Consider a firm whose production function is Y = AK LM. Moreover, a + +y ...

 

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Consider a firm whose production function is Y = AK LM. Moreover, a + +y <1, i.e., the production function is decreasing returns to scale. At time t = 0, the firm rents capital K, hires labor L, and buys raw materials M. At t = 1, the firm sells its output at price p, pays capital rent at rate r, pays wages at rate w, and pays for raw materials at price pm. 1. Formulate the profit maximization problem. 2. Write the first order conditions, i.e., set the derivatives of profit with respect to the 3 inputs equal to zero. 3. Log-linearize the results from last part, and present the system of equations in matrix form. 4. Solve the matrix equation using MATLAB to find the firm's optimal level of the 3 inputs. Consider a firm whose production function is Y = AK LM. Moreover, a + +y <1, i.e., the production function is decreasing returns to scale. At time t = 0, the firm rents capital K, hires labor L, and buys raw materials M. At t = 1, the firm sells its output at price p, pays capital rent at rate r, pays wages at rate w, and pays for raw materials at price pm. 1. Formulate the profit maximization problem. 2. Write the first order conditions, i.e., set the derivatives of profit with respect to the 3 inputs equal to zero. 3. Log-linearize the results from last part, and present the system of equations in matrix form. 4. Solve the matrix equation using MATLAB to find the firm's optimal level of the 3 inputs.