Cost Functions and Minimization

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Questions and Answers

A firm aims to produce a target output level. How does it determine the optimal combination of factors to use?

  • By selecting the combination that uses the most of each factor.
  • By choosing the combination that produces the target output at the lowest possible cost. (correct)
  • By randomly selecting a combination of factors.
  • By choosing the combination that maximizes output regardless of cost.

What does the cost function of a firm measure?

  • The minimum cost of producing a given level of output. (correct)
  • The average cost of producing a given level of output.
  • The total revenue earned from selling a given level of output.
  • The maximum cost of producing a given level of output.

In the context of cost minimization, what are the key decision variables for a firm?

  • The quantities of factors 1 and 2 used in production. (correct)
  • The level of target output and the wage rate.
  • The prices of factors 1 and 2.
  • The prices of the output and the level of advertising.

What is the objective of the cost minimization problem?

<p>To minimize the total cost of production, given a target output level. (D)</p> Signup and view all the answers

A firm's production function is given by $f(x_1, x_2) = y$, where $x_1$ and $x_2$ are the amounts of factor 1 and factor 2 used, respectively. The prices of the factors are $w_1$ and $w_2$. What is the total cost equation?

<p>$C = w_1x_1 + w_2x_2$ (D)</p> Signup and view all the answers

What does it mean for a firm to have a short-run cost function?

<p>The firm can adjust only a few factors of production. (B)</p> Signup and view all the answers

What is the key difference between the short-run and long-run cost functions?

<p>The short-run cost function considers some factors as fixed, while the long-run cost function considers all factors as variable. (A)</p> Signup and view all the answers

In the short run, with factor 2 fixed at $\bar{x_2}$, what is the firm's cost minimization problem?

<p>$min_{x_1} w_1x_1 + w_2\bar{x_2}$ subject to $f(x_1, x_2) = y$ (B)</p> Signup and view all the answers

What does the solution to the short-run cost minimization problem provide?

<p>The demand curve for factor 1. (A)</p> Signup and view all the answers

A firm's short-run cost function, given factor 2 is fixed at $\bar{x_2}$, is $C(w_1, w_2, y, \bar{x_2}) = w_1x_1^(w_1, w_2, y, \bar{x_2}) + w_2\bar{x_2}$. What does the expression $x_1^(w_1, w_2, y, \bar{x_2})$ represent?

<p>The cost-minimizing quantity of factor 1. (C)</p> Signup and view all the answers

In the long run, how does a firm determine the optimal levels of factors 1 and 2 ($x_1$ and $x_2$) to use?

<p>By considering factor prices ($w_1$ and $w_2$) and the production function $f(x_1, x_2) = y$. (A)</p> Signup and view all the answers

Given the long-run conditional factor demands $x_1(w_1, w_2, y)$ and $x_2(w_1, w_2, y)$, how is the long-run cost function, $C(w_1, w_2, y)$ determined?

<p>$C(w_1, w_2, y) = w_1x_1(w_1, w_2, y) + w_2x_2(w_1, w_2, y)$ (B)</p> Signup and view all the answers

What is the cost function for a perfect complements production function where $f(x_1, x_2) = min{x_1, x_2}$?

<p>$C(w_1, w_2, y) = (w_1 + w_2) * y$ (A)</p> Signup and view all the answers

For a production function $f(x_1, x_2) = min{x_1, x_2}$, which statement is correct about the conditional factor demands?

<p>If $w_1 &lt; w_2$, then $x_1(w_1, w_2, y) = y$ and $x_2(w_1, w_2, y) = y$ (C)</p> Signup and view all the answers

For a production function $f(x_1, x_2) = x_1 + x_2$, representing perfect substitutes, how does the firm decide which factor to use?

<p>The firm uses only the factor with the lower price. (D)</p> Signup and view all the answers

For a production function $f(x_1, x_2) = x_1 + x_2$, what does the cost function equal if $w_1 < w_2$?

<p>$C(w_1, w_2, y) = w_1 * y$ (A)</p> Signup and view all the answers

What is the general form of a Cobb-Douglas production function?

<p>$f(x_1, x_2) = Ax_1^a x_2^b$ (A)</p> Signup and view all the answers

Suppose a firm's long-run cost function is given by $C(w_1, w_2, y)$. If factor 2 is fixed in the short run, how can the short-run cost function be expressed?

<p>$C_s(w_1, w_2, \bar{x_2}, y) = C(w_1, w_2, x_2(w_1, w_2, y), y) + w_2\bar{x_2}$ (B)</p> Signup and view all the answers

What is the relationship between the short-run and long-run cost-minimizing amounts of the variable factor if the fixed factor is set at its long-run cost-minimizing level?

<p>The short-run and long-run cost-minimizing amounts of the variable factor are always equal. (D)</p> Signup and view all the answers

Define variable costs.

