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To find the profit-maximizing output, take the derivative of the profit function with respect to quantity ( ) and set it to zero:
). At this point, the market clears, establishing the equilibrium price ( P*cap P raised to the * power ) and equilibrium quantity ( Q*cap Q raised to the * power microeconomics with simple mathematics pdf
TR=P⋅Q=(100−Q)Q=100Q−Q2cap T cap R equals cap P center dot cap Q equals open paren 100 minus cap Q close paren cap Q equals 100 cap Q minus cap Q squared Step 2: Determine Marginal Revenue by differentiating TRcap T cap R To find the profit-maximizing output, take the derivative
−PxPynegative the fraction with numerator cap P sub x and denominator cap P sub y end-fraction To find the profit-maximizing output
is simply the first derivative of the demand function with respect to price (the slope parameter −bnegative b
are the partial derivatives of the utility function (Marginal Utilities): The Budget Constraint Consumers are constrained by their income ( ). If the price of good PXcap P sub cap X and the price of good PYcap P sub cap Y , the budget equation is:
P*=a−b(a−cb+d)cap P raised to the * power equals a minus b open paren the fraction with numerator a minus c and denominator b plus d end-fraction close paren Concrete Example Suppose a market is defined by the following functions: Set them equal to find equilibrium quantity: 100−2Q=10+3Q100 minus 2 cap Q equals 10 plus 3 cap Q 90=5Q90 equals 5 cap Q Q*=18cap Q raised to the * power equals 18 Substitute back into the demand equation to find equilibrium price:
