Hint for part i: The market price, $p_0$, and the corresponding quantity, $x_0$, satisfy the condition $D(x_0) = S(x_0)$. In this case, this equation can be rearranged into a quadratic equation which can easily be solved to find $x_0$. Then, $p_0$ is simply equal to $D(x_0)$ (which, by definition of $x_0$, is equal to $S(x_0)$). Keep in mind that a quadratic equation can have two solutions; it is up to you to determine which one is valid.
", "customMarkingAlgorithm": "", "scripts": {}, "type": "information", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "marks": 0, "unitTests": []}], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "prompt": "Consider a market with one product, and let $x$ denote the number of units of this product and $p$ denote the price per unit. You may assume that $x$ and $p$ are both continuous variables and that $x \\in [0, \\var{{b}}]$.
\nLet the inverse demand function be given by $p = D(x) = \\simplify{{b}^2} - x^2$, and let the inverse supply function be given by $p = S(x) = \\simplify{({b}-{a})*x +({b}-{a})*{b}}$
\ni) Find the market price, $p_0$, and the corresponding quantity, $x_0$.
$x_0 =$ [[0]]
$p_0 =$ [[1]]
ii) Now find the consumer' surplus and the producer's surplus. For the first and fifth gap below, you will need to write an explicit equation, and for the second and fourth gaps you will need to write an explicit quantity. (We recommend that you look at section 9.3 of the lecture notes before continuing with the question). As usual, if you want to write something like $2x^2$ then please write 2*x^2 and not 2x^2, as the latter may not be interpreted correctly by the system.
\nThe consumer's surplus is
$\\int_{0}^{x_0}$ [[2]] $\\mathrm{d} x - $ [[3]] $=$ [[4]] .
The producer's surplus is
[[5]] $- \\int_{0}^{x_0}$ [[6]] $\\mathrm{d} x =$ [[7]] .
Hint for part i: Remember that when you integrate $R'$ and $C'$ (to obtain $R$ and $Q$, respectively) you will obtain integration constants. You must determine the exact value of these constants; they are not arbitrary. For $R$, it is helpful to note that when you have no quantity, there will be no revenue. For $C$, it is helpful to note that even when you have no quantity, you still have the fixed costs.
\nHint for part ii: Try finding the stationary point/points of $\\pi$.
", "customMarkingAlgorithm": "", "scripts": {}, "type": "information", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "marks": 0, "unitTests": []}], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "prompt": "A firm has fixed costs of $\\var{{h}}$.
\nIts marginal revenue function is $R' : [0, \\infty ) \\longrightarrow \\mathbb{R}$ defined by $R'(q) = \\simplify{-2*{c}*q + 2*{m}*{f}}$ for all $q \\in [0, \\infty )$.
\nIts marginal cost function is $C' :[0, \\infty ) \\longrightarrow \\mathbb{R}$ defined by $C'(q) = \\simplify{2*{f}*q - 2*{m}*{c}}$ for all $q \\in [0, \\infty )$.
\n$q$ represents the quantity sold.
\ni) Find the revenue function $R$ and the cost function $C$.
(As usual, if you want to write something like $2q^2$ then please write 2*q^2 and not 2q^2, as the latter may not be interpreted correctly by the system.)
$R(x) =$ [[0]] .
\n$C(x) =$ [[1]] .
\nii) The profit function is $\\pi : [0, \\infty ) \\longrightarrow \\mathbb{R}$ defined by $\\pi (q) = R(q) - C(q)$ for all $q \\in [0, \\infty )$. Find the value of $q$ which maximises $\\pi (q)$.
\n$q=$ [[2]]
\nMake sure you fully justify (to yourself) that this is really the global maximum point of $\\pi$
(and not the global minimum for example).
The remainder of this question will look at matrices.
\nLet $M$ be a matrix. What does it mean to say that $M$ is symmetric?
\n[[0]]
\nLet $N$ be a matrix. We say that $N$ is skew symmetric if and only if $N = - N^{\\text{T}}$. Which one of the following matrices is skew symmetric?
\n[[1]]
\n", "sortAnswers": false, "marks": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "gaps": [{"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "matrix": [0, 0, 0, "1", 0], "distractors": ["", "", "", "", ""], "type": "1_n_2", "displayColumns": "1", "showFeedbackIcon": true, "scripts": {}, "showCorrectAnswer": false, "minMarks": 0, "choices": ["$M$ is equal to its inverse. That is, $M = M^{-1}$.
