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Let's go through each part of this monopoly problem one by one.

### Problem Analysis
The demand and cost functions for a monopoly in the diamond market are given as follows:

1. **Demand function**: \( P = 20 - Q \)
2. **Total cost function**: \( CT = 8 + 2Q + 0.5Q^2 \)
3. **Marginal cost**: \( CMg = 2 + Q \)

The tasks are:

**(a)** Explain how the monopolist chooses the equilibrium quantity and price.  
**(b)** Calculate the equilibrium price, quantity, and the monopolist’s profit.  
**(c)** Graph the demand, marginal revenue, marginal cost, and average total cost curves, marking equilibrium quantity, price, and profit.  
**(d)** Calculate consumer and producer surplus.  
**(e)** Analyze the effect of an increase in the elasticity of demand.

### Solutions

#### (a) Explanation of Monopoly Equilibrium
A monopolist maximizes profit by setting its marginal revenue (MR) equal to marginal cost (MC). The marginal revenue function for a linear demand can be derived from the inverse demand function. For the demand function \( P = 20 - Q \), total revenue \( TR \) is:

\[
TR = P \cdot Q = (20 - Q)Q = 20Q - Q^2
\]

Differentiating \( TR \) with respect to \( Q \) gives the marginal revenue:

\[
MR = \frac{d(TR)}{dQ} = 20 - 2Q
\]

The monopolist sets \( MR = MC \) to determine the equilibrium quantity.

#### (b) Calculating Equilibrium Price, Quantity, and Profit
1. **Equilibrium Quantity (Q)**: Set \( MR = MC \).
   \[
   20 - 2Q = 2 + Q
   \]
   Solving for \( Q \):
   \[
   20 - 2 = 3Q \Rightarrow Q = \frac{18}{3} = 6
   \]

2. **Equilibrium Price (P)**: Substitute \( Q = 6 \) into the demand function.
   \[
   P = 20 - Q = 20 - 6 = 14
   \]

3. **Profit Calculation**:
   - **Total Revenue (TR)**: \( TR = P \cdot Q = 14 \cdot 6 = 84 \).
   - **Total Cost (TC)**: Substitute \( Q = 6 \) into \( CT = 8 + 2Q + 0.5Q^2 \).
     \[
     CT = 8 + 2(6) + 0.5(6)^2 = 8 + 12 + 18 = 38
     \]
   - **Profit**: \( \text{Profit} = TR - TC = 84 - 38 = 46 \).

#### (c) Graphing the Functions
1. **Demand Curve**: This is given by \( P = 20 - Q \).
2. **Marginal Revenue Curve**: Derived as \( MR = 20 - 2Q \).
3. **Marginal Cost Curve**: Given as \( MC = 2 + Q \).
4. **Average Total Cost (ATC)**: Calculated as \( ATC = \frac{CT}{Q} = \frac{8 + 2Q + 0.5Q^2}{Q} \).

Graphically, equilibrium is at the intersection of MR and MC, where quantity is 6 and price is 14. The profit area (difference between price and ATC at \( Q = 6 \)) can be shaded.

#### (d) Calculating Consumer and Producer Surplus
1. **Consumer Surplus (CS)**: The area above the price line and below the demand curve.
   \[
   CS = \frac{1}{2} \times (20 - 14) \times 6 = \frac{1}{2} \times 6 \times 6 = 18
   \]

2. **Producer Surplus (PS)**: The area below the price line and above the MC curve, which can also be interpreted as the monopolist’s profit:
   \[
   PS = \text{Profit} = 46
   \]

#### (e) Effect of Increased Demand Elasticity
If the demand becomes more elastic, consumers are more responsive to price changes, which can lead to a lower monopoly price and higher output. The monopolist would potentially have to reduce the markup over marginal cost, decreasing its profit margin.

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Would you like further details on any part of this solution or need assistance with graphing? 

### Related Questions
1. How would the equilibrium change if the marginal cost increased?
2. What if the demand curve shifted upward?
3. How is consumer surplus different in a monopoly versus perfect competition?
4. Can you explain why MR has twice the slope of the demand curve?
5. What would happen to the monopolist's profit if a tax were introduced?

### Tip:
Remember, for monopolies, marginal revenue is always less than price except at zero output, due to the downward-sloping demand curve.

Suponga que lo contratan para analizar el mercado de diamantes, donde hay una única empresa. La demanda, el costo total y el costo marginal de la empresa son: Demanda: P = 20 - Q, Costo total: CT = 8 + 2Q + 0.5Q^2, Costo marginal: CMg = 2 + Q. a) Explique cómo el monopolista elige la cantidad y el precio de equilibrio. b) Calcule el precio, la cantidad de equilibrio y el beneficio que obtiene el monopolista. c) Grafique a mano la demanda, la curva de ingreso marginal, la curva de costo marginal y la curva de costo total medio de un monopolio. Señale en el dibujo la cantidad y precio de equilibrio. A su vez, marque con un color los beneficios de la empresa. d) Calcule el excedente del productor y el consumidor. e) ¿Cómo cambiaría el precio y la cantidad si la demanda fuera más elástica?

Analyze how a monopoly determines the equilibrium price and quantity in the diamond market. Learn to calculate the total revenue, marginal cost, and profits using the given demand and cost functions. Includes graphing the demand, marginal revenue, and cost curves, as well as calculating consumer and producer surplus. Understand how elasticity affects pricing strategies.

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