Answers
Answer: 1st one : degree = 5; number of terms = 3
2nd one: degree = 6; number of terms = 4
3rd one: degree = 4; number of terms = 2
Step-by-step explanation: First you have to simplify everything.
So for the first one you would get 14x^5 -30x^4 + 2x^3; After that you just say that the degree is 5, and number of terms is 3. For the second one you do the same thing; simplify and then find the degree and number of terms. If your confused on how to simplify then look at the work that I've showed below (my work) And if your confused on how to find the degree, it's just what term has the highest power, to be more clear lets say we have x^3 + x^4*y^7. We have 2 terms, and the degree of the 1st is 3, and the degree of the second term is 11. Meaning the degree of the polynomial is 11 (the larger one) If your confused on how to find the amount of terms, then it's just counting them, they're divided by subtraction and addition.
Related Questions
Does this sequence converge or diverge? \left\{\frac{10n-6}{2n-3}\right\} { 2n−3 10n−6 }
Answers
Answer:
The sequence converges
Step-by-step explanation:
Given the sequence Un = 10n-6/2n-3
To determine whether the sequence converges or diverges, we will have to take the limit of the sequence as n goes large.
\(\lim_{n \to \infty} \left\{\frac{10n-6}{2n-3}\right}\\\)
Divide through by n
\(= \lim_{n \to \infty} \left\{\frac{10n/n-6/n}{2n/n-3/n}\right}\\= \lim_{n \to \infty} \left\{\frac{10-6/n}{2-3/n}\right}\\= \left\{\frac{10-6/\infty}{2-3/\infty}\right}\\= \frac{10-0}{2-0}\\= 10/2\\= 5\)
Since the limit of the sequence gives a finite value, hence the sequence converges
40 POINTS FOR ANYONE THAT CAN Answer
n is the middle integer of three consecutive positive integers.
The three integers are multiplied to give a product.
n is then added to the product.
Prove that the result is a cube number.
n must be used in your proof.
Your final line must consist of only,
= n^3
Answers
Answer :
\(\boxed{\textsf{ The final answer is \textbf{n}$^{\textbf{3}}$ .}}\)
Step-by-step explanation:
Its given that n is the middle out of the three consecutive integers . So ,
The last integer will be :-
\(\sf\implies Last \ Integer \ = \ n - 1 \)
The next Integer will be :-
\(\sf\implies Next \ Integer \ = \ n + 1 \)
Now the Question says that the three integers are multipled to give a product . So that would be.
\(\sf\implies Product_{(three\ consecutive\ integers)}= (n-1)n(n+1) = (n^2-1)(n) = \pink{n^3-n}\)
Now thirdly it's given that n is added to the given integer . That would be ,
\(\sf\implies Adding\ n = \ n^3 - n + n = \pink{n^3} \)
Here - n and +n gets cancelled. So we are ultimately left out with n³.
Hence the final number is a cube of some number.
write equation in the standard form then identify the terms and the values of a,b and c
Answers
3 less than or equal to p/12
Answers
Answer:
p greater than or equal to 36
Four congruent right triangles are cut off the corners of a square. The diagram below shows
the dimensions of thesquare and the congruent right triangles.
What is the area of the shaded part of the diagram?
A. 18 sq. in
B. 20 sq. in
C. 28 sq. in
D. 34 sq. in
Answers
Answer:
C
Step-by-step explanation:
Find area of all trinagles and subtract that by total area of square.
Evaluate the expression:
(5/7)³ = ?
Answers
Answer:
125 over 343 as a fraction and 0.364431 as a decimal
Step-by-step explanation:
hope this helps dont know if its right
Answer:
125/343
Step-by-step explanation:
Solve for x: 9(x + 4) = 1 + 2x *
Answers
Answer:
x = -5
Step-by-step explanation:
9(x+4)=1+2x
9x+36 = 1 + 2x
35 = -7x
Divide
-5 = x
x= -5
\( \: \: \: \: \: \: \: 9(x + 4) = 1 + 2x \\ \\ = > 9x + 36 = 1 + 2x \\ \\ = > 9x - 2x = 1 - 36 \\ \\ = > 7x = ( - 35) \\ \\ = > x = (- \frac{35}{7} ) \\ \\ = > x = \green{\boxed{ - 5}}\)
If the cone’s radius is 1/2 what is its height
Answers
The term "directrix" refers to the circumference of a cone's base, and "generatrix" or "generating line" refers to each line segment between the directrix and apex.
