# Cost minimizing and calculus

## Homework Statement

Mary Jane grows herbs in her attic. The relationship between her inputs and outputs is given by the production function is Q = 3(K*L)^0.5 , where Q is value added (value of gross output less cost of electricity, fertilizer, seed, and food for a guard dog), K is the value of capital goods (lamps, sprinklers, pots, and security systems), and L is Mary’s hours of labor. The isoquants corresponding to this production function have slope -L/K . The rental cost of capital (interest rate plus depreciation rate) is 0.20. The opportunity cost of Mary’s labor is $7.20 per hour. Mary’s goal is to obtain a value added of$18,000 as cheaply as possible. a) Find the optimal values of K and L. b) Can Mary make a profit?

## The Attempt at a Solution

First of all, I don't understand why the function's derivative is -L/K. I know that to minimize the cost I must solve the equation -L/K = Relative Price of L & K. I think relative price of L & K is (Cost of Labour Per Hour)/(Cost of Rent Per Hour), but I don't know the price of Rent Per Hour. And if anyone could explain me the reason why cost is minimized when the isoquant's derivative equals the relative price of inputs, I'd be grateful.

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## The Attempt at a Solution

First of all, I don't understand why the function's derivative is -L/K.
The isoquants are curves of constant Q. Along these curves, the variation in Q is zero, so

$$0 = dQ = 3 \left( \sqrt{\frac{L}{K}} dK + \sqrt{\frac{K}{L}} dL \right).$$

This means that at constant Q,

$$\frac{dL}{dK} = - \frac{L}{K} .$$

I know that to minimize the cost I must solve the equation -L/K = Relative Price of L & K. I think relative price of L & K is (Cost of Labour Per Hour)/(Cost of Rent Per Hour), but I don't know the price of Rent Per Hour. And if anyone could explain me the reason why cost is minimized when the isoquant's derivative equals the relative price of inputs, I'd be grateful.
Our cost of capital is $$R=0.20$$ while our opportunity cost of labor is $$C= 7.20$$/hr. Our total cost is therefore $$T = R K + CL$$. We want to minimize this subject to the condition $$Q= 18k$$. This means that we want to minimize the function. You should find the minimum yourself, it's not quite what you wrote.