Production Function: Maximizing Profits

Production Function

• Producers, like consumers, possess a function; their goal is to maximize profits, not production.
• Profit maximization is subject to technological constraints which affect the production plan's feasibility, and the market structure

Simplifying the Producer Model

• Model assumes two inputs which include labor (L) and capital (K), to produce one output (q).
• Inputs or factors of production, are converted into output using the firm's technology, represented by the production function f(.).
• The output quantity is reliant on labor, capital, and technology i.e. q = f(L, K).

Classifying Production Factors

• Factors can be variable,changeable or fixed, hard to change.
• Production can occur in the short-run (one or more factors are fixed) or long-run (all factors are variable).
• Capital is assumed to be fixed in the short-run for analysis.

Returns to Scale

• Returns to scale determine how much output increases with increased input.
• Constant returns occur when outputs increase proportionally with inputs.
• Increasing returns occur when outputs increase more than inputs.
• Decreasing returns occur when outputs increase less than inputs.
• Specific Returns to Scale:
  • CRS: f(2L, 2K) = 2 ⋅ f(L, K)
  • IRS: f(2L, 2K) > 2 ⋅ f(L, K)
  • DRS: f(2L, 2K) < 2 ⋅ f(L, K)
• Implications of increasing inputs:
  • IRS results from further specialization.
  • DRS may arise from coordination difficulties.

Short-Run Production

• Focuses on a firm's short-run decision to optimize labor use for profit given fixed capital.
• Expressed mathematically as K = K̄, where the bar indicates a fixed quantity.
• The firm's production function becomes q = f(L, K̄), with labor as the only variable.
• As input increases, total output increases at a decreasing rate, exhibiting diminishing marginal productivity.
• The marginal product measures the additional output from hiring additional workers
• Crucial assumptions: perfectly competitive market for goods and factors of production, where firms cannot influence market prices or wages.

Firm's Profit Function

• Defined as the difference between total revenue and total cost: π = Total Revenue - Total Cost
• Total revenue is calculated by multiplying price by quantity: Total Revenue = p × q.
• Total cost equals the cost of labor and capital: Total Cost = w × L + r × K.
• Profit maximization problem:
  • max π = pq - wL - rK, s.t. q = f(L, K)

Isoprofit Lines

• Illustrate combinations of inputs/outputs generating the same profit level.
• Derived by rewriting the profit function as q = (π + rK)/p + (w/p) ⋅ L.
• The intercept is (π + rK)/p, and the slope is w/p.

Profit Maximization Conditions

• Optimal profit achieved where the isoprofit curve's slope equals the marginal productivity of labor MPL.
• Mathematical condition: p × ∂f(L, K)/∂L = w, where p × MPL is the marginal revenue product of labor (MRPL).
• At optimal levels, MRPL = w
• Equation restated: MPL = w/p, where w/p is the real wage
• The real wage should equal the quantity of goods the worker produces

Long-Run Production

• Involves optimizing both labor and capital.
• The profit-maximization problems firms face are similar to utility maximization problems.
• Given by:
  • max π = pq - wL - rK subject to q = f(L, K)

Cost Minimization

• Firms are constrained and unable to operate under conditions of perfect competition.
• Focuses on minimizing costs subject to technology and desired output levels i.e. min wL + rK, s.t. f(L, K) = q.

Isoquants

• Represent combinations of Labor and Capital producing the same output quantity.
• Isoquant is like an indifference curve, where higher isoquants represent a higher level of output.
• Marginal productivity of capital and labor determine extra output from extra input of capital or labor.
• The isoquant's slope (marginal rate of technical substitution) MRTS, equals the ratio of marginal productivity of labor to capital.

Isocost Lines

• Indicate combinations of labor and capital for a given cost level.
• The goal is to choose the lowest isocost line allowing desired output.
• Cost Equation: C = wL + rK rewritten as K = C/r - w/r
• Isocost line condition: MRTSL,K = w/r or MPL/MPK = w/r, indicates the market's trade-off rate between capital and labor.
• "Bang-for-buck equation" means that MPL/w = MPK/r, so dollar spent on labor should equal the marginal product per dollar spent on capital.
• Cost Function Formula: C= C(w,r,q), in terms of input prices and output quantity.

