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Let's go through this economics problem step by step:

We are given:
- The firm's **total cost function**:  
  $ C(Q) = 20 + 4Q - 2Q^2 + Q^3 $
- The firm's **marginal cost (MC) function**:  
  $ MC(Q) = 4 - 4Q + 3Q^2 $
- The **market price** of gum: $ P = 8 \, \text{dollars/kg} $
  
The firm operates in a **perfectly competitive market**, meaning it takes the market price as given. To maximize its profit, the firm will produce where **marginal cost equals market price**.

### a) ¿Cuánto producirá si desea maximizar sus beneficios?
To maximize profit, set marginal cost (MC) equal to price:

\[
MC(Q) = P
\]

So, we set:
\[
4 - 4Q + 3Q^2 = 8
\]

Solve for $ Q $:
\[
3Q^2 - 4Q + 4 - 8 = 0
\]
\[
3Q^2 - 4Q - 4 = 0
\]

Now, solve this quadratic equation using the quadratic formula:
\[
Q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, $ a = 3 $, $ b = -4 $, and $ c = -4 $. Plugging these into the formula:

\[
Q = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(3)(-4)}}{2(3)}
\]
\[
Q = \frac{4 \pm \sqrt{16 + 48}}{6}
\]
\[
Q = \frac{4 \pm \sqrt{64}}{6}
\]
\[
Q = \frac{4 \pm 8}{6}
\]

So we have two solutions:
\[
Q = \frac{4 + 8}{6} = 2 \quad \text{or} \quad Q = \frac{4 - 8}{6} = -\frac{2}{3}
\]

Since the quantity produced cannot be negative, the firm will produce $ Q = 2 \, \text{kg} $ to maximize its profit.

### b) ¿Cuál será su beneficio?
The firm's **profit** is given by:

\[
\text{Profit} = \text{Total Revenue} - \text{Total Cost}
\]
Total revenue (TR) is:
\[
TR = P \times Q = 8 \times 2 = 16
\]
Total cost (TC) is given by the cost function evaluated at $ Q = 2 $:
\[
TC(2) = 20 + 4(2) - 2(2)^2 + (2)^3
\]
\[
TC(2) = 20 + 8 - 8 + 8 = 28
\]
Thus, the profit is:
\[
\text{Profit} = 16 - 28 = -12
\]

The firm incurs a loss of 12 dollars.

### c) ¿Cuál es la curva de oferta de corto plazo de esta empresa perfectamente competitiva?
In a perfectly competitive market, the firm's short-run supply curve is the portion of the **marginal cost (MC) curve** that lies **above the average variable cost (AVC)**.

Since we are not explicitly given the AVC, for simplicity in this context, the short-run supply curve can be approximated as the marginal cost curve for the range where $ MC \geq P $. Therefore, the firm's supply curve is:

\[
MC(Q) = 4 - 4Q + 3Q^2 \quad \text{for } Q \geq 2
\]

This is the supply curve for quantities greater than or equal to $ Q = 2 $.

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Let me know if you would like any further details or explanations!

### Related Questions:
1. How does a firm determine its shutdown point in the short run?
2. What is the role of marginal cost in determining supply in a perfectly competitive market?
3. What happens if the market price falls below the firm's average variable cost?
4. How do fixed costs affect a firm's supply decisions in the short run?
5. Can a firm have positive production in the short run while making a loss?

#### Tip:
In perfectly competitive markets, firms are price takers, meaning they cannot influence the price and must adjust their production to maximize profits or minimize losses.

Una empresa perfectamente competitiva que produce chicles enfrenta un precio de mercado de $8 el kg. Su curva de costos totales es C(Q) = 20 + 4Q - 2Q^2 + Q^3. Su curva de costo marginal es: CMg(Q) = 4 - 4Q + 3Q^2. a) ¿Cuánto producirá si desea maximizar sus beneficios? b) ¿Cuál será su beneficio? c) ¿Cuál es la curva de oferta de corto plazo de esta empresa perfectamente competitiva?

Learn how to solve an economics problem involving the maximization of profits for a perfectly competitive firm. The problem covers cost functions, marginal cost, and profit calculation with detailed solutions.

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