What is the parameter for cone radius calculation?(For a relationship between the directrix of a conic section and the name "directrix" in this sense, see Dandelin spheres.
A cone's volume is equal to one-third of the sum of the base's area and the cone's height. Cubic units are used to measure volume.
A cone's radius affects how tall it is. A cone with a radius of 1 /2 has a height of 0.26. Volume is the amount of space that is occupied by a three-dimensional object.
A cone's volume (V) is determined by:
V = (1/3)πr²h
Where r is the radius and h is the height.
For a radius of 1/2:
=(1/3)π*(1/2)²*h
=(1/3)(22/77)(1/4)h
=0.26
Therefore, its height is 0.26.
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Suppose the volume of air inhaled by a person during respiration is given by V(t) = 3/5 pi(1 - cos pi t/2) liters at a time t (in seconds). When is the volume of inhaled air at a maximum? - t = 1, 2, 3, ... - t = 1, 4, 7, ... - t = 2, 4, 6, ... - t = 1, 3, 5, ... - t = 2, 6, 10, ...
Answers
The volume of inhaled air at a maximum at t = 2, 4, 6, ...
The volume of inhaled air is at a maximum when the function V(t) is at a maximum. This occurs when the derivative of V(t) is equal to zero.
The derivative of V(t) is:
V'(t) = (3/5) pi * (pi/2) * sin(pi t/2)
Setting this equal to zero gives:
(pi/2) * sin(pi t/2) = 0
This occurs when pi t/2 is equal to a multiple of pi. This means that t/2 is equal to a multiple of 1, or t is equal to a multiple of 2. Therefore, the volume of inhaled air is at a maximum when t = 2, 4, 6, ...
The correct answer is: t = 2, 4, 6, ...
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QUESTION 3 Find the general solutions of the following differential equations using D-operator methods: -2x 3.1 (D² −5D+6)y= e¯ + sin 2x (8) 3.2 (D² + 2D + 4) y = e²x sin 2x (8) [16]
Answers
To find the general solutions of the given differential equations using D-operator methods, we will use the fact that D-operator (D) represents differentiation with respect to x.
3.1 For the differential equation (D² - 5D + 6)y = e^(-x) + sin(2x), we can factorize the characteristic equation as (D - 2)(D - 3)y = e^(-x) + sin(2x). Solving each factor separately, we have: (D - 2)y = e^(-x) => y₁ = Ae^(2x) + e^(-x) (where A is a constant). (D - 3)y = sin(2x) => y₂ = Bsin(2x) + Ccos(2x) (where B and C are constants). The general solution is y(x) = y₁ + y₂ = Ae^(2x) + e^(-x) + Bsin(2x) + Ccos(2x).
3.2 For the differential equation (D² + 2D + 4)y = e^(2x)sin(2x), the characteristic equation is (D + 2i)(D - 2i)y = e^(2x)sin(2x). Solving each factor separately, we have: (D + 2i)y = e^(2x)sin(2x) => y₁ = Ae^(-2ix)e^(2x)sin(2x) = Ae^(2x)sin(2x)
(D - 2i)y = e^(2x)sin(2x) => y₂ = Be^(2ix)e^(2x)sin(2x) = Be^(2x)sin(2x)
The general solution for the first differential equation is y(x) = Ae^(2x) + e^(-x) + Bsin(2x) + Ccos(2x), and the general solution for the second differential equation is y(x) = Ae^(2x)sin(2x) + Be^(2x)sin(2x), where A, B, and C are constants.
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PLEASE HELP IM GIVING BRAINLIEST!
Answers
Answer:
-1/7
Step-by-step explanation:
Mark me as brainliest
Answer:
m = 1/7
Step-by-step explanation:
(0 - 1)/(-7 - 0)
-1/-7 = 1/7
Find the equation of the line passingthrough the points (3, 3) and (4, 5).y = [? ]x + [AEnter
Answers
Answer:
y = 2x - 3
Explanation:
The slope-intercept form of the equation of a line is generally given as;
\(y=mx+b\)where m = the slope of the line
b = the y-intercept of the line
The slope(m) of the line can be determined using the below formula;
\(m=\frac{y_2-y_1}{x_2-x_1}\)Given the points (3, 3) and (4, 5), so our x1 = 3, x2 = 4, y1 = 3, and y2 = 5.
