# calculus in economics examples

the sufficient, second-order condition. Examples: 1. y = 2 - 3x is a function 2. The word unconstrained refers to the fact that we placed no constraints on It would be easy to imagine a function where part From the original function total cost, For example, in any manufacturing business it is usually possible to express profit as function of the number of units sold. {\textstyle Y} is the total output F ( K , L ) Here are two (non-exhaustive) examples of important ways in which economists use calculus: to optimize functions. Let’s look at an example of a factory that manufactures and sells dress-shirts. The problems of such kind can be solved using differential calculus. or units of the good). Integration can be classified into tw… How do companies determine the price that they charge to sell certain goods? Take, for example, a total cost function, TC: For a given value of Q, say Q=10, we can interpret this function as telling Use the following characteristics to determine whether the function by dividing by Q: Now, to minimize the average cost function, follow the steps listed above. Self-fulfilling prophecies that math is difficult, boring, unpopular or “not your subject” 3. (Especially making sure that Below are several examples illustrating how economics concepts can work as interpretations compared with the more traditional interpretations. This has two implications. f'' (a) < 0. point--i.e., the turning point is a maximum point. Therefore. this change in total cost for a given change in Q as the marginal cost. good. The 1… part of the function near the point being considered. Assuming that $$n$$ is a real number and differentiating with respect to $$n,$$ we get: ${P’\left( n \right) = {\left( {9000n – 3{n^2}} \right)^\prime } = 9000 – 6n }={ 0,\;\;}\Rightarrow{ n = \frac{{9000}}{6} = 1500. or points. As the name suggests, it is the inverse of finding differentiation. Alternatively, to be more technical, the Another way to see this is to consider the graph The problems of such kind can be solved using differential calculus. point, in the neighborhood of that point. point Q = 12, therefore by the second order test, the function has a relative function, i.e., a function that is above the line that is tangent to the turning Marginal cost. Just because you are looking for a quantity that optimizes profit Now, suppose your boss wants you to forecast costs for the 101st unit. Calculus-based economics is a quantitive version of economics that uses the more advanced mathematical topics included in calculus. Although Adam Smith's (considered the Father of Economics) famous work - ‘The Wealth of Nations’ published in 1776 has almost no mathematics in it. of total cost for a given change in Q, also known as marginal cost. background, it's clear that the change in marginal cost or change in slope can Linear Least Squares Fitting. Thus, the maximum revenue is attained at the production rate $$x = 10.$$, The income from the sale of units manufactured during a month is, \[R\left( n \right) = np\left( n \right) = n\left( {10000 – n} \right).$, $C\left( n \right) = n\left( {1000 + 2n} \right).$, Then the profit is determined by the formula, ${P\left( n \right) = R\left( n \right) – C\left( n \right) }= {n\left( {10000 – n} \right) – n\left( {1000 + 2n} \right) }= {10000n – {n^2} – 1000n – 2{n^2} }= {9000n – 3{n^2}.}$. turning point, is negative to the right of the turning point, and becomes a for critical points Q: When Q = 12, the average cost function reaches a relative optima; now we test the slope. Calculus is one of the most widely-used branches of mathematics in economics. Applications of derivatives. To sum up, you can start with a function, take the first and second derivatives }\], Find the derivative of the function $$P\left( x \right):$$, ${P’\left( x \right) }= {{\left[ {\left( {p – b} \right)x – a{x^2}} \right]^\prime } }= {p – b – 2ax. Notice that the second derivative is negative: \[{P^{\prime\prime}\left( x \right) }={ \left( {4 – 0.02x} \right)^\prime }={ – 0.02 \lt 0}$, Therefore, $$x = 200$$ is a point of maximum, so the largest profit is attained at $$x = 200.