I. (50%)1. A monopolist uses one input, X, which she purchases at the fixed price r=2 to produce her output, Q. Her demand and production functions are p=80-2q and q=3√x, respectively. Please determine the values of p, q, and x at which the monopolist maximizes her profit. (25%)

2. Assuming that Mr. Lee consumes three goods- hot dog, cake, and automobiles. Denote the quantities of these goods consumed by X₁, X₂, and X₃, respectively. Suppose that Mr. Lee's utility function is given by U = 5 logX₁ + 4 logX₂ + log(0.1 + X₃), that the prices of the goods are given by P₁ = $1, P₂ = $2, and P₃ = $2000, and that his total income is $9000. Note that automobiles must be bought in discrete units (that is, X₃ must equal 0, 1, 2, and so on). Given these constraints, how will he allocate income so as to maximize utility? And what is the maximized utility Mr. Lee gains from this simple model? Please round your answers off to 2 decimal places. (25%)

I. (50%)
1. A monopolist uses one input, X, which she purchases at the fixed price r=2 to produce her output, Q. Her demand and production functions are p=80-2q and q=3√x, respectively. Please determine the values of p, q, and x at which the monopolist maximizes her profit. (25%)

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