Let's substitute these values into our slope formula and solve for m;
\(m=\frac{5-3}{4-3}=\frac{2}{1}=2\)Let's go ahead and use point (3, 3) where x = 3 and y = 3 with slope(m) = 2 to determine the y-intercept(b);
\(\begin{gathered} y=mx+b \\ 3=2(3)+b \\ 3=6+b \\ b=3-6 \\ b=-3 \end{gathered}\)We now have m = 2 and b = -3, therefore the required equation of a line can be written as;
\(y=2x-3\)Please can someone answer this question, I never understand these and would like to know how to do them... QUESTION ATTACHED
Answers
Answer:
sqrt (2) * sqrt(45) + sqrt (2) * sqrt (80)
45 = 9 *5 "9" is a perfect square so
sqrt (45) = 3 * sqrt (5) * sqrt (2)
sqrt (2) * sqrt (80)
80 = 16 * 5 "16" is a perfect square so
sqrt (2) * sqrt (80) = sqrt (2) * 4 * sqrt(5)
Adding both answers we get
3 * sqrt (5) * sqrt (2) = 3 * sqrt(10)
PLUS
4 * sqrt (10)
which equals
7 * sqrt (10)
SO, the value of a is 7
Step-by-step explanation:
c) After this tax is collected you can assume that these funds are gone and that no goods or services are purchased with them, and no government employees are paid with this tax revenue. Determine the impact the tax has on the steady state levels of capital per worker \& consumption per worker. Sketch a diagram showing the impact of this shock. Explain what impact the shock has on the level and growth rate of the standard of living (as measured by output per worker) in steady state. ( 8 points)
d) Suppose instead, after the tax is collected, the government is able to use these funds to create and implement plans that cause the growth rate of labour augmenting technological change to rise to 3% per year. Determine the impact the tax has on the steady state levels of capital per effective worker, output per effective worker \& consumption per effective worker. Sketch a diagram showing the impact of this shock. Explain what impact the shock has on the level and growth rate of the standard of living (as measured by output per worker) in steady state. ( 10 points)
Answers
The shock in part (c) leads to a decrease in capital per worker and consumption per worker, potentially affecting the standard of living. In contrast, the shock in part (d) leads to an increase in output per effective worker, which can positively impact the standard of living.
(c) When the tax funds are assumed to be gone without any goods or services purchased or government employees paid, it implies that the tax revenue is completely removed from the economy. In this case, the impact on the steady state levels of capital per worker and consumption per worker would depend on the specific economic model and assumptions.
Generally, the removal of tax revenue would lead to a reduction in both capital per worker and consumption per worker. The exact magnitude of the impact would depend on various factors, such as the marginal propensity to consume and the saving behavior of individuals. In steady state, the reduction in capital per worker could lead to lower productivity and potentially lower output per worker, affecting the standard of living.
To sketch a diagram showing the impact of this shock, you would typically have the levels of capital per worker and consumption per worker on the y-axis and time or steady state on the x-axis. The diagram would show a downward shift in both the capital per worker and consumption per worker curves, indicating a decrease due to the removal of tax revenue.
(d) When the tax funds are used by the government to implement plans that increase the growth rate of labor-augmenting technological change to 3% per year, it implies that the tax revenue is directed towards productivity-enhancing investments or policies. In this case, the impact on the steady state levels of capital per effective worker, output per effective worker, and consumption per effective worker can be analyzed.
The increase in the growth rate of labor-augmenting technological change would lead to higher productivity and potentially higher output per effective worker in steady state. This increase in output per effective worker could also translate into higher consumption per effective worker, depending on the saving and consumption behavior.
To sketch a diagram showing the impact of this shock, you would typically have the levels of capital per effective worker, output per effective worker, and consumption per effective worker on the y-axis and time or steady state on the x-axis. The diagram would show an upward shift in the output per effective worker curve, indicating an increase due to the improved technological change.
Overall, the shock in part (c) leads to a decrease in capital per worker and consumption per worker, potentially affecting the standard of living. In contrast, the shock in part (d) leads to an increase in output per effective worker, which can positively impact the standard of living.
