Key Concepts: Linear Cost/Revenue/Profit; Demand & Supply (Brief Notes)

Cost, Revenue, and Profit Functions

  • Let x = number of units produced or sold
  • C(x) = total cost
  • R(x) = total revenue
  • P(x) = total profit
  • Fixed costs F: constant in each period
  • Variable costs: costs that vary with production, per-unit cost c
  • Formulas:
    • C(x)=c x+FC(x) = c\,x + F
    • R(x)=s xR(x) = s\,x
    • P(x)=R(x)−C(x)=s x−(c x+F)=(s−c) x−FP(x) = R(x) - C(x) = s\,x - (c\,x + F) = (s - c)\,x - F
  • Important note: distribute the negative sign when computing P(x) from R(x) - C(x)
  • Example: Puritron
    • Fixed cost F = 20{,}000; variable cost per unit c = 20; selling price s = 30
    • C(x)=20x+20,000C(x) = 20x + 20{,}000
    • R(x)=30xR(x) = 30x
    • P(x)=30x−(20x+20,000)=10x−20,000P(x) = 30x - (20x + 20{,}000) = 10x - 20{,}000
  • Example with F = 100{,}000; c = 14; s = 20
    • C(x)=14x+100,000C(x) = 14x + 100{,}000
    • R(x)=20xR(x) = 20x
    • P(x)=20x−(14x+100,000)=6x−100,000P(x) = 20x - (14x + 100{,}000) = 6x - 100{,}000
    • P(12,000)=6⋅12,000−100,000=−28,000P(12{,}000) = 6\cdot 12{,}000 - 100{,}000 = -28{,}000 (loss)
    • P(20,000)=6⋅20,000−100,000=20,000P(20{,}000) = 6\cdot 20{,}000 - 100{,}000 = 20{,}000 (profit)
  • Key takeaway: profit is positive if revenue exceeds cost; profit grows with x if (s - c) > 0

Demand and Supply; Linear Models

Demand

  • Demand equation expresses the relation between unit price p and quantity demanded x
  • Demand curve = graph of the demand equation
  • Example data (x in thousands):
    • (x1, p1) = (48, 8); (x2, p2) = (32, 12)
    • Slope m = (p2 - p1) / (x2 - x1) = (12 - 8) / (32 - 48) = -\frac{1}{4}
    • Point-slope form: p - p1 = m (x - x1) → p - 8 = (-\frac{1}{4})(x - 48)
    • Solve to slope-intercept: p = -\frac{1}{4}x + 20
  • Interpretation: as price decreases by $1, quantity demanded increases by 4 (thousand units)
  • Key values:
    • For x = 40 (thousand): p = 10
    • For p = 14: solve 14 = -\frac{1}{4}x + 20 → x = 24 (thousand) = 24{,}000
  • Intercepts:
    • y-intercept: p(0) = 20
    • x-intercept: p = 0 → x = 80 (thousand)
  • Graphing tips
    • Label axes: price p in dollars per unit, quantity x in thousands
    • Domain: x ≥ 0, p ≥ 0
    • Use two distant points for a clean line

Supply

  • Supply equation expresses the relation between unit price p and quantity supplied x
  • Supply curve = graph of the supply equation
  • Example: 4p - 5x = 120, with x in units of 100
    • Solve for p: p=54 x+30p = \frac{5}{4}\,x + 30
    • y-intercept: p = 30 when x = 0
    • x-intercept: set p = 0 → x = -24 (not realistic; domain restricted to x ≥ 0)
    • Pick a feasible point: x = 20 → p = \frac{5}{4}\cdot 20 + 30 = 55
    • Interpretation: price 55 corresponds to 2{,}000 units (since x = 20 means 20×100)
  • Graphing notes
    • Positive slope, label axes and units, respect the given units for x

Equilibrium and Graphing Tips

  • Market equilibrium occurs where demand and supply intersect
  • Demands slope downward (negative), supplies slope upward (positive)
  • When graphing multiple lines, clearly label each line with its equation
  • Always label axes and units on graphs; indicate domain restrictions for infeasible regions
  • On tests, two well-separated points improve graph accuracy

Quick Reminders

  • Keep units consistent (x in thousands or hundreds as defined)
  • Be careful distributing negative signs in P(x) = R(x) - C(x)
  • For demand problems, interpret slope in context (one-liner descriptions)
  • When asked about intercepts, compute both intercepts from the equation and interpret them in context