7 appendix maths

• The inverse demand function is used to show the relationship between the price a firm charges for its product and the quantity of that product it sells.
• Total Revenue (TR) is the total amount of money a firm makes from selling its product.
• Marginal Revenue (MR) is the additional revenue earned when one more unit of output is sold. It's the derivative of total revenue with respect to quantity QQQ.
• MR has the same intercept as AR (or the demand curve), but its slope is twice as steep. This means the marginal revenue curve falls faster than the demand curve.
• The inverse demand curve shows how price decreases as quantity increases.
• The MR curve below the demand curve and has a steeper slope.
• Q = a/b is where total revenue reaches its maximum, and beyond this point, MR becomes negative.
• TR starts at 0 when Q=0 increases, and reaches a maximum at Q∗, then decreases as Q increases beyond Q∗
• Profit maximization occurs when MR = MC
• • First-order condition: We set the derivative of the profit function to zero to find the critical point where profit might be maximized.
  • Second-order condition: To confirm it’s a maximum, we need to check that the slope of the profit function is decreasing at this point. Mathematically, this means the second derivative of the profit function must be negative:
• What is the formula for the inverse demand function?

  P(Q) = a-bQ

  P= price

  Q= quantity

  b= slope
• how do you calculate total revenue?

  TR(Q) = aQ - bQ^2
• What is the marginal revenue & its formula?


  MR is the extra revenue from selling one more unit

  MR (Q) = a - 2bQ
• When does a firm maximize profit?

  When MR = MC.
• What are the first- and second-order conditions for maximum profit?

  First-order: MR=MC. Second-order: The slope of the profit function must be decreasing (second derivative is negative).