<p>Costs that are incurred from using the variable factor. (D)</p> Signup and view all the answers

Define fixed costs.

<p>Costs that are incurred regardless of the level of output. (B)</p> Signup and view all the answers

In the short run, what is the total cost of a firm equal to?

<p>Variable cost plus fixed cost. (B)</p> Signup and view all the answers

If the fixed cost does not depend on the output, then in the short run what is the change in the total costs for an additional unit of output equal to

<p>Marginal cost (B)</p> Signup and view all the answers

A firm's cost function is given by $C(y)$, variable cost function by $C_v(y)$, and fixed costs by $F$. Which of the following expressions is correct?

<p>$C(y) = C_v(y) + F$ (A)</p> Signup and view all the answers

What is the mathematical expression for Marginal Cost (MC)?

<p>$MC(y) = \frac{\Delta TC}{\Delta y}$ (D)</p> Signup and view all the answers

What is Average Cost (AC)?

<p>Total cost divided by quantity. (B)</p> Signup and view all the answers

In the context of cost curves, what does AFC represent?

<p>Average Fixed Cost. (D)</p> Signup and view all the answers

If total fixed costs are constant, what happens to Average Fixed Cost (AFC) as output (y) increases?

<p>AFC decreases. (D)</p> Signup and view all the answers

A textile factory has fixed plant size, but can vary its labor. If the cost of labor increases, what is the likely effect on the average variable cost (AVC)?

<p>AVC increases. (D)</p> Signup and view all the answers

Which statement describes the typical shape of the Average Fixed Cost (AFC) curve?

<p>Downward sloping. (A)</p> Signup and view all the answers

What is a typical shape of the Average Variable Cost (AVC) curve?

<p>U-shaped. (A)</p> Signup and view all the answers

How is the Marginal Cost (MC) curve related to the Variable Cost (VC) curve?

<p>MC is the slope of the VC curve. (A)</p> Signup and view all the answers

Marginal Cost (MC) represents:

<p>The cost of producing one additional unit of output. (B)</p> Signup and view all the answers

If $VC(y) = y^2$, what is the Marginal Cost (MC)?

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

How is the area beneath the Marginal Cost (MC) curve, up to a given quantity y, best described?

<p>The variable cost of producing y. (C)</p> Signup and view all the answers

The firm's demand curve in a perfectly competitive market is:

<p>Perfectly elastic. (A)</p> Signup and view all the answers

If a firm in a perfectly competitive market charges a price higher than the market price:

<p>It will sell none of its output. (C)</p> Signup and view all the answers

If $P\lt p^$ (where $p^$ is the existing market price) what happens, according to the theory of Perfect Competition?

<p>The firm cannot serve all consumers and loses money (C)</p> Signup and view all the answers

What does behavioral assumption say about firms?

<p>A firm makes economic decisions to maximize profit (C)</p> Signup and view all the answers

What is first step to know to maximize the profit?

<p>Devising the cost function (A)</p> Signup and view all the answers

In a perfectly competitive market, what is a firm's profit-maximizing decision?

<p>To set output such that marginal cost equals marginal revenue (D)</p> Signup and view all the answers

In a competitive market, a firm chooses the quantity, such that the marginal revenue = marginal cost. What happens if MR > MC?

<p>The firm should increase Output (D)</p> Signup and view all the answers

Flashcards

Cost Minimization

Choosing factor combinations to produce target output at the lowest possible cost.

Cost Function

A function that measures the minimum cost of producing a certain quantity of output when factor prices are given

Short run cost minimization

Cost minimization when at least one production factor is fixed.

Long run cost minimization

Cost minimization when all production factors are variable.

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Short-Run Cost Function

The minimal cost to produce a given level of output while adjusting only the variable factors.

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Long-Run Cost Function

The minimum cost of producing a given level of output, adjusting all factors of production.

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Fixed Costs

Costs that do not vary with the level of output in the short run.

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Variable Costs

Costs that vary with the level of output.

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Marginal Cost

The change in total costs for an additional unit of output.

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Average Cost

Total cost per unit of output.

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Average Variable Cost

The variable costs per unit of output.

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Average Fixed Cost

Fixed cost per unit of output.

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Perfect competition

Market structure with many buyers and sellers, easy entry and exit, and identical products.

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Monopoly

Market structure with a single firm in the market and no possibility for new firms to enter.

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Oligopoly

Market structure with a small number of firms that sell similar of differentiated products.

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Price taker

The firm sells its products at a predetermined “market price” that it cannot influence.

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Competitive Firm's Demand Curve

A firm's demand curve is horizontal at the market price.

