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:
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)
a-1) I agree
a-2) I agree, except I would express it here on Jiskha as (p^2)/3
a-3) I get S(p) = (p^2)/36. Check my work.
for b-2, shouldnt it be: (p^2)/(3*(w1^.5)*(w2^.5) ??
a) To find Earl's marginal cost function, we take the derivative of his cost function with respect to y:
MC(y) = ∂c/∂y = 3w1^(1/2) * w2^(1/2) * y^(1/2)
For the given values of w1 = $1 per pound, w2 = $1 per hour, and the price of lemonade p, Earl's supply function is:
S(p) = p^2/3
If lemons cost $4 per pound and the wage rate is $9 per hour, his supply function will be:
S(p) = p^2/18
b) Earl's marginal cost depends on the price of lemons (w1) and the wage rate (w2). When he is producing y units of lemonade, the marginal cost is:
MC(w1, w2, y) = 3w1^(1/2) * w2^(1/2) * y^(1/2)
The amount that Earl will supply depends on the three variables, p, w1, and w2. As a function of these three variables, his supply function is:
S(p, w1, w2) = p^2 / (3w1 * w2)
To find Earl's marginal cost (MC) function and supply function at different parameter values, we need to substitute the given values into the production function and cost function.
a) When lemons cost $1 per pound, the wage rate is $1 per hour, and the price of lemonade is p:
To find Earl's marginal cost function (MC), differentiate the cost function with respect to y (number of units of lemonade produced):
c(w1, w2, y) = 2w1^(1/2) * w2^(1/2) * y^(3/2)
MC(y) = ∂c/∂y = 3/2 * 2w1^(1/2) * w2^(1/2) * y^(1/2) = 3w1^(1/2) * w2^(1/2) * y^(1/2) = 3 * (1)^(1/2) * (1)^(1/2) * y^(1/2) = 3 * y^(1/2) = 3y^(1/2)
So, Earl's marginal cost function is MC(y) = 3y^(1/2).
To find Earl's supply function (S), we need to find y in terms of p. Substitute the given values into the production function:
f(x1, x2) = x1^(1/2) * x2^(1/3)
Since lemons cost $1 per pound, x1 = p/1 = p.
Since the wage rate is $1 per hour, x2 = 1.
Now, substitute these values into the production function:
f(p, 1) = p^(1/2) * 1^(1/3) = p^(1/2) * 1 = p^(1/2) = √p
So, Earl's supply function is S(p) = √p.
If lemons cost $4 per pound and the wage rate is $9 per hour, we can repeat the process to find Earl's supply function:
x1 = p/4
x2 = 9
f(p/4, 9) = (p/4)^(1/2) * 9^(1/3) = (p/4)^(1/2) * 3 = 3(p/4)^(1/2) = 3√(p/4)
So, Earl's supply function is S(p) = 3√(p/4).
b) To find Earl's marginal cost function and supply function as a function of w1, w2, and y:
Follow the same process as in part a, but keep w1 and w2 as variables instead of substituting specific values. The marginal cost function will be:
MC(w1, w2, y) = 3w1^(1/2) * w2^(1/2) * y^(1/2)
The supply function will be:
S(p, w1, w2) = p^2 / (3w1 * w2)