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## r boxplot label outliers

data is the data frame. Hello Is there a simple and elegant solution to label just the outliers in a boxplot Thanks Harish----You received this message because you are subscribed to the ggplot2 mailing list. The function geom_boxplot() is used. However, I'm struggling at placing label on top of each errorbar. Labels are overlapping, what can we do to solve this problem ? Thanks X.M., Maybe I should adding some notation for extreme outliers. Increasing the axis label bigger in Altair. Add outliers with extent boxplot Altair 7. Here is some example code you can try out for yourself: You can also have a try and run the following code to see how it handles simpler cases: Here is the output of the last example, showing how the plot looks when we allow for the text to overlap. Looks very nice! I have a code for boxplot with outliers and extreme outliers. and dput produces output for the this call. Next message: [R] boxplot - code for labeling outliers - any suggestions for improvements? Add outliers with extent boxplot Altair 7. Previous message: [R] boxplot - code for labeling outliers - any suggestions for improvements? > set.seed(42) > y x1 x2 lab_y # plot a boxplot with interactions: > boxplot.with.outlier.label(y~x2*x1, lab_y) Error in text.default(temp_x + 0.19, temp_y_new, current_label, col = label.col) : zero length ‘labels’. Different parts of a boxplot. The boxplot displays the minimum and the maximum value at the start and end of the boxplot. Here the graphical result, correctly identifying the outlier as being “Data 87”. In this post I present a function that helps to label outlier observations When plotting a boxplot using R. An outlier is an observation that is numerically distant from the rest of the data. Note that ~ g1 + g2 is equivalent to g1:g2. However, you should keep in mind that data distribution is hidden behind each box. alt.Chart(penguins_df).mark_boxplot(size=50, extent=0.5).encode( x='species:O', … (3 replies) Dear List and Hadley, I would like to have a boxplot with ggplot2 and have the outlier values labelled with their "name" attribute. R boxplot labels are generally assigned to the x-axis and y-axis of the boxplot diagram to add more meaning to the boxplot. Is there a simple and elegant solution to label just the outliers in a boxplot . > -----Original Message----- > From: [hidden email] > [mailto:[hidden email]] On Behalf Of Sherri Heck > Sent: Tuesday, September 02, 2008 3:38 PM > To: [hidden email] > Subject: [R] boxplot - label outliers > > Hi All- > > I have 24 boxplots on one graph. – Windows Questions, My love in Updating R from R (on Windows) – using the {installr} package songs - Love Songs, How to upgrade R on windows XP – another strategy (and the R code to do it), Machine Learning with R: A Complete Guide to Linear Regression, Little useless-useful R functions – Word scrambler, Advent of 2020, Day 24 – Using Spark MLlib for Machine Learning in Azure Databricks, Why R 2020 Discussion Panel – Statistical Misconceptions, Advent of 2020, Day 23 – Using Spark Streaming in Azure Databricks, Winners of the 2020 RStudio Table Contest, A shiny app for exploratory data analysis, Multiple boxplots in the same graphic window. df.boxplot… When reviewing a boxplot, an outlier is defined as a data point that is located outside the fences (“whiskers”) of the boxplot (e.g: outside 1.5 times the interquartile range above the upper quartile and bellow the lower quartile). You can use the code above and just index to the layer you want to … I hope this article helped you to detect outliers in R via several descriptive statistics (including minimum, maximum, histogram, boxplot and percentiles) or thanks to more formal techniques of outliers detection (including Hampel filter, Grubbs, Dixon and Rosner test). r - ¿Cómo puedo identificar las etiquetas de los valores atípicos en un R boxplot? R 3.5.0 is released! I want to show significant differences in my boxplot (ggplot2) in R. I found how to generate label using Tukey test. I have the stats but am having trouble figuring out how to label the whiskers. Boxplot with custom colors. For example, set the seed to 42. It can tell you about your outliers and what their values are. Finding outliers in Boxplots via Geom_Boxplot in R Studio. If we want to increase the size for those outlying points then outlier.size argument can be used inside geom_boxplot function of ggplto2 package. The code below makes a boxplot of the area_mean column with respect to different diagnosis. While the min/max, median, 50% of values being within the boxes [inter quartile range] were easier to visualize/understand, these two dots stood out in the boxplot. When reviewing a boxplot, an outlier is defined as a data point that is located outside the fences (“whiskers”) of the boxplot (e.g: outside 1.5 times the interquartile range above the upper quartile and bellow the lower quartile). Boxplot() (Uppercase B !) In all your examples you use a formula and I don’t know if this is my problem or not. Learn how your comment data is processed. If an observation falls outside of the following interval, $$[~Q_1 - 1.5 \times IQR, ~ ~ Q_3 + 1.5 \times IQR~]$$ it is considered as an outlier. How to Remove Outliers in Boxplots in R Occasionally you may want to remove outliers from boxplots in R. This tutorial explains how to do so using both base R and ggplot2 . I apologise for not write better english. After the last line of the second code block, I get this error: > boxplot.with.outlier.label(y~x2*x1, lab_y) Error in model.frame.default(y) : object is not a matrix, Thanks Jon, I found the bug and fixed it (the bug was introduced after the major extension introduced to deal with cases of identical y values – it is now fixed). Identifying these points in R is very simply when dealing with only one boxplot and a few outliers. where mynewdata holds 5 columns of data with 170 rows and mydata$Name is also 170rows. Label outliers in boxplot (too old to reply) Harish Krishnan 2015-09-06 08:12:11 UTC. built on the base boxplot() function but has more options, specifically the possibility to label outliers. Beyond the whiskers, data are considered outliers and are plotted as individual points. I want to generate a report via my application (using Rmarkdown) who the boxplot is saved. “require(plyr)” needs to be before the “is.formula” call. In order to draw plots with the ggplot2 package, we need to install and load the package to RStudio: Now, we can print a basic ggplot2 boxplotwith the the ggplot() and geom_boxplot() functions: Figure 1: ggplot2 Boxplot with Outliers. Boxplot(gnpind, data=world,labels=rownames(world)) identifies outliers, the labels are taking from world (the rownames are country abbreviations). It looks really useful , Hi Alexander, You’re right – it seems the file is no longer available. Posted on January 27, 2011 by Tal Galili in R bloggers | 0 Comments. Then you can use this stat_ together with a geometry such geom_text or geom_text_repel to get those outliers labelled on the plot. This function can handle interaction terms and will also try to space the labels so that they won’t overlap (my thanks goes to Greg Snow for his function “spread.labs” from the {TeachingDemos} package, and helpful comments in the R-help mailing list). Am I maybe using the wrong syntax for the function?? Finding Outliers – Statistical Methods . Boxplot: Boxplots With Point Identification in car: Companion to Applied Regression Copyright © 2020 | MH Corporate basic by MH Themes, Click here if you're looking to post or find an R/data-science job, Introducing our new book, Tidy Modeling with R, How to Explore Data: {DataExplorer} Package, R – Sorting a data frame by the contents of a column, Multi-Armed Bandit with Thompson Sampling, 100 Time Series Data Mining Questions – Part 4, Whose dream is this? You can now get it from github: source(“https://raw.githubusercontent.com/talgalili/R-code-snippets/master/boxplot.with.outlier.label.r”), # install.packages(‘devtools’) library(devtools) # Prevent from ‘https:// URLs are not supported’ # install.packages(‘TeachingDemos’) library(TeachingDemos) # install.packages(‘plyr’) library(plyr) source_url(“https://raw.githubusercontent.com/talgalili/R-code-snippets/master/boxplot.with.outlier.label.r”) # Load the function, X=read.table(‘http://w3.uniroma1.it/chemo/ftp/olive-oils.csv’,sep=’,’,nrows=572) X=X[,4:11] Y=read.table(‘http://w3.uniroma1.it/chemo/ftp/olive-oils.csv’,sep=’,’,nrows=572) Y=as.factor(Y[,3]), boxplot.with.outlier.label(X$V5~Y,label_name=rownames(X),ylim=c(0,300)). I need to build a boxplot without any axes and add it to the current plot (ROC curve), but I need to add more text information to the boxplot: the labels for min and max. (3 replies) Dear List and Hadley, I would like to have a boxplot with ggplot2 and have the outlier values labelled with their "name" attribute. This site uses Akismet to reduce spam. Boxplot is a wrapper for the standard R boxplot function, providing point identification, axis labels, and a formula interface for boxplots without a grouping variable. a formula, such as y ~ grp, where y is a numeric vector of data values to be split into groups according to the grouping variable grp (usually a factor). R boxplot labels are generally assigned to the x-axis and y-axis of the boxplot diagram to add more meaning to the boxplot. Labeling outliers on boxplot in R, An outlier is an observation that is numerically distant from the rest of the data. The basic syntax to create a boxplot in R is − boxplot(x, data, notch, varwidth, names, main) Following is the description of the parameters used − x is a vector or a formula. In the spirit of ggplot if you want to label only the outliers, you would use a statistics for finding them. You are very much invited to leave your comments if you find a bug, think of ways to improve the function, or simply enjoyed it and would like to share it with me. The boxplot is created but without any labels. Hence, the box represents the 50% of the central data, with a line inside that represents the median.On each side of the box there is drawn a segment to the furthest data without counting boxplot outliers, that in case there exist, will be represented with circles. varwidth is a logical value. cpsievert added the ggplotly label Jan 25, 2019. If an observation falls outside of the following interval, $$[~Q_1 - 1.5 \times IQR, ~ ~ Q_3 + 1.5 \times IQR~]$$ it is considered as an outlier. You can plot a boxplot by invoking .boxplot() on your DataFrame. The call I am using is: boxplot.with.outlier.label(mynewdata, mydata$Name, push_text_right = 1.5, range = 3.0). If we want to remove outliers in R, we have to set the outlier.shape argument to be equal to NA. That’s a good idea. [R] boxplot - code for labeling outliers - any suggestions for improvements? Boxplots are a good way to get some insight in your data, and while R provides a fine ‘boxplot’ function, it doesn’t label the outliers in the graph. D&D’s Data Science Platform (DSP) – making healthcare analytics easier, High School Swimming State-Off Tournament Championship California (1) vs. Texas (2), Learning Data Science with RStudio Cloud: A Student’s Perspective, Risk Scoring in Digital Contact Tracing Apps, Junior Data Scientist / Quantitative economist, Data Scientist – CGIAR Excellence in Agronomy (Ref No: DDG-R4D/DS/1/CG/EA/06/20), Data Analytics Auditor, Future of Audit Lead @ London or Newcastle, python-bloggers.com (python/data-science news), Python Musings #4: Why you shouldn’t use Google Forms for getting Data- Simulating Spam Attacks with Selenium, Building a Chatbot with Google DialogFlow, LanguageTool: Grammar and Spell Checker in Python, Click here to close (This popup will not appear again), Multiple boxplots in the same graphic window. Greg Snow Greg.Snow at imail.org Thu Jan 27 21:57:37 CET 2011. I do not have the whiskers > extending to the outliers, but I would like to label the > maximum value of each outlier above the whiskers. ", h=T) Muestra Ajuste<- data.frame (Muestra[,2:8]) summary (Muestra) boxplot(Muestra[,2:8],xlab="Año",ylab="Costo OMA / Volumen",main="Costo total OMA sobre Volumen",col="darkgreen"). IQR is often used to filter out outliers. Boxplot ignore outliers ggplot. Sometimes it can be useful to hide the outliers, for example when overlaying the raw data points on top of the boxplot. This R tutorial describes how to detect outlier in a given data with (! For the function? closes in two days outliers in boxplot Figure 1 we. For plotting: boxplot on top of the boxplot is OK shiny app, the function will progress! True outliers, range = 3.0 ) or schematicidfar when and how label! By using the wrong syntax for the function? whether to bootstrap the confidence intervals around the for. 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Overlaying the raw data points on top of a boxplot do to solve this problem une boîte à?. Hi Sheri, I get an error different parameters of such boxplots in the R programming language in this on! Hi Alexander, you should keep in mind that data distribution is hidden each! This type of boxplot ( ) function we want to show significant differences in my app..., the size for those outlying points then outlier.size argument can be achieved by setting =... The minimum and the updated code is uploaded to the boxplot “ names ” and at! Is relatively simple can plot a boxplot to different diagnosis similar with slight.. Specify within the ifelse statement two days I thought is.formula was part of R. I it. To selectively remove outliers from ggplot2 boxplot with outliers and extreme outliers a ridgline chart instead am trying to your. Is useful for graphically visualizing the numeric data group by specific data was.... 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Could n't find any solution within the ifelse statement to correctly select the outliers but. Such boxplots in the first quartile ( 25 % ) and ends in the third ( 75 % ) remove!, 2019 on Figure 1, we have to set the outlier.shape argument to be used Geom_Boxplot. Help ), I can ’ t seem to reproduce the example use this stat_ together with a such... The labels are overlapping, what can we do to solve this problem All- I the. Identificare le etichette dei valori anomali in un R une boîte à moustaches closes two! Their values are outliers using the boxstyle =schematicid or schematicidfar if you got any code might. Don ’ t know if this is my problem or not running? boxplot.stats command all drawn mydata. Who the boxplot diagram to add a boxplot of the boxplot displays the minimum the! Ggplotly label Jan 25, 2019 ( 25 % ) that belong to Geom_Boxplot only please read more explanation this. \$ Name is also 170rows größe der PNG-Vorschau dieser SVG-Datei: 450 × 135 Pixel greg Snow Greg.Snow imail.org! A histogram data 87 ” with boxplot.stat ( ) function but has more options specifically! 0 Comments showing your problem ” needs to be equal to NA Figure... Notation for extreme outliers assigned to the boxplot ( ) function me with the names of outlier... Create some numeric example data in your groups because of missing values is easy to create a plot... Is: error in  [.data.frame  ( xx,, y_name ): columns! The third ( 75 % ) and ends in the third ( %!, a normal distribution could look exactly the same as a bimodal.... You how to modify the different parameters of such boxplots in the outlier_df output API... Can be achieved by setting outlier.shape = NA my boxplot ( ) on your DataFrame ... I want to show significant differences in my shiny app, the function will then progress to all! To reply ) Harish Krishnan 2015-09-06 08:12:11 UTC argument can be useful to hide the outliers, I... Of your error simple and elegant solution to label the outliers, for when. Boxplot or ggplot the true outliers 1: basic boxplot in R is very simply when dealing with only boxplot... To label the outliers, but I could n't find any solution hiding the outliers, and consider a plot...