", "The transpose of $M$ is equal to its inverse. That is, $M^{\\text{T}} = M^{-1}$.
", "$M$ is equal to $-M$.
", "$M$ is equal to its transpose. That is $M = M^{\\text{T}}$.
", "$M$ is equal to \"minus\" its transpose. That is $M = -M^{\\text{T}}$.
"], "displayType": "radiogroup", "shuffleChoices": true, "marks": 0, "maxMarks": 0, "customMarkingAlgorithm": "", "unitTests": []}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "matrix": ["1", 0, 0], "distractors": ["", "", ""], "type": "1_n_2", "displayColumns": "1", "showFeedbackIcon": true, "scripts": {}, "showCorrectAnswer": false, "minMarks": 0, "choices": ["$\\begin{bmatrix} 0 & 1 & 5 \\\\ -1 & 0 & 9 \\\\ -5 & -9 & 0 \\end{bmatrix}$
", "$\\begin{bmatrix} 1 & 1 \\\\ -1 & -1 \\end{bmatrix}$
", "$\\begin{bmatrix} 0 & 1 & 5 \\\\ -1 & 1 & 9 \\\\ -5 & -9 & 0 \\end{bmatrix}$
"], "displayType": "radiogroup", "shuffleChoices": true, "marks": 0, "maxMarks": 0, "customMarkingAlgorithm": "", "unitTests": []}], "unitTests": [], "customMarkingAlgorithm": "", "type": "gapfill"}, {"steps": [{"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "prompt": "We know that $S = \\begin{bmatrix} \\var{{S11}} & a & \\var{{S13}} \\\\ \\var{{S21}} & \\var{{S22}} & c \\\\ b & \\var{{S32}} & \\var{{S33}} \\end{bmatrix}$. We recommend that you use this to find $S^{\\text{T}}$.
\nSince we also know that $S = S^{\\text{T}}$, we can compare the entries of $S$ and $S^{\\text{T}}$ to determine the values of $a,b,$ and $c$.
\nYou can do the same for $K$, but remember that $K$ is skew symmetric, meaning you want to find $-K^{\\text{T}}$.
", "customMarkingAlgorithm": "", "scripts": {}, "type": "information", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "marks": 0, "unitTests": []}], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "prompt": "Consider the matrices
\n$S = \\begin{bmatrix} \\var{{S11}} & a & \\var{{S13}} \\\\ \\var{{S21}} & \\var{{S22}} & c \\\\ b & \\var{{S32}} & \\var{{S33}} \\end{bmatrix}$
\nand
\n$K = \\begin{bmatrix} r & \\var{{K12}} & s \\\\ t & u & v \\\\ \\var{{K31}} & \\var{{K32}} & 0 \\end{bmatrix}$
\nfor some $a,b,c,r,s,t,u,v \\in \\mathbb{R}$. We are told that $S$ is symmetric and that $K$ is skew symmetric. Find the values of $a,b,c,r,s,t,u,$ and $v$.
\n$a=$ [[0]] ; $b=$ [[1]] ; $c=$ [[2]] ; $r=$ [[3]] ;
\n$s=$ [[4]] ; $t=$ [[5]] ; $u=$ [[6]] ; $v=$ [[7]] .
