specify the constants,the equations, set an interval for x to run and just set the two equations equal to each other. You may need a "for" loop and a logic function in the end (outside the loop) that is "yes" when they intersect and "no" when they don't. e.g. y1=ax^2+bx+c y2=ax+b a=.. b=.. c=.. for 0
Question:I am studying for upcoming exams. And in previous exams you need to calculate the horizontal speed of a ball, and later it asks "Identify the assumption needed to calculate vertical acceleration" Answers will be much appreciated

Answers:My guess is effect of air resistance can be ignored so vertical acceleration depends only on gravity.

Question:I do not know the function of the normal distribution curve used to calculate z- scores and their corresponding area under the curve as percentiles.. Is it something like f(x) = ln(x) or something --> i know thats not it just providing a template for the answer i have been searching for. From there i need to find the definite integral from - infinite; to -2, for x, i know this should be .05 i just need the function to perform the equation/proof for an upcoming test, and i know how to find the definite integral i just had a question about the function

Answers:Refer to this link to think about the normal distribution curve: http://www.netmba.com/statistics/distribution/normal/ You will need this to answer your problem! Make sure to know how you will approach this problem.

From Youtube

Horizontal and Vertical Tangent Lines to Polar Curves :This video explains how to determine the points on a polar curve where there are horizontal and vertical tangent lines. mathispower4u.yolasite.com

25. Adding Demand and Supply Curves Horizontally :This video demonstrates the concept of summing horizontally to obtain market demand and supply curves using a concrete example. In particular, I show how to aggregate the demands of two types of consumers to a market demand, and how to aggregate the supply curves of identical firms to a market supply. At the end, I find the equilibrium price and quantity. The video is a nice application of various tools in the microeconomist's toolkit. For a complete listing of my videos, check out my video website: sites.google.com