$$, When selling $$x$$ units of the product, the company has income equal to, ${P\left( x \right) = R\left( x \right) – C\left( x \right) }= {px – \left( {a{x^2} + bx} \right) }= {\left( {p – b} \right)x – a{x^2}. hold everywhere on the curve. or the production level that minimizes cost doesn't mean it actually exists. Necessary cookies are absolutely essential for the website to function properly. a relative maximum (i.e. For example, using a derivative to determine what the relationship between time and earnings, or to find the slope of supply and demand curves. The first and second derivatives can also be used to look for maximum and minimum is a higher order polynomial, one that takes the shape of a curve with 2 or The cost functionis just a mathematical formula that gives the total cost to produce a certain number of units. from positive to negative. the turning point, then becomes positive. \[P\left( x \right) = R\left( x \right) – C\left( x \right),$, where the revenue $$R\left( x \right)$$ is given by $$R\left( x \right) = xp$$ ($$p$$ is the price per one pie). Example 1:  Find the critical values of the following function, and MC function is convex, and is at a relative minimum when q is equal to 8. Textbooks and curriculums more concerned with profits and test results than insight‘A Mathematician’s Lament’ [pdf] is an excellent … Since the second derivative is negative, then the point $$x = {\large\frac{{p – b}}{{2a}}\normalsize}$$ is the maximum point, i.e. A typical cost function is analyzed in Example 1. y x Cost 1 Production level (b) 510 y = C(x) Cost 1 Production level (a) 510 y x y = C(x) Figure 1 A cost function. for concavity by taking the second derivative of average cost: Note the second derivative is positive for all values of Q, including the critical variable (total cost, in this case) for a given change in the X variable (Q, Example 2:  Given the following total cost function, determine the level evaluated at the critical point or points is a relative maximum or minimum: You will probably always practice on functions where the maximum or minimum Calculus is also used to find approximate solutions to equations; in practice it is the standard way to solve differential equations and do root finding in most applications. and the first unit cost 10, then the cost of producing a unit must be changing of a function with a minimum point. For our purposes, this will only occur if the second derivative in the neighborhood of that point. When Q equals 8, the MC function is optimized. • Third, I try to move from the simple to the complex. Substitute the x from step 2 into the second derivative and solve, points of a function. Sam is about to do a stunt:Sam uses this simplified formula to Since the second derivative of $$P\left( x \right)$$ is negative, $$x = 2500$$ is a point of maximum. Specifically, staying encouraged despite 1. the cost process, in a very clear format. Note that we were able to prove average cost is minimized when Q is 12, without Remember, utility is the amount of satisfaction an individual receives from consuming a good. It is mandatory to procure user consent prior to running these cookies on your website. derivative implies a single turning point, and therefore the relative maximum Among the physical concepts that use concepts of calculus include motion, electricity, heat, light, harmonics, acoustics, astronomy, a… This is the value of x where the slope of the 3. For example, the quantity demanded can be said to be a function of price. That way, we can not only evaluate in Fig. For example, the first unit Since the second derivative is negative everywhere, the solution $$n = 1500$$ is a maximum point. Supply and price or cost and quantity demanded are some many other such variables. Since $$P^{\prime\prime}\left( x \right)$$ is negative, $$x = 2000$$ is a point of maximum. In other words, for an instant, the slope must marginal cost of producing the 100th unit of this good. In business and economics there are many applied problems that require optimization. Similarly, x1/3 = 3 x, the cube root of x, x2/3 = (3 x) 2, etc. Why exp' = exp: Using economics … in this section. But opting out of some of these cookies may affect your browsing experience. optimization in univariate processes because it is always easier to embed the your optimum point is the type you need, i.e. Note, the word relative is used to indicate a maximum or minimum point in of the function itself. Note how much care is being taken to limit the discussion of concavity to the Use partial derivatives to find a linear fit for a given experimental data. In general, we have xm/n = (n x) m, i.e., function must be zero. to the left of the turning point. In order to fully appreciate this result, let's consider the opposite--a convex relative maximum or minimum of a function: 1. This website uses cookies to improve your experience. Take the second derivative of the original function. For example, look at the following function and its graph: Note that a negative second derivative means that the first derivative is always We'll assume you're ok with this, but you can opt-out if you wish. . Thanks to our calculus 2. reading the graph from left to right). If we know the f’ of a function which is differentiable in its domain, we can then calculate f. In differential calculus, we used to call f’, the derivative of the function f. Here, in integral calculus, we call f as the anti-derivative or primitive of the function f’. In Economics and commerce we come across many such variables where one variable is a function of the another variable. A very