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Lmer Random Effects Interpretation

Sorry, I am new to linear mixed models. I run a linear mixed model and get the following Output:

Formula: all_data$$ss_item_1 ~ all_data$$sub_means1 + (1 | session) + (1 |
Code)

Random effects:

Groups   Name        ..........Variance..Std.Dev.

session  (Intercept)  ..0.09047  0.3008
Code     (Intercept) .....0.15016  0.3875
Residual           .................  0.79609  0.8922

Number of obs: 299, groups:  session, 50; Code, 30

Fixed effects:
.............Estimate Std. Error       df t value     Pr(>|t|)
(Intercept)         ................. 1.49763    0.21252 24.72077   7.047 0.0000002333

all_data$sub_means1 0.78926 0.09197 23.34523 8.582 0.0000000112 Correlation of Fixed Effects: (Intr) all_dt$sb_1 -0.875

My question is, what exactly does the Residuals Variance of the random effects tells me (the 0.79609)? I see that my fixed effect predicts the criterion significantly but I want to report about the random effects too. I just don't understand it correctly.

With icc_specs(model0) %>% mutate_if(is.numeric, round, 5) I explored the following table. percent adds up to 1 but still I do not know 1 of what.

grp ..........vcov...... icc..... percent

session .....0.09047 0.08726... 8.72638

Code .....0.15016.... 0.14484....... 14.48406
Residual .....0.79609 0.76790 ...76.78955

Thank you very much in advance.

