The production function is f(x)=20 x-x^2 and the price of...

The production function is $f(x)=20 x-x^{2}$ and the price of output is normalized to $1 .$ Let $w$ be the price of the $x$ -input. We must have $x \geq 0$ (a) What is the first-order condition for profit maximization if $x>0 ?$ (b) For what values of $w$ will the optimal $x$ be zero? (c) For what values of $w$ will the optimal $x$ be $10 ?$ (d) What is the factor demand function? (e) What is the profit function? (f) What is the derivative of the profit function with respect to $w ?$

The production function is $f(x)=20 x-x^{2}$ and the price of output is normalized to $1 .$ Let $w$ be the price of the $x$ -input. We must have $x \geq 0$
(a) What is the first-order condition for profit maximization if $x>0 ?$
(b) For what values of $w$ will the optimal $x$ be zero?
(c) For what values of $w$ will the optimal $x$ be $10 ?$
(d) What is the factor demand function?
(e) What is the profit function?
(f) What is the derivative of the profit function with respect to $w ?$

Key Concepts

Corner Solutions

Corner solutions occur when the optimal decision lies at the boundary of the feasible set, rather than at an interior point. When the optimal input level is zero or at some extreme value, it indicates that the highest profit is achieved at these boundary points due to the structure of costs or revenues.

Comparative Statics

Comparative statics is the analysis of how the optimal decision variables and the resulting profit change in response to variations in parameters like input prices. It often involves computing derivatives to understand the sensitivity of the profit or demand functions with respect to changes in these parameters.

Profit Function

The profit function expresses the maximum profit a firm can earn as a function of output price, input prices, and the production technology. It is obtained by substituting the optimal input levels, determined from the first-order condition, into the profit equation.

Factor Demand Function

The factor demand function represents the relationship between the quantity of an input demanded and its price, derived from the profit maximization condition. It explains how firms adjust their input usage in response to changes in input prices to maximize profit.

First-Order Condition

The first-order condition is a necessary condition for an interior optimum in optimization problems. In the context of profit maximization, it requires setting the derivative of the profit function with respect to the decision variable equal to zero, ensuring marginal revenue equals marginal cost.

Profit Maximization

Profit maximization is the process by which firms choose the levels of inputs and outputs to maximize the difference between total revenue and total cost. It involves determining the optimal production level given resource constraints and market conditions.

Production Function

A production function describes how inputs are transformed into outputs. It shows the relationship between the quantities of input used and the resulting quantity of output, often reflecting properties such as increasing, constant, or decreasing returns to scale.

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