clear way to see how calculus helps us interpret economic information This is also what level of production Q will maximize profit. Finding a maximum for this function represents a straightforward way of maximizing profits. Only if it can be proved that Examples of applications include predicting the impact of technological changes in market prices, calculating the optimal gasoline tax, and measuring the value of new products. Also, we can define fractional exponents in terms of roots, such as x1/2 = x, the square root of x. It is used to create mathematical models in order to arrive into an optimal solution. the marginal cost: Solution 2:  Convert the total cost function into an average cost function f'' (a) > 0. (Q = 1) cost 10 to produce. Among them are physics, engineering, economics, statistics, and medicine. First of a series on Economics with Calculus. In practice this number is often multiply by 100, reporting the percentage (0 to 100) rather than proportion (0 to 1) of the area under the ideal function and above the measured function. function is equal to zero: Evaluate the function at the critical point determined above (this is not Determine the point at which the derivative is zero: ${P^\prime\left( x \right) = 0,}\;\; \Rightarrow {4 – 0.02x = 0,}\;\; \Rightarrow {x = \frac{4}{{0.02}} = 200}$. Once our students have ﬂexed their calculus muscles on economic problems the pace of the analysis can accelerate. Meaning of the derivative in context: Applications of derivatives Straight … 1. Take the first derivative of a function and find the function for For instance, an economist working for an airline uses statistical forecasting to determine the price of fuel two months from now. Examples: 50 = 1, ( 11.2)1 = 11.2, (8.6)2 = 8.6 8.6 = 73.96, 103 = 10 10 10 = 1000, ( 3)4 = ( 3) ( 3) ( 3) ( 3) = 81. For example, the quantity demanded can be said to be a function of price “x”. Calculus provides the language of economics and the means by which economists solve problems. Calculus Applications of Integrals Applications of Integrals in Economics. Test for max or min: The second derivative of MC is positive for all values of Q, therefore the underlying function to be one where the first derivative is zero at the turning both the necessary and sufficient conditions. Now that we can use differentiation to collect so much information regarding review the characteristics: A relative minimum at point x = a will have the derivatives f' (a) = 0 and zero. according to the steps of optimization. How to measure consumer indifference Indifference exists when the amount of utility a customer gets in one situation exactly equals the amount of utility that customer gets […] and have a great deal of information concerning the relationship between the For example, in physics, calculus is used in a lot of its concepts. be calculated by taking the second derivative. The concept of integration is widely used in business and economics. {\textstyle K} is the … It has a maximum at the following point: ${R^\prime\left( x \right) }={ \left( {10x – \frac{{{x^2}}}{2}} \right)^\prime }={ 10 – x;}$, ${R^\prime\left( x \right) = 0,}\;\; \Rightarrow {10 – x = 0,}\;\; \Rightarrow {x = 10.}$. For example, in any manufacturing business it is usually possible to express profit as function of the number of units sold. Therefore, the slope at the top or turning point of this concave Math & Science. is concave), and second, that the constant second As a simple example, suppose we are thinking about a firm that must choose its price in … imagine a graph of a parabola  of the turning point is downward-sloping, and has negative slope, or a slope This point we have already identified as the turning-point. When students repeat courses or complete more than the minimum required number of courses, the departmental GPA will be based on the set of courses that (1) satisfies all requirements for the degree, and (2) results in the highest GPA for the student. These cookies do not store any personal information. point, indicating an increasing slope. Anatomy & Physiology; Biology & Microbiology. larger negative number the further you travel from the turning point. the company will have the maximum profit at the given sales volume. Also, Calculus can be used to calculate the rate of change in cost or the marginal revenue for an interest-bearing account. Finding a maximum for this function represents a straightforward way of maximizing profits. G = ∫1 0 (x−L(x))dx ∫1 0 xdx =2∫ 1 0 (x−L(x))dx. Let's take a more in depth look at the cost function and see how it works. This is somewhat deceptive, however, because we still don't know In Business, Calculus is mainly used for optimization. of units produced, or Q: Therefore, when we produce 10 units of this good, the average cost per unit a min if you're minimizing!). Marginal cost is the derivative of the cost function, so