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A ladder leaning against a wall makes a 35° angle with the ground. The foot of the ladder is 5 meters from the wall. What is the length of ladder? (to the nearest tenth)
Answers
Answer:
L = length of the ladder;
K = distance between wall and the foot of the ladder;
a = angle between ladder and the ground
cos(a) = K/L,
L=K/cos(a),
L=K/cos(35°),
L=5/cos(35°),
L=6.1 m
Answer: 6.1
Step-by-step explanation:
Find the particular solution of the following differential equation: dxdy=1+x2+y2+x2y2, given that y = 1 when x = 0
Answers
The particular solution of the given differential equation is y = x2 + 1
The particular solution of the given differential equation is y = x2 + 1.
We can solve this differential equation by using separation of variables. First, we separate the equation into two equations:
dy/dx = 1 + x2 + y2
y2 + x2y = -1
We can integrate the first equation with respect to x:
∫ dy/dx dx = ∫ (1 + x2 + y2) dx
y = x2 + ∫ (1 + y2) dx
We can integrate the second equation with respect to y:
∫ y2 + x2y dy = ∫ -1 dy
y3/3 + x2y2/2 = -y + C
Substituting the value of y from the first equation into the second equation:
(x2 + 1)3/3 + x2(x2 + 1)2/2 = -(x2 + 1) + C
Given that y = 1 when x = 0, we can substitute the values into the equation to obtain the value of the constant:
(0 + 1)3/3 + 0(0 + 1)2/2 = -(0 + 1) + C
1/3 - 0/2 = -1 + C
C = 4/3
Therefore, the particular solution of the given differential equation is y = x2 + 1.
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Read this word problem.
A guppy was swimming 8 feet below the surface of the river. It swam down 6 more feet
to escape a large, hungry fish. What is the position of the guppy now relative to the
surface?
To solve the problem, Jenna subtracted 8 - 6 and came up with an answer of 2 feet. Is Jenna
correct? Why or why not?
Answers
Answer:No because once the guppy swam down 8 feet below the surface of the river the 8 is now negative since the surface of the ocean is 0 and 0-8=-8 and then he swam down 6 feet more so we have to subtract-8-6= -14 feet so the position of the guppy is now -14 feet below sea level.
Step-by-step explanation:
What is Theory X and Y and how it is different from Theory Z when talking about leadership?
Answers
People hate their jobs, want to get away from them, and hate having to be responsible, claims Theory X. People are independent and appreciate having obligations placed on them, claims Theory Y.
What are Theories X and Y?Theories of human labor motivation and management include Theory X and Theory Y.
They were invented by Douglas McGregor in the 1950s while he was a faculty member at the MIT Sloan School of Management, and they underwent further development in the 1960s.
Consider work a natural part of life, and come up with creative solutions for workplace issues.
According to Theory X, individuals despise their jobs, desire to escape them, and detest having to be accountable.
According to Theory Y, people are self-driven and enjoy having responsibilities placed on them.
Therefore, people hate their jobs, want to get away from them, and hate having to be responsible, claims Theory X. People are independent and appreciate having obligations placed on them, claims Theory Y.
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PLS HELP DUE SOON PLS
Answers
Answer:
Step-by-step explanation:
1. (11,9)
2.(8,6)
3. (6,7)
HELPPPPPPPPPPPP[PPPPPP
Answers
Answer:
26mm
Step-by-step explanation:
w+10 + w+ 20 = 144/3
3w+30 = 48
3w = 18
w=6
longest side is 6+20 = 26
Which equation is not a solution to the equation 2^t = sqrt10
Answers
The expression that is not a solution to the equation \(2^t\) = 10 is \(log_{10} 4\). The correct answer is 3.
In order for an expression to be a solution to the equation \(2^t\)= 10, it must yield the value of t that satisfies the equation when substituted into it. Let's evaluate each option to determine which one is not a valid solution:
(1) \(2/1 log 2\): This expression simplifies to log 2, which is not equal to the value of t that satisfies the equation \(2^t\) = 10.
(2) \(log_2\sqrt10\): This expression can be rewritten as \(log_2(10^{(1/2)}).\) By applying the property of logarithms, we can rewrite it as \((1/2)log_2(10)\). Since \(2^(1/2)\) is equal to the square root of 2, this expression simplifies to \((1/2)log_2(2^{(5/2)})\), which is equal to (5/4).
(3)\(log_{10}4\): This expression does not involve the base 2, so it is not a valid solution to the equation \(2^t\) = 10.
(4)\(log_{10} 4\): This expression simplifies to log 4, which is not equal to the value of t that satisfies the equation \(2^t\) = 10.
Therefore, the expression that is not a solution to the equation \(2^t\)= 10 is (3)\(log_{10}4.\)
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Question
Which expression is not a solution to the equation 2^t = 10 ?