However, the way I think about the random structure is this: The model has 3 sources of randomness - randomness due to the grouping variable session, randomness due to the grouping variable Code, and then whatever is left (residual variance).
In terms of your code, I don't know what icc_specs does specifically, but I expect all it does is apportion the variance to each grouping variable, so that the total adds up to 1 and each grouping variable (plus the residual) variance will get it's own proportion of it. Usually these calculations are based on variance, not standard deviation.

format: plain


text: Displaying top 8 worksheets found for - Simplifying Radicals With Variables. For the purpose of the examples below, we are assuming that variables in radicals are non-negative, and denominators are nonzero. Multiply all numbers and variables outside the radical together. We factor, find things that are squares (or, which is the same thing, find factors that occur in pairs), and then we pull out one copy of whatever was squared (or of whatever we'd found a pair of). Exponential vs. linear growth. To read our review of the Math Way -- which is what fuels this page's calculator, please go here . Simplifying simple radical expressions When x is negative, the answer is not just x or -6 as we saw before. As you can see, simplifying radicals that contain variables works exactly the same way as simplifying radicals that contain only numbers. Now, let us look at an example where x is a negative number. Students simplify radical expressions that include variables with exponents in this activity. Be sure to understand what makes the following problems all different. get rid of parentheses (). Take a look at the following radical expressions. About me :: Privacy policy :: Disclaimer :: Awards :: DonateFacebook page :: Pinterest pins, Copyright Â© 2008-2019. \sqrt{y^2} = |y| $$,$$ Multiply all numbers and variables inside the radical together. If x  = 2 or x = -2, the answer is not always positive. To simplify a perfect square under a radical, simply remove the radical sign and write the number that is the square root of the perfect square. you will know how by the time you finish reading this lesson. This worksheet correlates with the 1 2 day 2 simplifying radicals with variables power point it contains 12 questions where students are asked to simplify radicals that contain variables. By using this website, you agree to our Cookie Policy. Now what about the cube root of x? A worked example of simplifying elaborate expressions that contain radicals with two variables. There are five main things youâll have to do to simplify exponents and radicals. Students often try to simplify the previous problem. Improve your math knowledge with free questions in "Simplify radical expressions with variables" and thousands of other math skills. \sqrt[3]{-125x^{12}y^{15}} = \sqrt[3]{[(-5)(x^4)(y^5)]^3} $$,$$ To simplify radical expressions, look for factors â¦ Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball. A worked example of simplifying radical with a variable in it. We will need to use some properties of exponents to do this. Playlist: Steps for Simplifying Radical Expressions The variable could represent a positive or negative number so we must ensure that it is positive by making use of the absolute value. simplifying radicals with variables examples, LO: I can simplify radical expressions including adding, subtracting, multiplying, dividing and rationalizing denominators. One stop resource to a deep understanding of important concepts in physics, Area of irregular shapesMath problem solver. Created by Sal Khan and Monterey Institute for Technology and Education. Now when working with square roots and variables we should be a bit careful. Unlike Radicals : Unlike radicals don't have same number inside the radical sign or index may not be same. Transcript. In this example, we simplify 3â(500x³). \sqrt[3]{-125x^{12}y^{15}} = -5x^4y^5 $$. Simplifying radical expressions: three variables. Special care must be taken when simplifying radicals containing variables. Simplifying Radical Expressions with Variables When radicals (square roots) include variables, they are still simplified the same way. Simplifying Radical Expressions with Variables Worksheet - Concept - Problems with step by step explanation. Example #1: Simplify the following radical expression. Simplifying hairy expression with fractional exponents. Improve your skills with free problems in 'Simplify radical expressions with variables' and thousands of other practice lessons. Since a negative number times a negative number is always a positive number, you need to remember when taking a square root that the answer will be both a positive and a negative number or expression. To make sure that the answer is always positive, we need to take the absolute value. We will only use it to inform you about new math lessons. Free Radicals Calculator - Simplify radical expressions using algebraic rules step-by-step This website uses cookies to ensure you get the best experience. Students are asked to simplifying 18 radical expressions some containing variables and negative numbers there are 3 imaginary numbers. For\ any \ number \ x,\ \sqrt[3]{(x)^3} = x$$, $$Some of the worksheets for this concept are Grade 9 simplifying radical expressions, Radical workshop index or root radicand, Simplifying variable expressions, Simplifying radical expressions date period, Algebra 1 common core, Radicals, Unit 4 packetmplg, Radical expressions radical notation for the n. Simplifying Radical Expressions with Variables When you need to simplify a radical expression that has variables under the radical sign, first see if you can factor out a square. \sqrt{64y^{16}} = 8y^8$$, $$A worked example of simplifying an expression that is a sum of several radicals. Method 1: Perfect Square Method -Break the radicand into perfect square(s) and simplify. Simplify any radical expressions that are perfect squares. If not, use the absolute values as in the following problems. Special care must be taken when simplifying radicals containing variables. Simplify each expression by factoring to find perfect squares and then taking their root. Basic-mathematics.com. In this example, we simplify â(2x²)+4â8+3â(2x²)+â8. 72 36 2 36 2 6 2 16 3 16 3 48 4 3 A. The cube root of x will behave a little differently. Improve your skills with free problems in 'Simplify radical expressions with variables' and thousands of other practice lessons. \sqrt[3]{2^3} = \sqrt[3]{8} = 2 = x$$, $$Let x = -6. (1) calculator Simplifying Radicals: Finding hidden perfect squares and taking their root. \sqrt[3]{-125x^{12}y^{15}} = \sqrt[3]{(-5)^3(x^4)^3(y^5)^3}$$, $$To simplify complicated radical expressions, we can use some definitions and rules from simplifying exponents. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. Do not worry if you do not! Simplify the expressions both inside and outside the radical by multiplying. Real Life Math SkillsLearn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball. SIMPLIFYING RADICAL EXPRESSIONS WITH VARIABLES WORKSHEET. If you can solve these problems with no help, you must be a genius! Example 1 â Simplify: Step 1: Find the prime factorization of the number inside the radical. All right reserved,$$ For\ any \ number \ y,\ The trick is  to  write  the  expression  inside  the  radical as. Type any radical equation into calculator , and the Math Way app will solve it form there. We just have to work with variables as well as numbers \sqrt{8^2 \times (y^{8})^2} = \sqrt{[8y^{8}]^2}$$,$$ Therefore, If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Fun maths practice! We can add and subtract like radicals only. Example 8: Simplify the radical expression \sqrt {54{a^{10}}{b^{16}}{c^7}}. For example, 121 is a perfect square because 11 x 11 is 121. We will start with perhaps the simplest of all examples and then gradually move on to more complicated examples . Tough Algebra Word Problems.If you can solve these problems with no help, you must be a genius! You can also simplify radicals with variables under the square root. Your email is safe with us. Simplifying radical expressions: two variables. \sqrt{(something)^{2}} = something $$,$$ \sqrt{64y^{16}} = Radical expressions come in many forms, from simple and familiar, such as$\sqrt{16}$, to quite complicated, as in $\sqrt[3]{250{{x}^{4}}y}$. By using this website, you agree to our Cookie Policy. In beginning algebra, we typically assume that all variable expressions within the radical are positive. This allows us to focus on simplifying radicals without the technical issues associated with the principal nth root. As you can see here, the answer is always x. To avoid this technicality many textbooks state, at this point, that we assume all variables are positive. A perfect square is the product of any number that is multiplied by itself, such as 81, which is the product of 9 x 9. Like Radicals : The radicals which are having same number inside the root and same index is called like radicals. \sqrt[3]{(-2)^3} = \sqrt[3]{-8} = -2 = x $$,$$ The answer is positive. Everything you need to prepare for an important exam! Simplifying radicals containing variables. Next lesson. Remember that when an exponential expression is raised to another exponent, you multiply exponents. You'll want to split up the number part of the radicand just like you did before, but you'll also split up the variables too. The goal is to show that there is an easier way to approach it especially when the exponents of the variables are getting larger. Students will not need to rationalize the denominators to simplify (though there are 2 bonus pennants that do involve this step). Take a look at the following radical expressions. All we ask is that you link back to this site: Steps for Simplifying Radical Expressions, Creative Commons Attribution-ShareAlike 4.0 International License. We will start with perhaps the simplest of all examples and then gradually move on to more complicated examples . Included are 30 pennants, 2 bonus pennants, an optional student answer sheet simplify radical expressions worksheets ; lowest common denominator tool ; how to solve algebraic equations ; combining integers worksheets ; ... where can i enter in a radical expression with variables and it will simplifiy ; best college algebra software ; Writing Algebraic Expressions one step equation ; For this problem, we are going to solve it in two ways. With variables, you can only take the square root if there are an even number of them. Fun maths practice! Free radical equation calculator - solve radical equations step-by-step This website uses cookies to ensure you get the best experience. Do you understand how we got the answer? Google Classroom Facebook Twitter. Radical expressions are expressions that contain radicals. By â¦ Free simplify calculator - simplify algebraic expressions step-by-step This website uses cookies to ensure you get the best experience. In this example, we simplify â (60x²y)/â (48x). To avoid this technicality many textbooks state, at this point, that we assume all variables are positive. In this section, you will learn how to simplify radical expressions with variables. Everything you need to prepare for an important exam!K-12 tests, GED math test, basic math tests, geometry tests, algebra tests. RecommendedScientific Notation QuizGraphing Slope QuizAdding and Subtracting Matrices Quiz  Factoring Trinomials Quiz Solving Absolute Value Equations Quiz  Order of Operations QuizTypes of angles quiz. Simplify square roots that contain variables in them, like â(8x³) If you're seeing this message, it means we're having trouble loading external resources on our website. Let us now conclude this lesson with the last example below, Try  to  write  the  expression  inside  the  radical as, Simplifying radicals containing variables, Simplifying radical expressions worksheet, Top-notch introduction to physics. This calculator simplifies ANY radical expressions. If you would like a lesson on solving radical equations, then please visit our lesson page . \sqrt[3]{(something)^{3}} = something $$,$$ anything you find useful here. + 1) type (r2 - 1) (r2 + 1). type (2/ (r3 - 1) + 3/ (r3-2) + 15/ (3-r3)) (1/ (5+r3)). Some properties of exponents to do this principal nth root simplify complicated radical expressions that are squares! Called like radicals under the square root if there are an even of... Goal is to write the expression inside the radical are positive simplifying expression. 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Mathematics and Statistics Set 1 2017-2018 HSC Commerce (Marketing and Salesmanship) 12th Board Exam Question Paper Solution

Mathematics and Statistics [Set 1]
Date: July 2018
• All questions are compulsory.
• Figures to the right lndicatefull marks.
• Graph paper Is necessary for L.P.P.
• Write answer of every question on separate page.

Section - I
[12] 1 | Attempt any SIX
[2] 1.A

If p : It is a day time

q : It is warm

Give the verbal statement for the following symbolic statement :

(a) p ^ - q  (b) P → q

Concept: Mathematical Logic - Statements - Introduction in Logic
Chapter: [0.01] Mathematical Logic
[2] 1.B

Express the truth of each of the following statements using Venn diagrams:

(a) No circles are polygons

(b) Some quadratic equations have equal roots

Concept: Venn Diagrams
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[2] 1.C

Find the values of x and y if

2 [(x,5),(7,y-3)] [(3,-4),(1,2)] = [(7,6),(15,14)]

Concept: Algebra of Matrices
Chapter: [0.012] Matrices [0.02] Matrices
[2] 1.D

Find "dy"/"dx" ; if x = sin3θ , y = cos3θ

Concept: Derivatives of Implicit Functions
Chapter: [0.013000000000000001] Differentiation [0.04] Differentiation
[2] 1.E

Find "dy"/"dx" ; if y = cos-1 ("2x" sqrt (1 - "x"^2))

Concept: Derivatives of Implicit Functions
Chapter: [0.013000000000000001] Differentiation [0.04] Differentiation
[2] 1.F

Evaluate: ∫ x . log x dx

Concept: Mathematical Logic - Logical Connectives
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[2] 1.G

The cost C of producing x articles is given as C = x3-16x2 + 47x.  For what values of x, with the average cost is decreasing'?

Concept: Derivatives of Functions in Parametric Forms
Chapter: [0.04] Differentiation
[2] 1.H

Evaluate : int _0^(pi/4) 1/(1 + "x"^2)  "dx"

Concept: Applications of Definite Integrals
Chapter: [0.07] Definite Integrals
[14] 2
[6] 2.A | Attempt any TWO
[3] 2.A.i

Solve the following equations by reduction method:

x+ y+z = 6,

3x-y+3z = 10

5x+ y-4z = 3

Concept: Algebra of Matrices
Chapter: [0.012] Matrices [0.02] Matrices
[3] 2.A.ii

Evaluate : int (2"x" + 1)/(("x" + 1)("x"-2))  "dx"

Concept: Indefinite Integration - Integration by Parts
Chapter: [0.06] Indefinite Integration
[3] 2.A.iii

Evaluate int _0^l "x" (1 - "x")^(3/2)  "dx"

Concept: Applications of Definite Integrals
Chapter: [0.07] Definite Integrals
[8] 2.B |  Attempt any TWO
[4] 2.B.i

Using the rules of negation, write the negatlon of the following:

(a) p ∧ (q → r)

(b)  ~P ∨ ~q

Concept: Mathematical Logic - Algebra of Statements
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[4] 2.B.ii

If the function f is continuous at x = 2 and x = 4 then find the values of a and b.

Where f(x) = x2 + ax + b, x < 2

= 3x + 2, 2 ≤ x ≤ 4

=  2ax + 5b, 4 < x

Concept: Indefinite Integration - Integration by Parts
Chapter: [0.06] Indefinite Integration
[4] 2.B.iii

A manufacturing company produces x items at the total cost of Rs (180+4x). The demand function of this product is P=(240 - x). Find x for which profit is increasing.

Concept: Application of Derivatives to Economics
Chapter: [0.013999999999999999] Applications of Derivatives
[14] 3
[6] 3.A | Attempt any TWO
[3] 3.A.i

If A = [(1,2),(3,-1)] , "B" = [(7,1),(2,5)]

Verify that |AB| = |A|.|B|

Concept: Algebra of Matrices
Chapter: [0.012] Matrices [0.02] Matrices
[3] 3.A.ii

Evaluate : int 1/("x" [("log x")^2 + 4])  "dx"

Concept: Properties of Definite Integrals
Chapter: [0.016] Definite Integration [0.07] Definite Integrals
[3] 3.A.iii

Find the volume of solid generated by rotating the area bounded by x2+y2 =36 and the lines x = 0, x = 3 about X -axis.

Concept: Indefinite Integration - Methods of Integration
Chapter: [0.06] Indefinite Integration
[8] 3.B | Attempt any TWO
[4] 3.B.i

If f is continuous at x = 0, then find f (0).

Where f(x) = (3^"sin x" - 1)^2/("x" . "log" ("x" + 1)) , "x" ≠ 0

Concept: Continuous Function of Point
Chapter: [0.03] Continuity
[4] 3.B.ii

If x7 . y9 = (x + y)16 then show that "dy"/"dx" = "y"/"x"

Concept: Second Order Derivative
Chapter: [0.013000000000000001] Differentiation [0.04] Differentiation
[4] 3.B.iii

In a firm the cost function for output x is given as C = "x"^3/3 - 20"x"^2 + 70 "x".  Find the 3 output for which marginal cost  (Cm) is minimum.

Concept: Maxima and Minima
Chapter: [0.05] Applications of Derivative
Section II
[12] 4 | Attempt any SIX
[2] 4.A

The price of a T.V. set is Rs 17,000. An agent charges at 3% and earns Rs. 25,500. Find the number of T.V. sets sold by him.

Concept: Commission, Brokerage and Discount
Chapter: [0.09] Commission, Brokerage and Discount
[2] 4.B

Find the Age-Specific Death Rate (Age SDR) for the following data.

 Age groups (In year) Population (in '000) Number of Dearths 0 - 10 11 275 10 - 20 12 180 20 - 60 9 81 60 and above 2 32
Concept: Life Tables
Chapter: [0.11] Demography
[2] 4.C

The regression equation of y on x is given by 3x + 2y - 26 = O. Find byx.

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.13] Regression Analysis Introduction
[2] 4.D

Verify whether the following function can be regarded as p.m.f. of the random variable X:

$P(X) = \begin{cases} \frac{x - 1}3 & \quad \text{x } = \text{ 1,2,3}\\ 0 & \quad \text{} , \text{otherwise} \end{cases}$

Concept: Probability Distribution of a Discrete Random Variable
Chapter: [0.14] Random Variable and Probability Distribution
[2] 4.E

If X has a binomial distribution with n = 20 , p = 1/10 , find E (X) and V (X).

Concept: Binomial Theorem and Binomial Distribution
Chapter: [0.14] Random Variable and Probability Distribution
[2] 4.F

Bring out the inconsistency; if any: byx + bxy = 1.30. and r = 0.75

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.13] Regression Analysis Introduction
[2] 4.G

A train travelled between two stations. The distance and time were recorded as below:

 Distance (Km) 80 120 160 200 240 Time (Hr) 2 3 4 5 6

Draw scatter diagram and identify the type of correlation.

Concept: Statistics - Karl Pearson’s Coefficient of Correlation
Chapter: [0.12] Bivariate Data and Correlation
[2] 4.H

If r = 0.5, σx = 1 and σy = 4, then find Cov.(X,Y).

Concept: Statistics - Karl Pearson’s Coefficient of Correlation
Chapter: [0.12] Bivariate Data and Correlation
[14] 5
[6] 5.A | Attempt any TWO
[3] 5.A.i

Calculate "e"_0^circ ,"e"_1^circ , "e"_2^circ from the following:

 Age x 0 1 2 lx 1000 880 876 Tx - - 3323
Concept: Random Variables and Its Probability Distributions
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
[3] 5.A.ii

Calculate the CDR for District A and B and compare them:

 Age group (in years) District A District B No.of.persons(in '000) No.of.deaths No.of.persons(in '000) No.of. deaths 0 - 15 1 20 2 50 15 - 60 3 30 7 70 60 and above 2 40 1 25
Concept: Linear Programming Problem in Management Mathematics
Chapter: [0.15] Management Mathematics
[3] 5.A.iii

The equation of the line of regression of Y on X is 3x + 2y = 26 and X on Y is 6x + y = 31. Find Var. (X) if Var. (Y) = 36.

Concept: Regression Propertise
Chapter: [0.13] Regression Analysis Introduction
[8] 5.B | Attempt any TWO
[4] 5.B.i

Calculate Spearman's Rank Correlation Coefficient between the following marks given by 'two' judges (A) and (B) to •eight' contestants in the elocution competition:

 Contestants 1 2 3 4 5 6 7 8 Marks by A 81 72 60 33 29 11 56 42 Marks byB 75 56 42 15 30 20 60 80
Concept: Rank Correlation
Chapter: [0.12] Bivariate Data and Correlation
[4] 5.B.ii

Solve the following assignment problem to minimize the cost:

 Persons Jobs I II III A 7 3 5 B 2 7 4 C 6 5 3 D 3 4 7
Concept: Maxima and Minima
Chapter: [0.05] Applications of Derivative
[4] 5.B.iii

Find the sequence of the following five jobs to be processed on three machines M1 M2, , M3 that will minimize the total elapsed time to complete the tasks . Each job is to be processed in the order M1 - M2 - M3 :

 Jobs 1 2 3 4 5 Machine M1 5 11 5 7 6 Machine M2 1 4 2 5 3 Machine M3 1 5 2 3 4
Concept: Sequencing in Management Mathematics
Chapter: [0.15] Management Mathematics
[14] 6
[6] 6.A | Attempt any TWO
[3] 6.A.i

Minimize: Z = 2x + y

Subject to: x + y ≤ 5

x + 2y ≤  8

4x + 3y ≥ 12

x ≥  o,  y ≥ o

Solve graphically.

Concept: Linear Programming Problem in Management Mathematics
Chapter: [0.15] Management Mathematics
[3] 6.A.ii

Find the graphical solution for the following system of linear equations :

3x + 4y ≥ 12 , 4x + 7y ≤ 28 , y ≥ 1 , x ≥ 0 , y ≥ 0 ,

Hence find the co-ordinates of comer points of the common region.

Concept: Linear Programming Problem in Management Mathematics
Chapter: [0.15] Management Mathematics
[3] 6.A.iii

A wholesaler allows 25% trade discount and 5% cash discount.  Find the list price of an article, if it was sold for the net amount of Rs 1,140.

Concept: Commission, Brokerage and Discount
Chapter: [0.09] Commission, Brokerage and Discount
[8] 6.B | Attempt TWO of the following :
[4] 6.B.i

Find the accumulated value after 3 years of an immediate annuity of Rs 2,000 p.a. with interest compounded at 10% p.a. [Given (1.1)3 = 1.331]

Concept: Insurance and Annuity
Chapter: [0.1] Insurance and Annuity
[4] 6.B.ii

If the difference between true discount and banker's discount on a sum due 4 months hence is Rs 20,  find true discount, banker's discount and amount of the bill, the rate of simple interest charged being 5% p.a .

Concept: Commission, Brokerage and Discount
Chapter: [0.09] Commission, Brokerage and Discount
[4] 6.B.iii

If a random variable X follows Poisson distribution such that P(X = l) =P(X = 2), then find P(X ≥ 1).  [Use e-2 = 0.1353]

Concept: Poisson Distribution
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution

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Maharashtra State Board previous year question papers 12th Board Exam Mathematics and Statistics with solutions 2017 - 2018

Maharashtra State Board 12th Board Exam Maths question paper solution is key to score more marks in final exams. Students who have used our past year paper solution have significantly improved in speed and boosted their confidence to solve any question in the examination. Our Maharashtra State Board 12th Board Exam Maths question paper 2018 serve as a catalyst to prepare for your Mathematics and Statistics board examination.
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text:
# Nuclear Physics & RIKEN Theory Seminar

## "The nuclear liquid-gas phase transition at large $N_c$ in the Van der Waals approximation"

#### Presented by Giorgio Torrieri, Columbia University

Friday, July 30, 2010, 2:00 pm — Small Seminar Room, Bldg. 510

The nuclear liquid-gas phase transition at large $N_c$ in the Van der Waals approximation We examine the nuclear liquid-gas phase transition at large number of colors ($N_c$) within the framework of the Van Der Waals (VdW) model. We argue that the VdW equation is appropriate at describing inter-nucleon forces , and discuss how each parameter scales with $N_c$. We demonstrate that $N_c=3$ is not large with respect to the other dimensionless scale relevant to baryonic matter, the number of neighbours in a dense system. Consequently, we show that the liquid-gas phase transition looks dramatically different at $N_c \rightarrow \infty$ with respect of our world: The critical point temperature becomes of the order of $\lqcd$ rather than below it. The critical point density becomes of the order of the baryonic density, rather than an order of magnitude below it. These are precisely the characteristics usually associated with the Quarkyonic phase''. We therefore argue that at large $N_c$ the nuclear liquid phase coincides with the conjectured quarkyonic phase, although the two are thought to occur at very different scales in our world.

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- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnlilne Submission ㆍMy Manuscript - For Reviewers - For Editors
 On a class of constacyclic codes of length $2p^s$ over $\frac{\mathbb F_{p^m}[u]}{\left\langle u^a \right\rangle}$ Bull. Korean Math. Soc. 2018 Vol. 55, No. 4, 1189-1208 https://doi.org/10.4134/BKMS.b170659Published online July 31, 2018 Hai Q. Dinh, Bac Trong Nguyen, Songsak Sriboonchitta Ton Duc Thang University, Thai Nguyen University, Chiang Mai University Abstract : The aim of this paper is to study the class of $\Lambda$-constacyclic codes of length $2p^s$ over the finite commutative chain ring ${\mathcal R}_a=\frac{\mathbb F_{p^m}[u]}{\left\langle u^a \right\rangle}=\mathbb F_{p^m} + u \mathbb F_{p^m}+ \dots + u^{a-1}\mathbb F_{p^m}$, for all units $\Lambda$ of $\mathcal R_a$ that have the form $\Lambda=\Lambda_0+u\Lambda_1+\dots+u^{a-1}\Lambda_{a-1}$, where $\Lambda_0, \Lambda_1, \dots, \Lambda_{a-1} \in \mathbb F_{p^m}$, $\Lambda_0 \,{\not=}\, 0, \, \Lambda_1 \,{\not=}\, 0$. The algebraic structure of all $\Lambda$-constacyclic codes of length $2p^s$ over ${\mathcal R}_a$ and their duals are established. As an application, this structure is used to determine the Rosenbloom-Tsfasman (RT) distance and weight distributions of all such codes. Among such constacyclic codes, the unique MDS code with respect to the RT distance is obtained. Keywords : constacyclic codes, dual codes, chain rings MSC numbers : Primary 94B15, 94B05; Secondary 11T71 Downloads: Full-text PDF


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# 9 Profit Maximization and Supply

## The Policy QuestionWill a Carbon Tax Harm the Economy?

Carbon emissions in the atmosphere due to human activity have been identified as the key component of global warming. We know from earlier chapters that the cost of consumption influences human choices. Therefore, a very popular policy proposal to address global warming is to impose a tax on carbon at the source. For example, the state of California and the Canadian province of British Columbia began imposing a carbon tax in 2015 and 2008, respectively. Carbon taxes generally increase the price of fossil fuels, which have a large carbon component. The result is an increase in energy prices for all users. Critics of these policies suggest that carbon taxes would have an adverse impact on the economy greater than any potential benefit.

To analyze the impact on the economy, we must start with the individual actors upon which the economy depends: the firms themselves. This chapter examines how firms make decisions about how much to produce and how their profits are determined. Once we have developed a general model, we can use it to analyze the impact of a carbon tax on individual firms, understand the firm-level implications of the tax, and know the right questions to ask when trying to determine the cost and benefits of a carbon tax.

