The inverse demand for shampoo is given by P = 30 - 0.03Q, where P is the price per bottle in dollars and Q is bottles brought to market in hundreds. There are two manufacturers in the local market. Firm 1's cost function is given by C₁ = 0.05q₁², where q₁ is the number of bottles it brings to market. Firm 2's cost function is given by C₂ = 0.03q₂², where q₂ is the number of bottles it brings to market. The two firms are Cournot competitors who set output so that Q = q₁ + q₂. The profit maximizing level of output for Firm 1 is ____.

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The profit-maximizing output for Firm 1 is found by setting its marginal cost equal to the marginal revenue from the inverse demand curve. The marginal cost for Firm 1 is MC₁ = 0.1q₁. The marginal revenue for the inverse demand curve is MR = 30 - 0.06Q. Setting these two equations equal to each other, we get:0.1q₁ = 30 - 0.06(q₁ + q₂)0.1q₁ = 30 - 0.06q₁ - 0.06q₂0.16q₁ = 30 - 0.06q₂q₁ = (30 - 0.06q₂)/0.16q₁ = 187.5 - 0.375q₂Since the two firms are Cournot competitors, they will set their outputs simultaneously. This means that Firm 1 will choose its output based on the output of Firm 2. Substituting the above equation for q₁ into the inverse demand curve, we get:P = 30 - 0.03(q₁ + q₂)P = 30 - 0.03(187.5 - 0.375q₂ + q₂)P = 30 - 0.03(187.5 + 0.625q₂)P = 30 - 5.625 - 0.01875q₂P = 24.375 - 0.01875q₂Setting this price equal to Firm 1's marginal cost, we get:0.1q₁ = 24.375 - 0.01875q₂q₁ = 243.75 - 0.1875q₂Substituting this equation for q₁ into the equation for Q, we get:Q = q₁ + q₂Q = 243.75 - 0.1875q₂ + q₂Q = 243.75 + 0.8125q₂Setting this equal to the inverse demand curve, we get:243.75 + 0.8125q₂ = 30 - 0.03Q243.75 + 0.8125q₂ = 30 - 0.03(243.75 + 0.8125q₂)243.75 + 0.8125q₂ = 30 - 7.3125 - 0.024375q₂243.75 + 0.8125q₂ = 22.6875 - 0.024375q₂1.065625q₂ = 221.0625q₂ = 207.5Substituting this value for q₂ into the equation for q₁, we get:q₁ = 243.75 - 0.1875q₂q₁ = 243.75 - 0.1875(207.5)q₁ = 243.75 - 38.75q₁ = 205Therefore, the profit-maximizing level of output for Firm 1 is 205 bottles.

The Inverse Demand for Shampoo Is Given by P =