Book cover for Intermediate Microeconomics: A Modern Approach

Intermediate Microeconomics: A Modern Approach

Hal R. Varian

ISBN #9780393927023

7th Edition

224 Questions

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Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter on Technology covers the technical aspects of production technology by examining different production functions such as fixed proportions, perfect substitutes, and Cobb-Douglas. It emphasizes key production measures like marginal product and technical rate of substitution to analyze input efficiency and substitution possibilities. Additionally, the section distinguishes between short-run and long-run production and discusses returns to scale, which are crucial for understanding how changes in inputs affect overall production output.

Learning Objectives

1

Describe various production functions including fixed proportions, perfect substitutes, and Cobb-Douglas.

2

Explain key production measures such as marginal product and technical rate of substitution.

3

Differentiate between short-run and long-run production processes.

4

Analyze the concept of returns to scale and its effect on overall production.

5

Evaluate how changes in input levels influence production outcomes.

Key Concepts

CONCEPT

DEFINITION

Production Technology

The methods, processes, and techniques used to convert inputs into outputs in the production process.

Production Function

A mathematical representation expressing the relationship between inputs and outputs in production.

Fixed Proportions

A production function where inputs are used in a fixed ratio, meaning that the outputs cannot be increased by substituting one input for another.

Perfect Substitutes

A production scenario where one input can be replaced by another without affecting the output level, implying full interchangeability.

Cobb-Douglas Production Function

A specific functional form of production that implies a constant elasticity of substitution between inputs, often expressed as Q = A * L^α * K^β.

Marginal Product

The additional output produced by using one more unit of a specific input, keeping other inputs constant.

Technical Rate of Substitution (TRS)

The rate at which one input can be substituted for another while maintaining the same level of output.

Short-Run Production

A production period during which at least one input (such as capital) is fixed and cannot be changed.

Long-Run Production

A production period when all factors of production can be adjusted or varied.

Returns to Scale

A concept that describes how the output changes when all inputs are increased proportionately.

Example Problems

Example 1

Consider the production function $f\left(x_{1}, x_{2}\right)=x_{1}^{2} x_{2}^{2}$. Does this exhibit constant, increasing, or decreasing returns to scale?

Example 2

Consider the production function $f\left(x_{1}, x_{2}\right)=4 x_{1}^{2} x_{2}^{\frac{1}{3}}$. Does this exhibit constant, increasing, or decreasing returns to scale?

Example 3

The Cobb-Douglas production function is given by $f\left(x_{1}, x_{2}\right)=A x_{1}^{a} x_{2}^{b}$ It turns out that the type of returns to scale of this function will depend on the magnitude of $a+b .$ Which values of $a+b$ will be associated with the different kinds of returns to scale?

Example 4

The technical rate of substitution between factors $x_{2}$ and $x_{1}$ is $-4 .$ If you desire to produce the same amount of output but cut your use of $x_{1}$ by 3 units, how many more units of $x_{2}$ will you need?

Example 5

True or false? If the law of diminishing marginal product did not hold, the world's food supply could be grown in a flowerpot.
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Step-by-Step Explanations

QUESTION

If a production process requires 2 units of Input A and 3 units of Input B to produce 1 unit of output, how many units of each input are needed to produce 10 units of output?

STEP-BY-STEP ANSWER:

Step 1: Identify the fixed ratio – 2 units of Input A and 3 units of Input B per unit of output.
Step 2: Multiply the fixed input quantities by the desired output level (10).
Step 3: Calculate Input A: 2 units/output * 10 outputs = 20 units.
Step 4: Calculate Input B: 3 units/output * 10 outputs = 30 units.
Final Answer: 20 units of Input A and 30 units of Input B are required.

Fixed Proportions

QUESTION

Given the Cobb-Douglas production function Q = L^0.5 * K^0.5, calculate the marginal product of labor (MPL) when L = 16 and K = 9.

STEP-BY-STEP ANSWER:

Step 1: Recognize that the Cobb-Douglas function is Q = L^0.5 * K^0.5.
Step 2: The marginal product of labor (MPL) is the partial derivative of Q with respect to L. For L^0.5, differentiate to get (0.5) * L^(-0.5). Multiply by K^0.5, yielding MPL = 0.5 * L^(-0.5) * K^0.5.
Step 3: Substitute L = 16 and K = 9 into the formula. Calculate L^(-0.5) = 1/sqrt(16) = 1/4 and K^0.5 = sqrt(9) = 3.
Step 4: Compute MPL = 0.5 * (1/4) * 3 = 0.5 * 0.25 * 3 = 0.375.
Final Answer: The marginal product of labor is 0.375.

Cobb-Douglas Production Function

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Common Mistakes

  • Confusing the marginal product with the average product.
  • Assuming the technical rate of substitution remains constant across different production functions.
  • Overlooking the difference between short-run and long-run production settings.
  • Misinterpreting returns to scale by not accounting for proportional changes in all inputs.