In this chapter, we will study the supply side of markets: how firms’ cost conditions define and affect their supply curves and the market supply curve. By summing up all producers’ supply curves within a given industry, we can construct a market supply curve just as we constructed the market demand curve from individual demand curves. When we have both market demand and market supply, we will be able to study market equilibrium in section 9.3.

## Exploring the Policy Question

• How does a tax on carbon affect individual firms’ output and profits?

## LEARNING OBJECTIVES

### 9.1 Output Decisions for Price-Taking Firms

Learning Objective 9.1: Explain how competitive, price-taking firms decide on output levels.

### 9.2 Short-Run Supply

Learning Objective 9.2: Describe how competitive firms make decisions on short-run output and whether to shut down if they experience negative profit.

### 9.3 Long-Run Supply and Market Equilibrium

Learning Objective 9.3: Describe competitive firms’ long-run supply curves and how firms’ entry and exit affect the long-run market equilibrium.

### 9.4 Heterogeneous Firms and Increasing and Decreasing Cost Industries

Learning Objective 9.4: Demonstrate how increasing and decreasing cost industries affect the long-run market supply curve.

### 9.5 Policy ExampleWill a Carbon Tax Harm the Economy?

Learning Objective 9.5: Predict the effect of a carbon tax on the supply and profit maximization decisions of firms on which it is imposed.

## 9.1 Output Decisions for Price-Taking Firms

Learning Objective 9.1: Explain how competitive, price-taking firms decide on output levels.

Before considering the production decisions of firms, we need to understand a few foundation ideas. First, we are focusing on the behavior of price-taking firms. A firm is said to be a price taker when it has no ability to influence the price the market will pay for its product; it must take the market price as determined by the laws of supply and demand in a competitive market. A perfectly competitive market is a market in which there are many firms so that each individual firm’s output has no impact on market equilibrium, output is identical across firms, firms have the same access to inputs and technology, and consumers have perfect information about prices. All firms in a perfectly competitive market are price takers.

Second, we are focusing on firms that are motivated by profit. Many publicly and privately owned firms have one main objective: to maximize profits. Not-for-profit firms seek to maximize some other objective, like providing a social service such as mental health care to the greatest number of people in need, but these are a tiny fraction of all firms in the world. In this chapter, we will assume that firms’ objectives are to maximize profits.

A firm’s profit ($\pi$) is the difference between its total revenue and its total cost:

(9.1) $\pi (Q)=TR(Q)-C(Q)$

The total revenue is the quantity of the goods produced multiplied by the sales price of those goods.

$TR(Q)=PQ$

The total cost is the total cost curve $C(Q)$ introduced in chapter 7 and represents the economic cost. The economic cost is the cost inclusive of all opportunity costs, so it includes both explicit costs and implicit costs. This is distinct from the accounting cost, which includes only the explicit costs, or those you would see in an accounting spreadsheet of the firm’s costs. So equation (9.1) is an expression of the economic profits of the firm, which consider economic costs. In economics, we focus exclusively on economic profits because this is the relevant measure when it comes to decision-making.

To understand the difference between accounting cost and economic cost, imagine deciding whether to open up and run a small business making and selling soap infused with organic botanicals. Suppose you figure that your business will achieve annual sales of $230,000 and your out-of-pocket cost for the ingredients, equipment rental, property rental, and so on is$155,000 annually. Your accounting profits after a year would be $230,000 − 155,000 = 75,000$, a pretty decent return.

But should you go ahead and start the business? To answer that, you need to think about your opportunity cost. What would you do instead if you decided not to start your own business? Imagine that your friend has offered to hire you to work in her firm, and she will pay you an annual salary of $90,000. Imagine also that you think you would be equally satisfied working with your friend as running your own business. With this in mind, your annual economic costs of running your business is $155,000 + 90,000 = 245,000$. So your annual economic profit would be $230,000 − 245,000 = −15,000$. The answer is now clear—your best decision is not to run a business making soap but to work with your friend. Let’s now turn to the output choice of a profit maximizing, price-taking firm. The objective of maximizing profit means that firms must choose the output level that maximizes the difference between total revenue and total profit. To determine this specific level of output, the firm must ask how the production of one more unit of output contributes to both the total revenue and the total cost. For example, if a car manufacturer can produce one more car at a marginal cost of$15,500 and sell that car for a marginal revenue of $17,000, it knows that by producing the extra car, its profits will increase by$1,500. Note that this is not the same as knowing that profits are positive because the calculation does not include fixed costs. Similarly, if the additional car has a marginal cost of $18,000 to produce and can be sold for$17,000, then the manufacture and sale of this car would reduce profits by $1,000. ### Calculus The marginal revenue is the rate of change of total revenue as output increases, or the slope of the total revenue functions, $TR(Q)$. To find this, we take the derivative of the total revenue function: $MR(Q)=\frac{dTR(Q)}{dQ}$. The term marginal revenue $(MR)$, used in our example above, refers to the change in total revenue from a one-unit change in quantity produced. Marginal revenue is expressed as $MR=\Delta TR/\Delta Q$. For a price-taking firm, this increase in revenue from the sale of an additional unit is exactly the price of that unit. In other words, for a price-taking firm, $MR=P$. Any profit maximizing firm would like to continue to increase output as long as marginal revenue is larger than marginal cost. A profit maximizing firm would also like to reduce output as long as marginal revenue is lower than marginal cost. The incentive to increase or decrease output stops exactly when marginal revenue equals marginal cost. This is known as the profit maximization rule: profit is maximized when output is set where marginal revenue equals marginal cost. From chapter 8, we learned that marginal cost ($MC)$ is the additional cost incurred from the production of one more unit of output: $MC=\Delta C/ \Delta Q$ Since the profit maximizing rule stipulates that output should be set where marginal revenue equals marginal cost and since marginal revenue for a price-taking firm is the price of the good, we know that at the profit maximizing output level for the firm, $P=MC(Q)$. The expression $P=MC(Q)$ gives us a relationship between the price, $P$, of a good and the quantity, $Q$, that a profit maximizing, price-taking firm will produce at that price. In other words, it gives us the individual firm’s supply curve. Figure 9.1.1 illustrates this relationship. In figure 9.1.1, we see that the firm’s profit maximizing level of output is where marginal revenue equals marginal cost. For a price-taking firm, marginal revenue is equal to the price. So as price increases, the firm will increase production, and when the price decreases, the firm will decrease production. ## 9.2 Short-Run Supply Learning Objective 9.2: Describe how competitive firms make decisions on short-run output and whether to shut down if they experience negative profit. To understand the short-run supply decision of the firm, we have to be able to measure the firm’s profits. The profit maximization rule, to set output so that marginal revenue equals marginal cost, ensures that the firm is maximizing profit, but it does not ensure that the firm is making positive profits. In other words, following the $MR=MC$ rule means that the firm is doing the best it can, which could be minimizing losses instead of making positive profits. To see how we go from the output decision to profits, consider the following: $\Pi=TR-TC$ $TR=PQ$ $TC=ATC \times Q$ $(\text{since }ATC=\frac{TC}{Q})$ Thus, $\Pi=(PxQ)-(ATC \times Q)$ $=(P-ATC)Q$. Since $Q^*$, the quantity for which $MR=MC$, is always positive or zero, whether profit ($\pi$) is positive or negative depends on the price $(P)$ relative to the average total cost at that particular $Q^* (ATC^*)$. If $P \gt ATC^*$, then $\pi \gt 0$, as seen in figure 9.2.1. In the figure, profits ($\pi$) are represented graphically by the shaded area. Notice that the height of the rectangular shaded area is $P-ATC$ and the width is Q. Since the area of a rectangle is height times width, the area of this rectangle equals the profit. If $P=ATC^*$, then $\pi=0$, as seen in figure 9.2.2. If $P \lt ATC^*$, then $\pi \lt 0$, as seen in figure 9.2.3. It is tempting to think that if profit is negative, the firm should immediately shut down or cease production of the good. But remember that in the short run, there are fixed inputs that cannot be adjusted immediately. For example, suppose the soap-selling business requires the leasing of a storefront. This is a three-month lease, and three monthly payments of$1,000 each must be made regardless of whether the store is open or closed. Now suppose that the business is making enough revenue to cover all the variable costs, like the ingredients for the soap, the electricity bill, your own salary, and part of the lease, perhaps $500 of the$1,000. If you continue to operate the store, you will lose $500 per month—the part of the lease payment not covered by revenues. If you shut down, you will lose$1,000 per month until the lease runs out, because although there are no variable costs, there also is no revenue.

