Cost Functions: Variable, Fixed, MC and AC

Examples of Costs Functions

• Considering the cost function C(y) = y^2 + 1, variable costs are y^2, and fixed costs are 1.
• AVC = y, AFC = 1/y, AC = y + 1/y, and MC = 2y.
• MC and AVC curves intersect at y = 0.
• The MC and AC intersect at y = 1 where AC(y) is at its lowest point, y^2 =1

More Cost Function examples

• Considering the cost function C(y) = y^3 - 8y^2 + 30y + 5, variable costs include y^3 - 8y^2 + 30y, and fixed costs are 5.
• AVC = y^2 - 8y + 30, AFC = 5/y, and AC = y^2 - 8y + 30 + 5/y.
• Marginal Cost (MC) is 3y^2 - 16y + 30
• You can determine where MC and AVC curves intersect by plotting data.

Short-Run and Long-Run Average Total Cost Curves

• The long-run total cost curve always yields the lowest possible total production cost for any output level y.
• To produce y units of output, a firm considers its plant size, denoted by k.
• In the long run, plant size can be optimally adjusted to k(y) for each level of output y, represented as c(y) = c(y, k(y)).
• In the short run, production occurs with the existing size (k*), represented as c_s(y) = c(y, k*).
• Producing y units with a plant of size k* is more expensive than doing so with a plant of size k(y), in other words: c(y) <= c_s(y).

Short-Run and Long-Run Marginal Cost Curves

• If a firm producing 100 t-shirts has a plant that was best suited to produce 1,000 t-shirts
• Short-run average costs and the plant size chosen to produce 100 t-shirts at different levels Y.

Firm Supply

• A firm's product supply decision is influence by production function and the market environment.
• Market environmental factors include the number of firms and if other firms' decisions affect its payoffs.

Market Environments

• Monopoly: Single seller that determines supplied quantity and market-clearing price.
• Oligopoly: A few firms whose decisions influence each other.
• Monopolistic Competition: Many firms sell slightly different products, each with a small output level relative to the total market.
• Pure Competition: Numerous firms offering the same product, each's output level insignificantly affecting the total.

Pure Competition

• A firm operating in a perfectly competitive market has no influence over the market price as a market price-taker.
• If a firm sets its price above the market price, the quantity demanded from it is zero.
• Conversely, if the firm sets its price below the market price, it faces the entire market demand.
• If a price is at p', then the amount needed from the firm is zero
• At any price below pe, the firm gets the entire demand curve

A firms role

• Each firm seeks to maximize profits in the short run.
• Firms decides y * in order to take prices as given therefore maximizing profits i.e T(y)
• Revenue is the product of the price and quantity: R(y) = p * y.

Maximizing Profit

• Solutions may be when y* > 0, it is known as the interior solution
• Solutions may be when y* = 0, maximum profit isn't obtained by producing at all

Maximizing Profit more

• For the interior case of y* > 0, firms should choose the level of output, y*, where market price equals the marginal cost or dπ(y)/Δy = p – MC(y) = 0.

Example

• A competitive firm producing stained glass windows with a cost function C(y) = 4y^2 + 291 produces 26 windows per week to maximize profits.
• If the firms open a second factory with the cost function C(y) = 8y^2 + 291, and the price remain the same they will also produce at the new factory to maximize profits.