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)
To find Earl's marginal cost function and supply function, we need to substitute the given values into the production and cost functions.
a) Given that lemons cost $1 per pound (w1 = $1), the wage rate is $1 per hour (w2 = $1), and the price of lemonade is p, we can find the marginal cost function and supply function.
Marginal Cost (MC):
To find the marginal cost (MC), we 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)
Differentiating with respect to y:
MC = ∂c/∂y = 3/2 * 2 * w1^(1/2) * w2^(1/2) * y^(1/2)
MC = 3 * w1^(1/2) * w2^(1/2) * y^(1/2)
MC = 3y^(1/2) [since w1 = $1 and w2 = $1]
Therefore, Earl's marginal cost function is MC(y) = 3y^(1/2).
Supply Function (S):
To find the supply function (S), we need to find the number of units of lemonade produced (y) as a function of the price (p). We can do this by inverting the production function.
f(x1, x2) = x1^(1/2) * x2^(1/3)
Solving for x2 in terms of x1 and substituting x1 for y, we get:
x2 = y^(3/2)
x2^(1/3) = y^(1/2)
Substituting these values into Earl's production function:
f(y^(1/2), y^(3/2)) = (y^(1/2))^(1/2) * (y^(3/2))^(1/3)
f(y^(1/2), y^(3/2)) = y^(1/4) * y = y^(5/4)
Simplifying further:
f(y^(1/2), y^(3/2)) = y^(5/4)
p = y^(5/4)
To find the supply function, we need to solve for y in terms of p:
p = y^(5/4)
y = p^(4/5)
Therefore, Earl's supply function is S(p) = p^(4/5).
Now, if lemons cost $4 per pound (w1 = $4) and the wage rate is $9 per hour (w2 = $9), we can determine the new supply function.
Marginal Cost (MC):
MC(w1, w2, y) = 3w1^(1/2) * w2^(1/2) * y^(1/2)
MC($4, $9, y) = 3(4)^(1/2) * (9)^(1/2) * y^(1/2)
MC($4, $9, y) = 6 * 3 * y^(1/2)
MC($4, $9, y) = 18y^(1/2)
Supply Function (S):
S(p, w1, w2) = p^2 / (3w1 * w2)
S(p, $4, $9) = p^2 / (3 * 4 * 9)
S(p, $4, $9) = p^2 / 108
Therefore, when lemons cost $4 per pound and the wage rate is $9 per hour, Earl's supply function is S(p) = p^2 / 108.