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1. In economics, a Cobb-Douglas production function takes the form Y = AL³ Kº, where Y the output of goods produced by a company in a year, A is a variable capturing technological progress called Total Factor Productiv- ity, L captures the amount of labor going into production (measured in thousands of person-years), and K captures the amount of capital (e.g. machinery, raw material) going into production. Assume throughout this problem that a and 3 are both greater than or equal to zero. dY dt (a) Your company's production goals include making equal to .5. Suppose a and 3 are fixed at 0.5 each, and furthermore right now dA d.K dL you have L = K = A= 1 and = .1. What value of do dt dt you need to meet your company's production goals? dt (b) In words, what does the answer to the previous question mean? =

1. In economics, a Cobb-Douglas production function takes the form Y = AL³ Kº, where Y the output of goods produced by a company in a year, A is a variable capturing technological progress called Total Factor Productiv- ity, L captures the amount of labor going into production (measured in thousands of person-years), and K captures the amount of capital (e.g. machinery, raw material) going into production. Assume throughout this problem that a and 3 are both greater than or equal to zero. dY dt (a) Your company's production goals include making equal to .5. Suppose a and 3 are fixed at 0.5 each, and furthermore right now dA d.K dL you have L = K = A= 1 and = .1. What value of do dt dt you need to meet your company's production goals? dt (b) In words, what does the answer to the previous question mean? =

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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1. In economics, a **Cobb-Douglas production function** takes the form: \[ Y = AL^\beta K^\alpha, \] where \( Y \) is the output of goods produced by a company in a year, \( A \) is a variable capturing technological progress called **Total Factor Productivity**, \( L \) captures the amount of labor going into production (measured in thousands of person-years), and \( K \) captures the amount of capital (e.g. machinery, raw material) going into production. Assume throughout this problem that \( \alpha \) and \( \beta \) are both greater than or equal to zero. (a) Your company’s production goals include making \(\frac{dY}{dt}\) equal to 0.5. Suppose \( \alpha \) and \( \beta \) are fixed at 0.5 each, and furthermore right now you have \( L = K = A = 1 \) and \(\frac{dA}{dt} = \frac{dK}{dt} = 0.1\). What value of \(\frac{dL}{dt}\) do you need to meet your company’s production goals? (b) In words, what does the answer to the previous question mean? (c) Suppose the production function displays **constant returns to scale**, meaning that \( \alpha + \beta = 1 \). Suppose further that we have \( L = K = A = \frac{dA}{dt} = \frac{dK}{dt} = \frac{dY}{dt} = 1\). Which value of \( \alpha \) will maximize \(\frac{dL}{dt}\)?
Transcribed Image Text:1. In economics, a **Cobb-Douglas production function** takes the form: \[ Y = AL^\beta K^\alpha, \] where \( Y \) is the output of goods produced by a company in a year, \( A \) is a variable capturing technological progress called **Total Factor Productivity**, \( L \) captures the amount of labor going into production (measured in thousands of person-years), and \( K \) captures the amount of capital (e.g. machinery, raw material) going into production. Assume throughout this problem that \( \alpha \) and \( \beta \) are both greater than or equal to zero. (a) Your company’s production goals include making \(\frac{dY}{dt}\) equal to 0.5. Suppose \( \alpha \) and \( \beta \) are fixed at 0.5 each, and furthermore right now you have \( L = K = A = 1 \) and \(\frac{dA}{dt} = \frac{dK}{dt} = 0.1\). What value of \(\frac{dL}{dt}\) do you need to meet your company’s production goals? (b) In words, what does the answer to the previous question mean? (c) Suppose the production function displays **constant returns to scale**, meaning that \( \alpha + \beta = 1 \). Suppose further that we have \( L = K = A = \frac{dA}{dt} = \frac{dK}{dt} = \frac{dY}{dt} = 1\). Which value of \( \alpha \) will maximize \(\frac{dL}{dt}\)?
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