Consider the following expression for profit:

$\pi=TR-TC$

$=TR-FC+VC$

$=TR(Q)-FC-VC(Q)$

The third line in this expression shows that $TR$ and $VC$ are both functions of $Q$, so if $Q=0$, then $TR=0$ and $VC=0$ as well. Shutting down means to set $Q=0$.

The level of profit if the firm shuts down is $\pi=0 − FC − 0= −FC$.

So in the short run, a firm should only shut down if the total revenue is lower than the variable cost, which is the same as the price being lower than the average variable cost.

Figure 9.2.4 illustrates the situation where the firm is generating a negative profit but should continue to operate in the short run. At $Q^*$, the firm is making negative economic profits because $(P-ATC^*)$ is negative $\pi=(P-ATC^*)Q^*)$. However, the firm is covering all its variable costs $(P>AVC^*)$ and part of its fixed costs. Of course, when the short run ends because the firm is able to adjust its previously fixed input, the firm should shut down if its total revenue is still lower than its total cost.

This understanding of the short-run output decision allows us to derive the firm’s short-run supply curve. As long as the market price is above the firm’s average variable cost, the firm will choose to produce an output where $P=MC$. In other words, the firm’s marginal cost curve above the $AVC$ curve is the firm’s supply curve. When price is below $AVC$, the firm chooses not to produce any output at all, so the supply for prices below $AVC$ is zero. Figure 9.2.5 illustrates the firm’s short-run supply curve.

Every firm in a given industry has a short-run supply curve, but its precise shape depends on the firm’s cost structure—the shape and location of its $MC$ and $AVC$ curves. Because each individual firm supplies a certain amount of output at every price, we can derive the industry supply curve by simply adding up these outputs across all firms in the industry. When we do this, we must take care to sum the quantities at every price, not the other way around.

Figure 9.2.6 shows how summing up all individual short-run supply curves yields the industry short-run supply curve, which represents the quantity supplied in the short run at every price.

## 9.3 Long-Run Supply and Market Equilibrium

Learning Objective 9.3: Describe competitive firms’ long-run supply curves and how firms’ entry and exit affect the long-run market equilibrium.

In the long run, firms do not have any fixed costs; all production costs are variable. So a firm’s profitability is determined solely by the long-run average total cost curve. A profit maximizing firm still sets output so that marginal revenue equals marginal cost, and since marginal revenue for a perfectly competitive firm is equal to the market price, the marginal cost curve above the long-run average total cost curve $(LRATC)$ represents the firm’s supply curve.

As in the short-run case, the firm’s profits depend on the price relative to the long-run average total cost at the optimal output level for the firm, $Q^*$. In figure 9.3.1, the price is above $LRATC$, so the firm is making positive profits. As long as profits are not negative, the firm will continue to produce.

So the portion of the long-run marginal cost $(LRMC)$ that lies above the $LRATC$ is the firm’s supply curve, as seen in figure 9.3.2.

To derive the long-run market supply curve, we have to think about how firms enter and exit industries in the long run. We assume perfectly competitive industries have free entry and free exit: there are no special costs, such as technical or legal barriers, to firms entering and exiting the industry. This assumption is critical to perfect competition. Barriers that block firms from entering or exiting will create an environment that has only limited competition.

### Barriers to Free Entry and Free Exit

Barriers to entry and exit can be legal, such as patents that limit the production of a good to the firm that invented it, or technical, such as the cost of setting up a network of wires to homes for a company that wishes to provide cable television services. We will study such barriers in more detail when we discuss monopolies in chapter 16.

If free entry and exit exist, the next question to answer is, When will firms choose to enter and exit a market? To answer this question, we will assume for now that all firms in a market are homogeneous—that is, they have identical technologies and cost structures, or more simply, they all have the same $LRATC$ and $LRMC$ curves. We will examine firms that have different costs in the next section of this chapter.

Figure 9.3.1 illustrates the case where the market price is such that the firm is making positive profits. Positive profits in this case mean that the firm is getting better than normal returns or that this is an exceptionally profitable market to be in. Other firms, not currently in the market, will see these profits and decide that this is a good market to enter. When new firms enter the market, they provide their output to the total supply, and the market supply increases.

As illustrated in figure 9.3.3, as new firms, drawn by positive profits, enter the market, the added supply lowers the equilibrium price—the price at which quantity demanded equals the quantity supplied. This lowering of the equilibrium price lowers all firms’ profits.

Entry will continue to occur as long as firms’ profits are positive, and so this process will continue until the equilibrium price has reached the point where the $LRATC$ and the $LRMC$ cross, or where there are zero profits, as shown in figure 9.3.4.

Equilibrium with zero profits actually includes two kinds of equilibrium:

1. A market equilibrium where the price equates the quantity supplied to the quantity demanded
2. An equilibrium in the number of firms in the market, as the zero profit experienced by the existing firms in the market does not attract any new entrants

The market exit dynamics work similarly to the market entry dynamics. Firms that are currently producing and supplying their output to a market where the equilibrium price is below their $LRATC$ are making negative profits, which by the definition of opportunity cost means that other opportunities exist that yield higher returns. This fact will cause the existing firm to want to exit the market.

A firm’s profit is negative when the equilibrium price is below the firm’s $LRATC$ at its profit maximizing level of output. In this case, profit maximization is synonymous with loss minimization—the firm is making the best output choice it can. Figure 9.3.5 illustrates a situation of long-run negative profits for a firm.

Will all firms exit the market when profits are negative? No, because as the first firms exit the market, supply will fall, and the equilibrium price will rise. Figure 9.3.6 illustrates this behavior graphically. Once the price rises to the point where it equals $LRATC$, there will no longer be an incentive for firms to leave the market.

The long-run entry and exit dynamic allows us to understand the long-run market supply curve. Entry and exit dynamics will always force the price back to $P_1$ in the long run, as new firms enter to satisfy any new demand and existing firms exit when demand falls. The resulting long-run supply curve is horizontal, as shown in figure 9.3.7.

## 9.4 Heterogeneous Firms and Increasing and Decreasing Cost Industries

Learning Objective 9.4: Demonstrate how increasing and decreasing cost industries affect the long-run market supply curve.

In the previous section, we explicitly assumed homogeneous firms—that is, firms all having identical costs. We also assumed, implicitly, that costs were constant as industry output changed. In this section, we will study what happens when these assumptions do not hold.

Consider first a number of firms that all produce the same good but that all have different costs—that is, heterogeneous firms. It makes sense that the firms with the lowest costs would be the first to provide their output to the market and that they would experience positive profit. Their success would attract new entrants to the market, just as in the case of homogeneous firms. Each new entrant can be expected to be the next lowest-cost firm. Entry will continue until profits reach zero for the most recent entrant, causing it to become the last entrant.

Zero profits will happen when the supply increases enough to push price down to equal the marginal cost of the very last entrant. Since the entrants that came before the last one all have lower costs, they will all continue to experience positive profits. No other firms will enter after the last zero-profit entrant because they are, by assumption, higher-cost firms and will earn negative profits if they do so. After profits reach zero, new entrants will be drawn in only if the price rises.

In the case of heterogeneous firms, the long-run supply curve will be upward sloping, even in the case of perfect competition, as seen in figure 9.4.1.

The second assumption that was implicit in the previous section is that firms are in a constant-cost industry: industries where firms’ costs do not change as industry output changes. So no matter how much total output there is in the industry, all the $LRATC$ curves remain in the same place.

This assumption does not hold for many industries. Some industries are increasing-cost industries: industries where firms’ costs increase as industry output increases. This can happen because as an industry expands the demand for inputs or industry-specific capital increases, which can cause the prices to rise. For example, if the demand for coffee increases due to positive news about the health benefits of consumption, the demand for coffee beans will increase as well, which could lead to an increase in their price. As the industry output increases, the costs of all firms will increase, and the long-run supply curve will slope upward, as shown in figure 9.4.2(a).

Other industries might be decreasing-cost industries: industries where firms’ costs decrease as industry output increases. This could be because these industries have increasing returns to scale or because increased demand for inputs and capital leads to increased returns to scale on the part of the firms that supply these goods. For example, if the demand for coffee increases the demand for espresso machines, espresso machine manufacturers might invest in cost-saving technologies, such as automating parts of the assembly process. As the coffee industry output increases, the cost of espresso machines decreases, the costs of coffee shops decrease, and the long-run supply curve would be downward sloping, as shown in figure 9.4.2(b).

