Calculus Expert Required
Budget: $10 – $30 USD
im looking for someone who can help me explain some concepts of cobb douglas.
Suppose the quantity, q, of a product manufactures depends on the number of workers.
W , and the amount of capital invested, K, and is represented by the Cobb-Douglas
function
q = 64W^3/4 K^14 .
Suppose further that labor costs $18 per worker and capital costs $28 per unit, and the budget is $4600.
B. What is the optimum number of workers and the optimum number of units of capital?
C.. Show that at the optimum values of W and K, the ratio of the marginal productivity of labor
( ∂q/∂W) to the marginal productivity of capital
( ∂q/∂K) is the same as the ratio of the cost of a unit of labor to the cost of a unit of capital.
D. Now assume that the product manufactured can be sold for $10 per unit. Give a function P (K, W ) for the profit associated with producing q units.
E. What optimum combination of capital and labor corresponds to maximum profit if
there are no budget constraints?
F. What is the optimal profit when we apply the budget constraint of $4600 to the
formulation?
G.Recompute the optimum values of W , K and profit P (K, W ) when the budget is
increased by one dollar.
H. Let λ be the Lagrange multiplier. Does increasing the budget by $1 allow the production of λ extra units of the product? Explain why. Why are the values equal to each other (1.81=1.81)
I. Does the value of λ change if the budget changes from $4600 to $5600? (Ans: NO, but need help to explain using the G and F equations)
J. What condition must a Cobb-Douglas production function q = cKαW β satisfy to
ensure that the marginal increase of production is not affected by the size of the
budget?
Suppose the quantity, q, of a product manufactures depends on the number of workers.
W , and the amount of capital invested, K, and is represented by the Cobb-Douglas
function
q = 64W^3/4 K^14 .
Suppose further that labor costs $18 per worker and capital costs $28 per unit, and the budget is $4600.
B. What is the optimum number of workers and the optimum number of units of capital?
C.. Show that at the optimum values of W and K, the ratio of the marginal productivity of labor
( ∂q/∂W) to the marginal productivity of capital
( ∂q/∂K) is the same as the ratio of the cost of a unit of labor to the cost of a unit of capital.
D. Now assume that the product manufactured can be sold for $10 per unit. Give a function P (K, W ) for the profit associated with producing q units.
E. What optimum combination of capital and labor corresponds to maximum profit if
there are no budget constraints?
F. What is the optimal profit when we apply the budget constraint of $4600 to the
formulation?
G.Recompute the optimum values of W , K and profit P (K, W ) when the budget is
increased by one dollar.
H. Let λ be the Lagrange multiplier. Does increasing the budget by $1 allow the production of λ extra units of the product? Explain why. Why are the values equal to each other (1.81=1.81)
I. Does the value of λ change if the budget changes from $4600 to $5600? (Ans: NO, but need help to explain using the G and F equations)
J. What condition must a Cobb-Douglas production function q = cKαW β satisfy to
ensure that the marginal increase of production is not affected by the size of the
budget?