Cost Minimization in Economics
22 Questions
0 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to Lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is the formula for the average cost (AC) based on total cost (C) and quantity (Q)?

  • AC = (Q + 2 + 4/Q)
  • AC = ∂C/∂Q
  • AC = C/Q (correct)
  • AC = Q^2 + 2Q + 4

    • What does marginal cost (MC) represent?

  • The fixed costs involved in production
  • The cost of producing the last unit of a good
  • The change in total cost from producing one additional unit (correct)
  • The total cost divided by quantity

    • At what quantity (Q) does the average cost reach its minimum according to the provided example?

  • 4
  • 3
  • 2 (correct)
  • 1

    • What happens when marginal cost is below average cost?

    <p>Average cost decreases with each additional unit produced (D)</p> Signup and view all the answers

    Given the total cost function C = Q^2 + 2Q + 4, what is the expression for marginal cost?

    <p>MC = 2Q + 2 (C)</p> Signup and view all the answers

    What is the objective of a firm's cost minimization problem?

    <p>Minimizing the cost of inputs while producing a certain output level (A)</p> Signup and view all the answers

    Which of the following expressions represents the total cost of inputs for a firm?

    <p>$wL + vK$ (A)</p> Signup and view all the answers

    What does the constraint $Q̄ = F(K, L)$ represent in the firm's cost minimization problem?

    <p>The level of production the firm aims to achieve (C)</p> Signup and view all the answers

    In setting up the Lagrangian for the cost minimization problem, what does the term $λ(Q̄ − F(K, L))$ account for?

    <p>The penalty for not meeting the production constraint (A)</p> Signup and view all the answers

    What do the first-order conditions indicate when solving the Lagrangian in this context?

    <p>The wage rate should equal the marginal cost of labor (A)</p> Signup and view all the answers

    What happens when the quantities of labor and capital that minimize costs are determined?

    <p>The firm reaches the desired production level at minimal costs (B)</p> Signup and view all the answers

    Which term represents the change in production relative to changes in capital in the first-order condition?

    <p>$ rac{∂F}{∂K}$ (B)</p> Signup and view all the answers

    What does the ratio $\frac{\partial F / \partial L}{\partial F / \partial K}$ represent in the context of production?

    <p>Marginal Rate of Technical Substitution (MRTS) (C)</p> Signup and view all the answers

    In the Cobb-Douglas production function $F = K^{1/2}L^{1/2}$, what is the assumed output level when plotting the isoquant?

    <p>4 units (D)</p> Signup and view all the answers

    If the price of labor is represented by 'w' and the price of capital by 'v', how is the ratio of prices related to MRTS?

    <p>It will be equal to MRTS at the optimal allocation. (B)</p> Signup and view all the answers

    Based on the information provided, what does the contingent demand function for labor, $L^* = L(w, v, \bar{Q})$, depend on?

    <p>Wages, rental rate of capital, and desired output level (B)</p> Signup and view all the answers

    What is the main graphical representation used to analyze the firm's cost minimization problem?

    <p>Isoquant and isocost lines (B)</p> Signup and view all the answers

    In the context of contingent demand, which statement correctly describes how to derive the firm's contingent demand for inputs?

    <p>By solving the Lagrangian for optimal input mix. (B)</p> Signup and view all the answers

    What do optimal allocations indicate in the context of labor and capital pricing?

    <p>A balance where the ratio of MPL to MPK equals the ratio of labor to capital prices. (B)</p> Signup and view all the answers

    Which equation reflects the relationship between minimum cost and the output produced?

    <p>$\bar{Q} = F(K, L)$ (B)</p> Signup and view all the answers

    What do the parameters 'w' and 'v' necessarily represent in the firm's cost minimization scenario?

    <p>Prices of inputs (C)</p> Signup and view all the answers

    How does changing the price of labor ('w') affect the firm's labor demand according to the contingent demand function?

    <p>An increase in 'w' decreases labor demand. (D)</p> Signup and view all the answers

    Flashcards

    Average Cost (AC)

    The total cost of production divided by the quantity produced. It represents the per-unit cost of producing a good.

    Marginal Cost (MC)

    The additional cost incurred by producing one more unit of a good. It represents the cost of producing the next unit.

    Relationship between AC and MC

    When marginal cost is below average cost, average cost decreases. When marginal cost is above average cost, average cost increases. When AC and MC are equal, average cost is at its minimum.

    Minimizing Average Cost

    To find the quantity that minimizes average cost, set the derivative of the average cost function equal to zero and solve for the quantity.

    Signup and view all the flashcards

    Total Cost Function

    A function that expresses the total cost of production as a function of input prices (wage and rental rate) and output quantity.

    Signup and view all the flashcards

    Cost Minimization Problem

    A firm's problem of finding the least expensive way to produce a given quantity of output.