Short-Run Cost Minimization

• Short-run fixed capital and cost includes fixed cost FC and a variable cost VC.
• Cost-minimization problems:
  • min wL + rK subject to √L x K = q
• Short-run function:
  • C = wq²/K + rK; based on wage rate (w), rental rate (r), and fixed capital (K).

Cost Types

• Marginal cost: additional cost incurred with producing the next unit
  • Formula: MC = dC/dq
• Average Total Cost: the average cost of production across all outputs i.e. AC = C/q
• Average Fixed Cost: the average cost of production across the fixed cost components i.e. AFC = FC/q
• Average Variable Cost: the average cost of production across the variable cost components i.e. AVC = VC/q
• Marginal cost rises, average total cost is U-shaped with marginal curve crossing at the minimum

Production Function Equation

• The form allows rewriting the marginal cost equation as: -MC = w(dL/dq) = w/MPL,

Long-Run Cost

• Long-run can be considered once we have derived the cost functions
• Long-run cost are always lower than short-run because all factors of production are variable For the production function, we:
  • min 5L + 10K subject to √L x K = q
  • K = 1/2 ⋅ L
  • L = √2q
  • K = √2/2 ⋅ q
• The function becomes C = 10√2q
• The cost indicates how cost varies with quantity, is useful in conditions less than perfectly competitive

Perfect Competition

• Each producer accepts the market price which is beyond control
• Characterized by a large number of buyers and sellers, homogeneous products, no entry barriers, perfect information, and zero transaction costs
• Firm will face perfectly elastic demand

Profit Maximization

• Total revenue minus total costs Total Cost = P x Q
• In perfectly competitive firms MR =MC
• Short run is defined only as : P=MC
• Condition to operation: price must equal marginal cost

Shut-Down Conditions

• Firms should shut down when P < AVC
• Graph in Black region which represents firms negative profit

Supply Decision

• The firms supply decision is quite straight forward
• We can determine the firms point, thus, we can find some profit- maximising quantities
• Firm set the price which is at least equal to the marginal cost for production
• So what does the marginal cost curve show? It shows the firms supply curve
• Generally it considers the possibility when the firm would shut down at some lower prices
• Thus the firms supply curve is determined by the marginal curve above the average variable cost curve

The Supply Curve

• Curve shows the firms supply by solving the cost-minimization problem and the profit maximising problems when we assume there is perfect competition
• The firms supply curve is the marginal cost curve above its average variable cost curve which allows a positive relationship between prices and quantities supplied
• This relationship determines that the law of supply an the increase price leads to increase in quantities supplied vice Versa

Deriving the Supply Schedule

To derive the supply schedule to solve his/Her problem at different levels We can then plot each of those point which show, thus the curve is called the supply Curve

Market Supply

Curve shows Ben's individual supply of Ice cream, The curve includes more sellers than just Ben which allows use to find the market supply

• The process or finding the process of finding the market is identical to find with markets demand, Lets assume both sides of the Market has to
• We find supply and plot the curve, this help us understand each price Lets also see how to drive the Market supply curves

Change in supply

The supply curve shifted due to several number of factors These factors includes,

• Input prices
• Technology
• Expectations
• Number of Sellers

Perfects competitions In the long Run

• The long run is usually represented by the characterises that does not contain barrier for the entries ,
• If the profits is to be made in a perfectly competitive market , then more firms will start to enter the market i.e The market increases which increases the supply curve which drives the market down , as the prices decreases the firms profit decrease .as Firms leaves the market
• As firms leaves the market, this drives to an increase In the prices And increases each individual firms lose less ,
• As price increased the level such level such each individual will make profits Graph in the black will shows firms revenue in the long run
• This implies , shows the long run should be flat, we can that the individual make profits in the long run , the curve in the long-run its pure defined by its minimum Average cost.
  • and when the MC=AC its minimum, Is the Morale toward cost minimization .
  • If some relax about the perfect Competitions then the prices would move upwards,