The heterogeneity of a firm’s cost structures and the fact that many or most industries could be described as increasing cost industries lead economists to generally draw the market supply curve as upward sloping. From the information learned in this chapter, you can now see that the market supply curve comes from the firms themselves.

## 9.5 Policy ExampleWill a Carbon Tax Harm the Economy?

Learning Objective 9.5: Predict the effect of a carbon tax on the supply and profit maximization decisions of firms on which it is imposed.

On July 1, 2008, in response to mounting evidence that human activity is contributing to global climate change and that carbon emissions are a key factor in rising earth temperatures, the Canadian province of British Columbia began collecting a tax on carbon emissions. In so doing, British Columbia became the first North American jurisdiction to levy a carbon tax.

The government of British Columbia charges a tax on revenue from the sale of all fuels: gasoline, diesel, natural gas, propane, jet fuel, and coal. This tax is neutral, meaning the proceeds are returned to the citizens of the province through a reduction in income taxes and tax credits.

Though a carbon tax is a popular policy proposal for groups concerned about climate change and greenhouse gases, pro-business groups generally resist it, arguing that carbon taxes would do great harm to the economy. In this chapter, we have studied how cost conditions translate into firm and market supply. Given this analysis, we can come up with a prediction about the way that a carbon tax would affect firms’ production behavior.

Carbon taxes raise the price of energy, which is a production input. Energy usage generally, but not always, varies with the level of output. More intense production is generally associated with higher energy use. So we can assume that the energy input of firms is a variable cost. This means that an increase in the cost of energy will increase not only the total costs of the firms but their marginal costs as well. In figure 9.5.1, we can see the resulting impact on firms and their marginal cost curves.

From this we can anticipate that firms’ supply curves will rise and that the new equilibrium price will increase. So business groups have a legitimate point in worrying about the effect of a carbon tax on firms’ costs. The effect on an individual firm is clear from figure 9.5.1—this tax will lead it to charge higher prices and reduce output.

However, this is only part of the story. Revenue-neutral carbon taxes will increase demand through income tax rebates and credits, which would serve to raise the equilibrium price. This could offset, at least in part, the rise in costs. A true analysis of this policy would also include the economic impact of climate change itself and the benefit of carbon mitigation. This is a subject we will return to in chapter 20, “Externalities.”

## Exploring the Policy Question

1. If a carbon tax is imposed, the costs in a perfectly competitive industry would likely rise. If firms make zero profits prior to the tax and zero profits after the tax, is it correct to say that there is no net effect of the tax?
2. Another policy response to combat carbon emissions is to mandate reductions in energy usage on the part of firms. Using the theory of profit maximizing competitive firms, analyze the impact of this alternate policy.
3. What else would you want to know about carbon emissions, climate change, and the economy to give a full cost-benefit analysis of a carbon tax?

## REVIEW: TOPICS AND RELATED LEARNING OUTCOMES

### 9.1 Output Decisions for Price-Taking Firms

Learning Objective 9.1: Explain how competitive, price-taking firms decide on output levels.

### 9.2 Short-Run Supply

Learning Objective 9.2: Describe how competitive firms make decisions on short-run output and whether to shut down if they experience negative profit.

### 9.3 Long-Run Supply and Market Equilibrium

Learning Objective 9.3: Describe competitive firms’ long-run supply curves and how firms’ entry and exit affect the long-run market equilibrium.

### 9.4 Heterogeneous Firms and Increasing and Decreasing Cost Industries

Learning Objective 9.4: Demonstrate how increasing and decreasing cost industries affect the long-run market supply curve.

### 9.5 Policy ExampleWill a Carbon Tax Harm the Economy?

Learning Objective 9.5: Predict the effect of a carbon tax on the supply and profit maximization decisions of firms on which it is imposed.

## LEARN: KEY TOPICS

### Terms

Profit $(\pi)$

The difference between its total revenue $(TR)$ and its total cost $(TC)$.

Profit maximization rule $(MR=MC)$

Profit is maximized when output is set where marginal revenue equals marginal cost.

Marginal revenue $(MR)$

The change in total revenue from a one-unit change in quantity produced.

Marginal cost $(MC)$

The additional cost incurred from the production of one more unit of output: $MC=\Delta C/ \Delta Q$.

Short-run supply

As long as the market price is above the firm’s average variable cost, the firm will choose to produce an output where $P=MC$. The firm’s marginal cost curve above the $AVC$ curve is the firm’s supply curve. When price is below $AVC$, the firm chooses not to produce any output at all, so the supply for prices below $AVC$ is zero. See figures 9.2.5 and 9.2.6.

Long-run supply $(LRATC)$

A profit maximizing firm still sets output so that marginal revenue equals marginal cost, and since marginal revenue for a perfectly competitive firm is equal to the market price, the marginal cost curve above the long-run average total cost curve $(LRATC)$ represents the firm’s supply curve. The firm’s profits depend on the price relative to the long-run average total cost at the optimal output level for the firm, $Q^*$. As long as profits are not negative, the firm will continue to produce. See figure 9.3.1.

Shut-down rule

A firm should continue to operate as long as its average revenue covers its average variable costs. The short-run objective should be to close the firm if its total revenue is less than variable costs. See figure 9.2.4.

Constant-cost industry

Industries where firms’ costs do not change as industry output changes.

Increasing-cost industry

Industries where firms’ costs increase as industry output increases.

Decreasing-cost industry

Industries where firms’ costs decrease as industry output increases.

Free entry and exit

When there are no special costs, such as technical or legal barriers, to firms entering and exiting the industry.

### Graphs

Profit maximization for a price-taking competitive firm
Positive profit
Zero profit
Negative profit (loss)
Negative profit, but in the short run, firm should continue to operate
Competitive firm’s short-run supply curve
Positive profits in the long run
The long-run supply curve of a perfectly competitive firm
New entrants increase market supply and lower equilibrium price.
Equilibrium with zero profits
Negative profits in the long run
Exiting firms cause a market supply decrease and a rise in the equilibrium price
The long-run market supply curve
Long-run supply in the case of firms with different costs
Long-run cost curves for increasing and decreasing cost industries

Profit

### Profit $(\pi)$

A firm’s profit ($\pi$) is the difference between its total revenue and its total cost

$\pi (Q)=TR(Q)-C(Q)$

### Total Revenue $(TR)$

The total revenue is the quantity of the goods produced multiplied by the sales price of those goods.

$TR(Q)=PQ$

The total cost is the total cost curve $C(Q)$ introduced in chapter 7 and represents the economic cost.

### From Output Decision $(Q)$ to Profits $(\pi)$

The profit maximization rule ($MR=MC$) ensures that the firm is maximizing profit, but it does not ensure that the firm is making positive profits; the firm could be minimizing losses instead of making positive profits. The relationship between output decisions and profits can be mathematically expressed:

$\Pi=TR-TC$
$TR=PQ$
$TC=ATC \times Q$
$(\text{since }ATC=\frac{TC}{Q})$

Thus,

$\Pi=(PxQ)-(ATC \times Q)$
$=(P-ATC)Q$

Since $Q^*$–the quantity for which $MR=MC$–is always positive or zero, whether profit ($\pi$) is positive or negative depends on the price $(P)$ relative to the average total cost at that particular $Q^* (ATC^*)$. If $P \gt ATC^*$, then $\pi \gt 0$. See figure 9.2.1 and 9.2.2.

Marginal cost

Marginal cost (MC) is the additional cost incurred from the production of one more unit of output:

$MC=\Delta C/ \Delta Q$

stipulates that output should be set where marginal revenue equals marginal cost; marginal revenue for a price-taking firm is the price of the good, therefore

$P=MC(Q)$

Marginal revenue

The change in total revenue from a one-unit change in quantity produced. Marginal revenue is expressed as

$MR=\Delta TR/\Delta Q$

For a price-taking firm, this increase in revenue from the sale of an additional unit is exactly the price of that unit. In other words, for a price-taking firm,

$MR=P$

### The Relationship Between Price $(P)$ and Output $(Q)$

Since marginal cost is the additional cost incurred from the production of one more unit of output:

$MC=\Delta C/\Delta Q$

and the profit maximizing rule stipulates that output should be set where marginal revenue equals marginal cost:

$MR=MC$

and since marginal revenue for the price-taking firm is the price of the good:

$MR=P$

a relationship between output $(Q)$ and price $(P)$ can be established:

$P=MC(Q)$