", "sortAnswers": false, "marks": 0, "showFeedbackIcon": true, "stepsPenalty": 0, "showCorrectAnswer": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "gaps": [{"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "correctAnswerFraction": false, "showFeedbackIcon": true, "mustBeReduced": false, "allowFractions": false, "minValue": "S21", "correctAnswerStyle": "plain", "scripts": {}, "type": "numberentry", "maxValue": "S21", "marks": "0.5", "customMarkingAlgorithm": "", "unitTests": []}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "correctAnswerFraction": false, "showFeedbackIcon": true, "mustBeReduced": false, "allowFractions": false, "minValue": "S13", "correctAnswerStyle": "plain", "scripts": {}, "type": "numberentry", "maxValue": "S13", "marks": "0.5", "customMarkingAlgorithm": "", "unitTests": []}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "correctAnswerFraction": false, "showFeedbackIcon": true, "mustBeReduced": false, "allowFractions": false, "minValue": "S32", "correctAnswerStyle": "plain", "scripts": {}, "type": "numberentry", "maxValue": "S32", "marks": "0.5", "customMarkingAlgorithm": "", "unitTests": []}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "correctAnswerFraction": false, "showFeedbackIcon": true, "mustBeReduced": false, "allowFractions": false, "minValue": "0", "correctAnswerStyle": "plain", "scripts": {}, "type": "numberentry", "maxValue": "0", "marks": "0.5", "customMarkingAlgorithm": "", "unitTests": []}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "correctAnswerFraction": false, "showFeedbackIcon": true, "mustBeReduced": false, "allowFractions": false, "minValue": "-K31", "correctAnswerStyle": "plain", "scripts": {}, "type": "numberentry", "maxValue": "-K31", "marks": "0.5", "customMarkingAlgorithm": "", "unitTests": []}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "correctAnswerFraction": false, "showFeedbackIcon": true, "mustBeReduced": false, "allowFractions": false, "minValue": "-K12", "correctAnswerStyle": "plain", "scripts": {}, "type": "numberentry", "maxValue": "-K12", "marks": "0.5", "customMarkingAlgorithm": "", "unitTests": []}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "correctAnswerFraction": false, "showFeedbackIcon": true, "mustBeReduced": false, "allowFractions": false, "minValue": "0", "correctAnswerStyle": "plain", "scripts": {}, "type": "numberentry", "maxValue": "0", "marks": "0.5", "customMarkingAlgorithm": "", "unitTests": []}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "correctAnswerFraction": false, "showFeedbackIcon": true, "mustBeReduced": false, "allowFractions": false, "minValue": "-K32", "correctAnswerStyle": "plain", "scripts": {}, "type": "numberentry", "maxValue": "-K32", "marks": "0.5", "customMarkingAlgorithm": "", "unitTests": []}], "type": "gapfill", "customMarkingAlgorithm": "", "unitTests": []}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "prompt": "In this part of the question we will explore the concept of an inverse of a $2 \\times 2$ matrix.
\nLet $a,b,c,d \\in \\mathbb{R}$ with $ad-bc \\neq 0$, and consider the matrix $M=\\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix}$. Suppose there exists a matrix $M^{-1} = \\begin{bmatrix} w & x \\\\ y & z \\end{bmatrix}$, for some $w,x,y,z \\in \\mathbb{R}$, such that $\\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\begin{bmatrix} w & x \\\\ y & z \\end{bmatrix} = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}$. We want to express $w,x,y,z$ in terms of $a,b,c,d$.
\nBy expanding the left-hand-side of the matrix equation above, and comparing the entries to the right-hand-side, we obtain four simultaneous equations.
\n(In the following, if you wish to write something like $cw$ then please write c*w and not cw, as the latter may not be interpreted correctly by the system).
By comparing the top left entries of both sides, we obtain the simultaneous equation [[0]] $=1$.
By comparing the top right entries of both sides, we obtain the simultaneous equation [[1]] $=0$.
By comparing the bottom left entries of both sides, we obtain the simultaneous equation [[2]] $=0$.
By comparing the bottom right entries of both sides, we obtain the simultaneous equation [[3]] $=1$.
We will treat these as a system of simultaneous equations with the unknows being $w,x,y,z$. That is, we wish to solve the system to obtain $w,x,y,z$ in terms of $a,b,c,d$. Note that two of the equations involve the unknowns $w$ and $y$ only, and the other two equations involve the unknowns $x$ and $z$ only. This makes it easier to solve the system:
(Again, if you wish to write something like $cw$ then please write c*w and not cw, as the latter may not be interpreted correctly by the system).
$w=$ [[4]] .
$x=$ [[5]] .
$y=$ [[6]] .
$z=$ [[7]] .
Please press the \"Show feedback\" button when you have completed this part of the question.
Week 5 (lectures 9 and 10): In this question, you will apply integration and differentiation to economics-related concepts such as an inverse demand function, an inverse supply function, market price, consumer's surplus, producer's surplus, a revenue function, and a cost function. You will also look at symmetric matrices, skew-symmetric matrices, and $2 \\times 2$ inverse matrices.
\nIf you have not provided an answer to every input gap of a question or part of the question, and you try to submit your answers to the question or part, then you will see the message \"Can not submit answer - check for errors\". In reality your answer has been submitted, but the system is just concerned that you have not submitted an answer to every input gap. For this reason, please ensure that you provide an answer to every input gap in the question or part before submitting. Even if you are unsure of the answer, write down what you think is most likely to be correct; you can always change your answer or retry the question.
\nAs with all questions, there may be parts where you can choose to \"Show steps\". This may give a hint, or it may present sub-parts which will help you to solve that part of the question. Furthermore, remember to always press the \"Show feedback\" button at the end of each part. Sometimes, helpful feedback will be given here, and often it will depend on how correctly you have answered and will link to other parts of the question. Hence, always retry the parts until you obtain full marks, and then look at the feedback again.