    Signup and view all the flashcards

    Inputs

    Resources used in the production process, such as labor (L) and capital (K).

    Signup and view all the flashcards

    Wage Rate (w)

    The cost of hiring one unit of labor.

    Signup and view all the flashcards

    Rental Rate (v)

    The cost of renting one unit of capital.

    Signup and view all the flashcards

    Production Function (F(K, L))

    A mathematical relationship that shows the maximum output (Q) that can be produced with given inputs of labor (L) and capital (K).

    Signup and view all the flashcards

    Isoquant

    A curve that shows all combinations of labor (L) and capital (K) that produce the same level of output (Q).

    Signup and view all the flashcards

    Lagrangian

    A mathematical function used to solve constrained optimization problems, like the cost minimization problem.

    Signup and view all the flashcards

    Price Ratio

    The ratio of the price of labor (w) to the price of capital (v).

    Signup and view all the flashcards

    MRTS

    Marginal Rate of Technical Substitution. The rate at which one input can be substituted for another while keeping output constant.

    Signup and view all the flashcards

    Cost Line

    A line that represents all the possible combinations of labor and capital that can be purchased with a given budget.

    Signup and view all the flashcards

    Tangency Point

    The point on the isoquant where it is tangent to the cost line. This point represents the cost-minimizing combination of inputs.

    Signup and view all the flashcards

    Contingent Demand

    The firm's demand for inputs (labor and capital) as a function of wages, rental rate of capital, and the desired output level.

    Signup and view all the flashcards

    Contingent Demand Function

    Mathematical equations that describe the firm's demand for labor and capital.

    Signup and view all the flashcards

    Hicksian Demand

    The consumer's demand for goods when their income and the prices of other goods are fixed, but the price of the good in question is allowed to vary.

    Signup and view all the flashcards

    Expenditure Function

    A function that shows the minimum expenditure required to achieve a given level of utility, given prices and the consumer's utility function.

    Signup and view all the flashcards

    Cost Function

    A function that shows the minimum cost of producing a given level of output, given input prices and the production function.

    Signup and view all the flashcards

    Study Notes

    Cost Minimization

    • Firms aim to produce a given output level at the lowest possible cost.
    • This mirrors consumer expenditure minimization, but focuses on the firm's cost minimization.
    • Key concepts include production functions, returns to scale, marginal rate of technical substitution (MRTS), and isoquants.
    • The firm chooses inputs (labor and capital).
    • Each input has a price (wage rate for labor, rental rate for capital).
    • The firm's total cost is the sum of the payments for labor (wL) and capital (vK).
    • Mathematically, the firm seeks to minimize wL + vK, subject to the constraint that the firm produces at least a certain quantity (Q). This quantity is determined by the production function (Q = F(K, L)).
    • Solving the cost minimization problem involves setting up a Lagrangian and finding the first order conditions.
    • The first-order conditions lead to a crucial relationship: the price ratio (w/v) must equal the marginal rate of technical substitution (MRTS) at the optimal input combination.
    • Graphically, the solution is where the isoquant (representing the required output) is tangent to an isocost line (representing a given total cost).

    Graphical Analysis

    • Firms' cost minimization can be analyzed graphically using isoquants and isocost lines.
    • Isoquants show different input combinations producing the same output level.
    • Isocost lines show different input combinations that yield the same total cost.
    • The optimal input combination occurs at the tangency point between the isoquant and the lowest isocost line.
    • This point represents the least cost for producing the desired output.

    Contingent Demand

    • Contingent demand functions describe the optimal quantities of inputs (labor and capital) as a function of input prices and required output.
    • These functions show how much of each input the firm will demand at different prices and output levels.
    • They are derived from the cost minimization problem and are analogous to Hicksian demand functions in consumer theory.

    Cost Functions

    • The cost function (C(w, v, Q)) shows the minimum cost of producing a given output (Q) when input prices (w and v) are given.

    • The relationship between cost and output is crucial.

    • Cost function calculation involves substituting optimal input levels (derived from contingent demand) into the total cost formula (vK* + wL*).

    • Average cost (AC) is total cost divided by output.

    • Marginal cost (MC) is the change in total cost resulting from producing one additional unit of output.

    • Cost curves illustrate the relationship between cost and output.

    • Cost curves are used to study firm behavior.

    • Key considerations include average cost, marginal cost, and the relationship between cost curves.

    Studying That Suits You

    Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

    Quiz Team

    Related Documents

    Cost Minimization PDF

    Description

    Explore the concept of cost minimization in firms, focusing on how businesses aim to achieve the lowest possible cost while producing a given output level. Key concepts include production functions, input choices, and the relationship between input prices and the marginal rate of technical substitution. Test your understanding of the mathematical formulation and first-order conditions for optimal input allocation.

    More Like This

    Use Quizgecko on...
    Browser
    Open
    Browser
    Browser