(1) 2/1 log 2
(2) log_2\sqrt10
(3) log_104
(4) log_10 4
graham is hiking at an altitude of 14,040 feet and is descending 50 feet each minute. max is hiking at an altitude of 12,500 feet and is ascending 20 feet each minute. how many minutes will it take graham and max to meet at the same altitude? a. 18.5 minutes b. 36 minutes c. 51.3 minutes d. 22 minutes
Answers
Answer: D:22,
14,040-12,500=1540.
1540/70 = 22
If the mass of the empty container is 250 grams and the final mass of the container with liquid is 1,300 grams, calculate the mass of the liquid by itself.
Answers
Hello!
To find the mass of the liquid by itself, subtract the mass of the container from the mass of the liquid and container.
⇒ 1,300 - 250
⇒ 1,050 grams
∴ The mass of the liquid by itself is 1,050 grams.
Answer for brainliest
Answers
Answer:
x = 2
Step-by-step explanation:
The measure of minor arc FD is 51x + 1
The measure of major arc FD is 360 - (51x + 1)
m<E = ½[m(major arc FD) - m(minor arc FD)]
36x + 5 = ½[360 - (51x + 1) - (51x + 1)]
72x + 10 = 360 - 51x - 1 - 51x - 1
174x = 348
x = 2
Point S is on line segment \overline{RT}
. Given ST=3x-8,ST=3x−8, RT=4x,RT=4x, and RS=4x-7,RS=4x−7, determine the numerical length of \overline{RT}.
RT
.
.
Answers
Answer:
RT = 20
Step-by-step explanation:
Point S is on line segment
R------------S------------T
RS + ST = RT
Given
ST=3x-8
RT=4x
RS=4x-7,
Step 1
We find x
4x - 7 + 3x - 8 = 4x
4x + 3x -7 - 8 = 4x
7x - 15 = 4x
7x - 4x = 15
3x = 15
x = 15/3
x = 5
Step 2
We find RT
RT = 4x
x = 5
RT = 4 × 5
RT = 20
The numerical length of RT is 20
Tarush is a landscape architect. For his first public project he is asked to create a small scale drawing of a garden to be placed in the corner of a city park. The garden is a right triangle with base 25, and height 30.
Draw the garden such that 1 unit on the grid below represents 5
Answers
Answer:
Given that the garden is a right triangle with base 25m start and height 30 m The length of the hypothenus can be achieved by using pythagorean theorem. Please find the attached file for the diagram. I made a large unit for the sake of clarity.
Answer:
6.0 on the left side 5.0 at the bottom 7.8 on the right side
Step-by-step explanation:
Draw the following triangle after a translation 1 unit to the left and 1 unit up.
HELP
Answers
Step-by-step explanation:
move the shape 1 square to the left and then move it one square up
The triangle after a translation 1 unit to the left and 1 unit up will have vertices at (2, 2), (-4, 4), and (-2, 0).
Translation is a rigid transformation that moves each point of a figure a certain distance in a specified direction. For this triangle, we will translate all its vertices 1 unit to the left and 1 unit up.
Impact on the Coordinates:
Let's consider each vertex of the triangle and apply the translation:
a) Vertex (3, 1):
After the translation, this point will move 1 unit to the left and 1 unit up. Therefore, the new coordinates will be (3 - 1, 1 + 1) = (2, 2).
b) Vertex (-3, 3):
This point will also move 1 unit to the left and 1 unit up. The updated coordinates will be (-3 - 1, 3 + 1) = (-4, 4).
c) Vertex (-1, -1):
Similarly, this point will undergo the translation, resulting in (-1 - 1, -1 + 1) = (-2, 0).
New Triangle:
Now, we have the new coordinates for the vertices after the translation:
Vertex 1: (2, 2)
Vertex 2: (-4, 4)
Vertex 3: (-2, 0)
The transformed triangle will have vertices at (2, 2), (-4, 4), and (-2, 0).
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Type the correct answer in the box. Use numerals instead of words. If necessary, use/ for the fraction bar.
Given the figure, find the total area of the shaded region.
D
8-
6-
4-
2-
O
-2-
o
S
The area of the shaded region is
B
R
8
с
square units
Answers
The value of the total area of the shaded region are,
⇒ 42 units²
We have to given that;
Sides of rectangle are,
AB = 9
BC = 6
Hence, The area of rectangle is,
⇒ 9 x 6
⇒ 54 units²
And, Area of triangle is,
A = 1/2 × 4 × 6
A = 12 units²
Thus, The value of the total area of the shaded region are,
⇒ 54 - 12
⇒ 42 units²
So, The value of the total area of the shaded region are,
⇒ 42 units²
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What is the slope of the line through (-5, -10) and (-1, 5)?
Answers
Answer:
look at the photo ..............................
2-period production economy: Economy has two periods, = 0,1. There is a
representative household and a representative firm. Household utility is given as
U(Co,C1) = log(Co)+ß log(C1) where ß E (0,1) is a discount factor. Firm production
function is given as F(K,L) = K«L1-a, where a € (0,1) is a capital share. Household is
endowed with initial level of capital K o in period O and maximum labor hours L= 1 in
each period += 0,1. Firms rent capital and hire labor every period and maximize their
profit.
(a) Write down Household's problem
(b) Write down Firm's problem
(c) Write down market clearing conditions
(d) Write down Social Planner's Problem
(e) Define Competitive Equilibrium
(f) Solve Social Planner's Problem: Show your steps to solve it
(g) Solve Competitive Equilibrium: Show your steps to solve it(h) Write down First
Welfare Theorem. Does the theorem hold? Verify it.
(i) Write down Second Welfare Theorem. Does the theorem hold? Verify it.
Answers
The provided questions cover various aspects of a 2-period production economy, including the household's problem, firm's problem, market clearing conditions, Social Planner's Problem, competitive equilibrium, and welfare theorems.
(a) The Household's problem is to maximize its utility over two periods subject to its budget constraint. The household's problem can be formulated as follows:
Max U(Co, C1) = log(Co) + ß log(C1)
subject to the budget constraint:
Co + (1+r)C1 ≤ (1+r)Ko + W0 + W1,
where Co and C1 are consumption in period 0 and 1 respectively, ß is the discount factor, r is the interest rate, Ko is the initial capital endowment, W0 and W1 are the wages in periods 0 and 1 respectively.
(b) The Firm's problem is to maximize its profit by choosing the optimal combination of capital and labor. The firm's problem can be formulated as follows:
Maximize F(K, L) - RK - WL,
where F(K, L) is the production function, K is capital, L is labor, R is the rental rate of capital, and W is the wage rate.
(c) The market clearing conditions are:
Capital market clearing: K1 = (1 - δ)K0 + S - C0, where δ is the depreciation rate, S is savings, and C0 is consumption in period 0.
Labor market clearing: L = L0 + L1, where L0 and L1 are labor supplies in periods 0 and 1 respectively.
(d) The Social Planner's Problem is to maximize social welfare, which is the sum of the household's utility and the firm's profit. The Social Planner's Problem can be formulated as follows:
Maximize U(C0, C1) + F(K, L) - RK - WL,
subject to the production function F(K, L) and the market clearing conditions.
(e) A Competitive Equilibrium is a situation where all markets clear and agents (household and firm) make optimal decisions based on prices and market conditions. It is characterized by the following conditions:
Household's problem is solved optimally.
Firm's problem is solved optimally.
Market clearing conditions hold.
(f) To solve the Social Planner's Problem, we need to set up the Lagrangian and solve for the optimal values of consumption, capital, and labor. The Lagrangian can be written as:
L = U(C0, C1) + F(K, L) - RK - WL + λ1[(1+r)K0 + W0 + W1 - Co - (1+r)C1] + λ2[K1 - (1 - δ)K0 + S - C0] + λ3[L - L0 - L1],
where λ1, λ2, and λ3 are the Lagrange multipliers.
(g) To solve the Competitive Equilibrium, we need to determine the prices of capital (R) and labor (W) that clear the markets. This can be done by equating the demand and supply of capital and labor, and solving the resulting equations.
(h) The First Welfare Theorem states that under certain conditions, a competitive equilibrium is Pareto efficient. It implies that a competitive equilibrium is a socially optimal allocation of resources. To verify the theorem, we need to demonstrate that the competitive equilibrium allocation is Pareto efficient.
(i) The Second Welfare Theorem states that any Pareto efficient allocation can be achieved as a competitive equilibrium with appropriate redistribution of initial endowments.
To verify the theorem, we need to show that given an initial Pareto efficient allocation, we can find prices and redistribution of endowments that lead to a competitive equilibrium that achieves the same allocation.
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