Keep in mind that in order to see the feedback for a particular part of a question, you must provide a full (but not necessarily correct) answer to that part. Do not worry though, as you can look at the feedback and then ammend your answer accordingly.
Furthermore, as with all questions, choosing to reveal the answers will only show you the answers which change every time the question is loaded (i.e. answers to randomised questions); the fixed answers will not be revealed.
This is the question for week 5 of the MA100 course at the LSE. It looks at material from chapters 9 and 10.
The following describes how we define our revenue and cost functions for part b of the question.
We have variables c, f, m, h.
\nThe revenue function is R(q) = -c q^2 + 2mf q .
The cost function is C(q) = f q^2 - 2mc q + h .
The \"revenue - cost\" function is -(c+f) q^2 +2m(c+f) q - h
\nDifferentiating, we see that there is a maximum point at m.
\nWe pick each one of f, m, h randomly from the set {2, .. 6}, and we pick c randomly from {h+1 , ... , h+5}. This ensures that the discriminant of the \"revenue - cost\" function is positive, meaning there are two real roots, meaning the maximum point lies above the x-axis. I.e. we can actually make a profit.
"}, "advice": "", "variables": {"S32": {"templateType": "anything", "group": "Variables for part d", "name": "S32", "definition": "random(1..9)", "description": "This is a constant that will appear in our symmetric matrix, S, in part d.
"}, "a": {"templateType": "anything", "group": "Variables for part a", "name": "a", "definition": "random(2..6)", "description": "A constant that is used in our functions for part a.
"}, "K32": {"templateType": "anything", "group": "Variables for part d", "name": "K32", "definition": "random(1..9)", "description": "This is a constant that will appear in our skew symmetric matrix, K, in part d.
"}, "m": {"templateType": "anything", "group": "Variables for part b", "name": "m", "definition": "random(2..6)", "description": "This is a constant that appears in our functions for part b. See the description of this question for more details.
"}, "S21": {"templateType": "anything", "group": "Variables for part d", "name": "S21", "definition": "random(1..9)", "description": "This is a constant that will appear in our symmetric matrix, S, in part d.
"}, "b": {"templateType": "anything", "group": "Variables for part a", "name": "b", "definition": "random(a+1..a+5)", "description": "A constant that is used in our functions for part a.
"}, "S22": {"templateType": "anything", "group": "Variables for part d", "name": "S22", "definition": "random(1..9)", "description": "This is a constant that will appear in our symmetric matrix, S, in part d.
"}, "h": {"templateType": "anything", "group": "Variables for part b", "name": "h", "definition": "random(2..6)", "description": "This is a constant that appears in our functions for part b. See the description of this question for more details.
"}, "f": {"templateType": "anything", "group": "Variables for part b", "name": "f", "definition": "random(2..6)", "description": "This is a constant that appears in our functions for part b. See the description of this question for more details.
"}, "S33": {"templateType": "anything", "group": "Variables for part d", "name": "S33", "definition": "random(1..9)", "description": "This is a constant that will appear in our symmetric matrix, S, in part d.
"}, "S11": {"templateType": "anything", "group": "Variables for part d", "name": "S11", "definition": "random(1..9)", "description": "This is a constant that will appear in our symmetric matrix, S, in part d.
"}, "c": {"templateType": "anything", "group": "Variables for part b", "name": "c", "definition": "random(h+1 .. h+5)", "description": "This is a constant that appears in our functions for part b. See the description of this question for more details.
"}, "K31": {"templateType": "anything", "group": "Variables for part d", "name": "K31", "definition": "random(1..9)", "description": "This is a constant that will appear in our skew symmetric matrix, K, in part d.
"}, "S13": {"templateType": "anything", "group": "Variables for part d", "name": "S13", "definition": "random(1..9)", "description": "This is a constant that will appear in our symmetric matrix, S, in part d.
"}, "K12": {"templateType": "anything", "group": "Variables for part d", "name": "K12", "definition": "random(1..9)", "description": "This is a constant that will appear in our skew symmetric matrix, K, in part d.
"}}, "ungrouped_variables": [], "extensions": [], "preamble": {"js": "", "css": ""}, "type": "question", "contributors": [{"name": "Michael Yiasemides", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1440/"}]}]}], "contributors": [{"name": "Michael Yiasemides", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1440/"}]}