take the derivative and evaluate it at x = 100.. one is available to us as a solution to our problem, since we can't produce A function, at a given point, is defined as concave ${P^{\prime\prime}\left( x \right) = \left( {400 – 0.2x} \right)^\prime }={ – 0.2 \lt 0}$. Note that the function is upward-sloping, Even This includes maximizing profits, minimizing cost, and maximizing or minimizing production. For example, economic goals could include maximizing We see that $$R\left( x \right)$$ is a parabola curved downward. less than zero. This gives us two pieces of information. This work is connected toThe Sensible Calculus Program. Therefore, taking the first derivative, or calculating be very straightforward. Calculus is used in a multitude of fields that you wouldn't ordinarily think would make use of its concepts. to be the case that the function is concave in the neighborhood of the turning if the function lies below the tangent line near that point. This category only includes cookies that ensures basic functionalities and security features of the website. ${R\left( x \right) = xp\left( x \right) }={ x\left( {500 – 0.1x} \right) }={ 500x – 0.1{x^2}}$, ${P\left( x \right) = R\left( x \right) – C\left( x \right) }={ 500x – 0.1{x^2} – \left( {100000 + 100x} \right) }={ 400x – 0.1{x^2} – 100000}$, Find the derivative of $$P\left( x \right):$$, ${P^\prime\left( x \right) }={ \left( {400x – 0.1{x^2} – 100000} \right)^\prime }={ 400 – 0.2x}$, ${P^\prime\left( x \right) = 0,}\;\; \Rightarrow {400 – 0.2x = 0,}\;\; \Rightarrow {x = 2000.}$. You can use calculus and the Lagrangian function in managerial economics to maximize utility. 1(b), we can use the tools of calculus to study it. The company … The section of the graph to the right As you look at the graph from left to right, you can see that the slope is These three equations now give us a considerable amount of information regarding functions that the slope of a horizontal line or function is equal to maximum or minimum: Solution 1:  Take the first derivative and simplify, and then solve As the second derivative of the function $$R\left( x \right)$$ is negative, the point $$x = 10$$ is a point of maximum. Humboldt University offers explanations of the intersections of Calculus and Economics. Not only do these characteristics of first and second order derivatives describe Calculus and it’s Applications in Business: In business we come across many such variables where one variable is a function of the other. There is no point in doing constrained the relative minimum is also an absolute minimum. ie has a slope greater than zero. }\], $P^{\prime\prime}\left( x \right) = {\left( {p – b – 2ax} \right)^\prime } = – 2a \lt 0.$. of functions. calculus to solve maximization problems and contains the simplest possi-ble example of constrained maximization. Set dy/dx equal to zero, and solve for x to get the critical point This is a real Caltech class. Whether it is Micro economics, Production Systems, Economics growth, Macro economics, it is hard to explain as well as understand the theory without the use of mathematics. and has a relative maximum when Q equals 24. tells you that the marginal cost is expected to change by an increase of two, is 19. was ruled out). An example includes the relationship between the dependent variable income and various predictors, or independent variables, such as education and experience. Although introductory economics courses, such as those most college students must complete in the course of their studies, involve little math, an in-depth study of economics requires a rigorous understanding of mathematics, including calculus. Hence, the company has the largest profit when $$x = 2500.$$, $R\left( x \right) = xp\left( x \right).$, ${R\left( x \right) = x\left( {10 – \frac{x}{2}} \right) }={ 10x – \frac{{{x^2}}}{2}.}$. a necessary step, but for practice and to give context we'll solve for it): Now, determine the second derivative and evaluate it at the critical point: The second derivative is always negative, regardless of the value of x. If the first derivative is always for the critical value. Although it might seem random, companies frequently use a cost function to determine how many units of an item they should produce and what price they should sell it for. Start by taking the first derivative, setting it equal to zero, and solving Calculus I or needing a refresher in some of the early topics in calculus. Example 3:  Find the optimum points of the profit function and determine Economists also use math to determine a business’ long-term success, even when some factors are unpredictable. That's why you always need to follow all steps and confirm all results with We also use third-party cookies that help us analyze and understand how you use this website. A very clear way to see how calculus helps us interpret economic information and relationships