48. Production Function for a Furniture Company The manager of Dixie Furniture Company estimates that the daily output of her factory (in thousands of dollars) @ is given by Q= 5w1/4y3/4 where z is the amount spent on labor and y is the amount spent on capital (both measured in thousands of dollars). a. Find the daily output of the factory if $16,000 is spent on labor and $81,000 is spent on capital each day. b. Suppose that the output of the factory is to be maintained at the level found in part (a). By how much should the amount spent on capital be changed if the amount on labor is increased by $1000? What is the MRTS? Certainly! Let's break down the problem step by step. 1. Daily Output Calculation: The production function for the Dixie Furniture Company is given as: [ Q = 5x{1/41{3/4} | where: o (x) represents the amount spent on labor (in thousands of dollars). o (y) represents the amount spent on capital (in thousands of dollars). We want to find the daily output when $16,000 is spent on labor ((x)) and $81,000 is spent on capital ((y)). Substituting the given values: [ Q = 5(16000) {1/74)(81000)(3/4} | Let's calculate this: [ Q = 5 \\cdot 16*{1/4} \\cdot 81%{3/4} ] First, let's compute the individual terms: o (16M1/4} = 2) o (81%3/4) = 27) Now multiply these values: [ Q = 5 \\cdot 2 \\cdot 27 = 270 | Therefore, the daily output of the factory is $270,000. 2. Change in Capital: We want to maintain the same output level found in part (a) while increasing the labor expenditure by $1000. Let's denote the new labor expenditure as (x'): [ x' = 16000 + 1000 = 17000 ] To find the corresponding capital expenditure ((y")) that maintains the same output, we'll use the production function: [ Q = 5(x){"/4){3/4} ] Solving for (y"): [ 270 = 5 \\cdot 17*1/4} \\cdot (y')*{3/4} ] First, compute (17*{1/4} = 2.571). Now solve for (y'): [ (y)*{3/4} = \\frac{270}5 \\cdot 2.571} = 20.97 ] Taking the fourth root of both sides: [ y' = (20.97)*4 \\approx 194,000 ] Therefore, the new capital expenditure should be approximately $194,000 to maintain the same output level. . Marginal Rate of Technical Substitution (MRTS): The MRTS measures the rate at which one input can be substituted for another while keeping output constant. It is given by: [ MRTS = -\\frac{\\partial Q/\\partial x}{\\partial Q/\\partial y} ] Taking partial derivatives with respect to (x) and (y): [ \\frac{\\partial Q} {\\partial x} = \\frac{SH4}x{-3/4{3/4} ] [ \\frac{\\partial Q}{\\partial y} = \\frac{15} {ap1741{-1/4} | Now evaluate these derivatives at the given values of (x) and (y): [ MRTS = -\\frac{(5/4)(16000)-3/4H81000)(3/4}}{(1 5/4)(16000){1/4}(81000) {-1/4)} ] Calculate the individual terms: w o (\\frac{5}{4}(16000){3/4}81000)(3,4) \\approx 0.00023) o (\\frac{151{4}(16000){1/4H81000)({1 /4} \\approx 0.00069) Finally, compute the MRTS: [ MRTS \\approx -\\frac{0.00023}{0.00069} \\approx -0.333 ] Therefore, the MRTS is approximately -0.333. In summary: Daily output: $270,000 Change in capital to maintain output: Increase by $194,000 MRTS: -0.333