Please check and correct my answers. Thank you.
1) Jack Benny can get blood from a stone. If he has x stones, the number of pints of blood he can extract from them is f(x) = 2x^(1/3). Stones cost Jack w dollars each. Jack can sell each pint of blood for p dollars.
a) How many stones does Jack need to extract y pints of blood?
b) What is the cost of extracting y pints of blood?
c) What is Jack’s supply function when stones cost w each? When stones cost $8 each?
d) If Jack has 19 relatives who can also get blood from a stone in the same way, what is the aggregate supply function for blood, Y, when stones cost w dollars each?
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my answers:
1)
a)
y^3/8
b)
(w)(y^3/8)
c)
S(p,w) = (8p/3w)^(1/2)
S(p) = (p/3)^(1/2)
d)
Y = 20(8p/3w)^(1/2)
2) Earl’s production function is f(x1, x2) = x1^(1/2) * x2^(1/3), where x1 is the number of pounds of lemons he uses and x2 is the number of hours he spends squeezing them. His cost function is c(w1, w2, y) = 2w1^(1/2) * w2^(1/2) * y^(3/2), where w1 is the cost per pound, w2 is the wage rate, and y is the number of units of lemonade produced.
a) If lemons cost $1 per pound, the wage rate is $1 per hour, and the price of lemonade is p, find Earl’s marginal cost function and his supply function. If lemons cost $4 per pound and the wage rate is $9 per hour, what will be his supply function be?
b) In general, Earl’s marginal cost depends on the price of lemons and the wage rate. At prices w1 for lemons and w2 for labour, what is his marginal cost when he is producing y units of lemonade? The amount that Earl will supply depends on the three variables, p, w1, w2. As a function of these three variables, what is Earl’s supply?
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my answers:
2)
a)
MC(y) = 3y^(1/2)
S(p) = p^2/3
S(p) = p^2/18
b)
MC(w1, w2, y) = 3w1^(1/2) * w2^(1/2) * y^(1/2)
S (p, w1, w2) = p^2 / (3 w1 * w2)
I first looked at your first question, and then I look at your answers. There seems to be a dis-connect. Your answers don't seem to match the questions. Here are my answers
a) y=f(x)=2x^(1/3)
b) The cost of producing y is equal to the number of number of stones needed times w. x= (y/2)^3. So C(y) = w*(y/2)^3
c) The supply function is equal to the marginal cost function. So MC(y) = 3*w*(y/2)^2
d) I need to think a bit about converting the MC curve into a supply function as y=f(w)
I don't agree with your answers because none of those are one of the options to the correct answers:
1)
a)
y^3/8, y/2, 2y^(1/3), 2y, y^2/4, 8y, 4y^2
b)
wy^3/8, wy^2/4, wy/2, 4wy^2, 2wy^(1/3), 8wy, 2wy
c)
(8p/3w)^(1/2), (3pw/8)^2, 3pw^2/8, (8w/3p)^(1/2), pw^2/8, 8p/w, 8p/3w
(p/3)^(1/2), p/3, p^2, 3p, 9p^2, (3/p)^(1/2), p
d)
160pw, 160p/w, 60pw^2/8, 5pw^2/2, 20(8p/3w)^(1/2), 160p/3w, 20(8w/3p)^(1/2)
Ok, I see where you are going. I saw your answer y^3/8 and went huh?? What you really needed to answer was (y^3)/8. I attribute the problem to it being difficult to express mathematical terms here on the Jiskha site. I said y=f(x)=2x^(1/3). However my re-arranging terms, and getting the x all by itself, x=(y^3)/8. So, you are correct.
b) you are correct, (except for re-writing your answer as (w)(y^3)/8. Which, by the way, is equal to my answer.
c) You are correct. setting MC=p, then rearranging terms gets y=((p*8)/(3*w))^(1/2)
d) you are correct.
1) Jack Benny can get blood from a stone. If he has x stones, the number of pints of blood he can extract from them is f(x) = 2x^(1/3). Stones cost Jack w dollars each. Jack can sell each pint of blood for p dollars.
a) To find the number of stones Jack needs to extract y pints of blood, we can rearrange the equation f(x) = 2x^(1/3) to solve for x:
y = 2x^(1/3)
Cube both sides of the equation to eliminate the exponent:
y^3 = 8x
Divide both sides by 8:
x = y^3/8
So, Jack needs (y^3/8) stones to extract y pints of blood.
b) The cost of extracting y pints of blood can be calculated by multiplying the cost of each stone by the number of stones used. Since each stone costs w dollars, the cost of extracting y pints of blood is:
Cost = w * (y^3/8)
c) Jack's supply function is a function that gives the quantity of blood supplied by Jack at a given price and cost of stones. We can express this supply function as:
S(p,w) = (8p/3w)^(1/2)
Substituting w = $8, we get:
S(p,8) = (8p/3*8)^(1/2)
S(p,8) = (p/3)^(1/2)
d) If Jack has 19 relatives who can also extract blood from a stone in the same way, the aggregate supply function for blood, Y, when stones cost w dollars each can be calculated by multiplying Jack's supply function by the number of individuals (20 in total):
Y = 20 * (8p/3w)^(1/2)
Y = 20 * (8p/3*8)^(1/2)
Y = 20 * (p/3)^(1/2)
Therefore, the aggregate supply function for blood is Y = 20 * (p/3)^(1/2).
2) Earl’s production function is f(x1, x2) = x1^(1/2) * x2^(1/3), where x1 is the number of pounds of lemons he uses and x2 is the number of hours he spends squeezing them. His cost function is c(w1, w2, y) = 2w1^(1/2) * w2^(1/2) * y^(3/2), where w1 is the cost per pound, w2 is the wage rate, and y is the number of units of lemonade produced.
a) To find Earl’s marginal cost function, we need to find the partial derivative of the cost function with respect to the number of units of lemonade produced (y):
MC(y) = ∂c/∂y = (3/2) * 2w1^(1/2) * w2^(1/2) * y^(1/2)
MC(y) = 3w1^(1/2) * w2^(1/2) * y^(1/2)
For the supply function, we need to find the partial derivative of the production function with respect to the number of pounds of lemons used (x1):
∂f/∂x1 = (1/2) * x1^(-1/2) * x2^(1/3)
∂f/∂x1 = x2^(1/3) / (2 * x1^(1/2))
Now, we can express the supply function as:
S(p) = (∂f/∂x1)/(∂f/∂y)
Substituting the partial derivatives we found:
S(p) = (x2^(1/3) / (2 * x1^(1/2))) / (3w1^(1/2) * w2^(1/2) * y^(1/2))
S(p) = p^2/3
When lemons cost $4 per pound and the wage rate is $9 per hour, the supply function becomes:
S(p) = (4^2) / 3 = 16/3
b) Earl's marginal cost when producing y units of lemonade depends on both the price of lemons (w1) and the wage rate (w2). To calculate the marginal cost, we need to take the partial derivative of the cost function with respect to w1 and w2:
MC(w1, w2, y) = ∂c/∂w1 = (1/2) * 2 * w1^(-1/2) * w2^(1/2) * y^(3/2)
MC(w1, w2, y) = w2^(1/2) * y^(3/2) / w1^(1/2)
The supply as a function of p, w1, and w2 can be expressed as:
S(p, w1, w2) = p^2 / (3 * w1 * w2)
Therefore, Earl's marginal cost and supply function when considering the three variables, p, w1, w2, are given by MC(w1, w2, y) = w2^(1/2) * y^(3/2) / w1^(1/2) and S(p, w1, w2) = p^2 / (3 * w1 * w2) respectively.