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Cost Function

The cost function gives the least cost with which to produce every possible level of output

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Firm decisions in competitive market

Deciding how much of output to supply and how to combine different factors of production

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Profit Maximization in Perfect Competition

Firm chooses output level to maximize profit

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Condition for profits maximization

Marginal Revenue equals Marginal Cost

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Entry of New Firms

What happens in the long-run when firms are malting economic profits

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Firms Exit the Market

Industry reach long-run equilibrium point when firms have a making economic losses

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Supply Curve

A firm choose the number of units to produce to maximize economic benefit, depending of his characteristics

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Entry and Exit affect

How does entry and exit effects industry dynamic

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Perfect Efficiency

Output cannot be made cheaper, and there's value for it

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Monopolist Decision

Monopolist chooses both quantity and price over market demand curve

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Describe a monopolist

Can choose both price and quantity at market demand curve

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Social Optimum

Price which equates MC with market demand intersects

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Study Notes

  • Cost function is an equation that gives the cost to product some final amount
  • Cost minimization aims to find the lowest combination for total expense
  • Cost minimization problem involves choosing amounts of Factor 1 and Factor 2 to hit a target while minimizing total expense

Cost Minimization

  • How does the firm decide which combination to use?
  • The firm chooses the combination of factors that produce the target output at the lowest possible cost
  • min W1X1 + W2X2 subject to f(X1,X2) = y
  • X1 and X2 are the amounts of Factor 1 and Factor 2 used

Cost Function

  • The solution to a cost-minimization problem depends on how expensive W1, W2, and y are
  • Denoted as C(W1, W2, y)
  • It measures the minimum cost of producing y units when factor prices are W1 and W2
  • Short-run vs long-run cost minimization occurs based on adjustable costs

Short Run Cost Function

  • It is defined as the minimum cost to produce a given level of output while adjusting only the variable factors of production
  • Production Function = f(x1,x2) = y

Long Run Cost Function

  • A long run cost function gives the minimum cost of producing a given level of output, adjusting all the factors of productions
  • The conditional factor demands are
    • X1 = x1(W1, W2, y)
    • X2 = x2(W1, W2, y)
  • Long-run cost function, C(W1, W2, y) = W1X1 * (W1, W2, y) + W2X2 * (W1, W2, y)
    • This substitutes the conditional factor demands into the cost equation

Short Run Cost

  • Suppose factor 2 is fixed at X2
  • The firm's short-run cost minimization problem involves W1X1 + W2X2
  • Minimum cost is: min x1 W1X1 + W2X2 subject to f(x1, X2) = y

Types of Production Functions

Perfect Compliments Production Function

  • f(x1, x2) = min{x1, x2}
  • Cost function is C(W1, W2, y) = (W1 + W2)y

Perfect Substitutes Production Function

  • f(x1, x2) = x1 + x2
  • The conditional factor demands are
    • x1(W1, W2, y) = y
    • x2(W1, W2, y) = 0 if W1 < W2
    • x1(W1, W2, y) = 0
    • x2(W1, W2, y) = y if W1 > W2

Cobb-Douglas production function

  • f(x1, x2) = x^a * x^b where a > 0, b > 0

Firm Linkage

  • The long-run conditional demands are used to connect Short run and Long run cost function
  • Suppose X2 is fixed in the short run, then C(W1, W2, y) = Cs (W1, W2, X2 (W1, W2, y), y)
  • Variable = minimum cost when factor 2 is fixed at the level that minimizes long-run costs.

Cost Curves

  • In the short run some factors are Variable while others are fixed.
  • Variable cost is the cost incurred from using the Variable factor
  • Variable cost = W1 * X1(W1, W2, y) for factor 1
  • In the short run, the total cost of a firm is C(y) = Cv(y) + F, where Cv is variable cost and F is fixed cost

Long Run

  • In the long run, all factors are variable, which means F=0

Marginal Cost

  • Marginal cost is the slope of the variable cost curve

Average Cost Curves

  • Average Cost = AC(y) = C(y)/y, measures the cost per unit of output
  • Average Variable Cost function measures the variable costs per unit of output
  • Average Variable Cost = AVC(y) = Cv(y) / y
  • Average Fixed Cost measures the fixed cost per unit of output
  • Average Fixed Cost Formula, AFC(y) = f/y
  • Average cost= Average Variable Cost + Average Fixed Cost

Cost Curve

  • Average Fixed Cost AFC(y) downward sloping
  • When AVC (4) is graphed there is Upward Sloping section
  • The cost curve AC(y) is shown as AVC(y) to reflect the rate value change

Marginal Cost(MC) curve

  • It is the slope of the VC(y) curve
  • the increased Ve for MC for any addition put = "12.
  • Total cost is C(y) =Cu(y) + f, VC Curve is shifted up
  • Graphically Marginal Cost Is show to intersect total cost TC

Cost Minimization:

-The Area Beneath the MC curve-up to y gives the VC of producing of output M(;ly) = "34 , V(4) =- Cu(y)

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