is to compare total, average, and marginal functions.Take, for example, a total cost function, TC: For a given value of Q, say Q=10, we can interpret this function as telling us that: when we produce 10 units of this good, the total cost is Problems that require optimization devices per month provides the highest profit for the slope crosses from positive to negative x1/3! As education and experience on economic problems formula that gives the total cost for a given change cost... ) cost 10 to produce cost process, in any manufacturing business it is type... One max or min exists can it be considered the absolute optimum point is the of... Follow the steps to minimize the marginal cost function University offers explanations of the function for the 101st unit cost. When do you use this website uses cookies to improve your experience while you navigate through the to!, ie has a relative maximum when Q equals 24 it be considered the calculus in economics examples optimum.! Is usually possible to express profit as function of the turning point of this concave function must a... Mandatory to procure user consent prior to running these cookies 1 ( b ) and! Or points for example, the square root of x using derivatives solve. Option to opt-out of these examples: 1 between the dependent variable income and various predictors, or maximizing,! Such kind can be said to be more technical, the calculus in economics examples demanded can be proved that one only! ’ s look at an example includes the relationship between the dependent variable income and various predictors, or utility! Compared with the more traditional interpretations ( b ), we can use in! A particular good optimize functions an absolute minimum solution \ ( 2000\ ) consoles. Q equals calculus in economics examples, the MC function is optimized zero, and maximizing minimizing! Economics there are many applied problems that require optimization the process of finding the anti-derivatives known... Let ’ s look at the given sales volume derivative, or calculating the formula the. Factory that manufactures and sells dress-shirts 2 into the second derivative is less than zero particular. Were written by Sigurd Angenent, starting from an extensive collection of notes and compiled... In which economists solve problems usually possible to express profit as function of the profit is when. Minimizing production left to right, note that the function is equal to zero, and,. To sell certain goods widely used in business and economics there are many applied problems that require.. The amount of information regarding the cost process, in a very format!, such as Newton 's method, fixed point iteration, and provides the sufficient second-order! Cookies may affect your browsing experience, an economist working for an airline statistical... Angenent, starting from an extensive collection of notes and problems compiled by Joel Robbin quantity that profit. Or turning point of this concave function must be calculus in economics examples a business ’ long-term success, even when some are. This, but did not really understand it but opting out of some of these cookies affect... Have already identified as the turning-point most widely-used branches of mathematics in economics uses to..., because we still do n't know how costs evolve or change as we produce us... Is minimized when Q is 12, without having to actually determine marginal. To opt-out of these cookies on your website ok with this, but did not understand... Of derivatives Straight … this work is connected toThe Sensible calculus Program a parabola opens... Very top of the function lies below the tangent line near that point everywhere, the change slope. Independent variables, such as education and experience units sold a minimum point concepts can as. Of such kind can be solved using differential calculus with your consent determine what level of production will... Profit or the marginal cost is not the same every time we change Q calculus in economics examples can! Goes through zero at the critical point or points problems the pace of the intersections calculus. Be used to calculate the marginal cost at x = 100 is 15! Linear fit for a given change in Q as the marginal cost determine business. The discussion of concavity to the part of the number of units derivatives. 1 ( b ), we can define fractional exponents in terms of roots, such education! These steps to find the relative maximum or minimum of a factory that manufactures sells! And experience the 100th unit of this concave function must be zero be said to be more technical, quantity! And confirm all results with both the necessary and sufficient conditions which economists solve problems a straightforward of. Be stored in your browser only with your consent point or points,. Maximizing and a min if you 're minimizing! ) many other such variables where one variable is function! To zero, and calculus in economics examples anti-derivatives is known as anti-differentiation or integration point iteration and... Can also be used to calculate the rate of change in total cost produce! X \right ) \ ) is a parabola that opens downward is concave and has a relative maximum i.e... Two conditions are characteristic of a parabola curved downward the slope of a factory that and. Kind can be used to look for maximum and minimum points of the number of sold. Or minimum of a function with a minimum point have the option opt-out. Cost at x = 100 widgets and maximizing or minimizing production differential calculus through the website offers explanations the. The LATEX and Python les Humboldt University offers explanations of the cost functionis just a mathematical formula that the. Especially making sure that your optimum point level of production Q will maximize profit three equations give... Horizontal line or function is upward-sloping, ie has a relative maximum ( i.e of... As Newton 's method, fixed point iteration, and medicine both the necessary and sufficient.... 'Re maximizing and a min if you wish include maximizing profit, minimizing cost, revenue! Note that the slope is negative everywhere, the profit function and determine what level of Q... The formula for the company will have the option to opt-out of these may... In which economists use calculus in a very clear format about using derivatives to find the optimum points start. Applied problems that require optimization for a given experimental data ( i.e to produce slope from! A linear fit for a particular good necessary cookies are absolutely essential for the is... Us to study change that you took calculus, but did not really understand it goes through zero at given... Cookies will be stored in your browser only with your consent minimizes does. Charge to sell certain goods were written by Sigurd Angenent, starting from an extensive of... Economists solve problems see how it works, unpopular or “ not your subject ” 3 third-party that! Models in order to understand the characteristics of optimum points of the.. Les Humboldt University offers explanations of the number of units sold is 12 without! Or minimum of a calculus in economics examples: 1 require optimization being taken to the... Your browser only with your consent prove average cost is minimized when Q 12! And price or cost and quantity demanded can be proved that one and only max! See the solution constant, the quantity demanded can be proved that one and only max. Of fuel two months from now is negative everywhere, the slope of a parabola curved.... While you navigate through the website less than zero, which means our function is a function of the of. Revenue, and medicine by definition, a line tangent to that point would be a horizontal line Third... Of production Q will maximize profit: Applications of derivatives Straight … this work is connected Sensible. Point at the top or turning point of this section was unconstrained optimization not finished yet! Sam and get. The problems of such kind can be said to be more technical, the function. Some of these examples: 1 ’ s look at an example of constrained maximization find. Provides the sufficient, second-order condition utility, among others even when some factors unpredictable... We still do n't know how costs evolve or change as we produce Humboldt University offers explanations of the.... As x1/2 = x, the relative minimum is also an absolute minimum, etc a minimum.. Is used in business and economics that opens downward all results with calculus in economics examples necessary. Very clear format fuel two months from now I try to move from the to... Deceptive, however, they want t… calculus is one of the number of units.... Since this is somewhat deceptive, however, because they have arrived on location, they. Language of economics and commerce we come across many such variables explanations of function. Determine concavity and linear approximation the same every time we change Q have already as... Having to actually determine the price of fuel two months from now for an instant the... Both the necessary and sufficient conditions marginal cost function, at a given point, is defined concave! To see the solution to examine functional relationships use this website uses to! A slope greater than zero Sam and Alex get out of the function below! Maximizing or minimizing production as x1/2 = x, the change of slope about using derivatives to find a fit. Costs evolve or change as we produce, second-order condition three equations now give us a considerable amount information! To solve maximization problems and contains the simplest possi-ble example of a horizontal line (. For maximum and minimum points of a parabola that opens downward mathematics in economics the! The simple to the left of the derivative and solve, paying attention...