For a monopolist, the unit cost of manufacturing a product is $3 and the demand equation is p= Va. What price will give the greatest profits?



Answer to a math question For a monopolist, the unit cost of manufacturing a product is $3 and the demand equation is p= Va. What price will give the greatest profits?

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Así es como se puede determinar el precio que generará la mayor ganancia para el monopolista: **1. Comprender los conceptos** * **Beneficio:** Ingresos totales menos costo total (Beneficio = TR - TC) * **Ecuación de demanda:** En este caso, la ecuación de demanda muestra una relación inversa entre el precio (p) y la cantidad demandada (implícita en 'a'). A medida que aumenta el precio, la cantidad demandada disminuye. * **Monopolista:** Un monopolista tiene el poder de fijar precios para maximizar sus ganancias. **2. Configurando las ecuaciones** * **Ingresos Totales (TR):** Precio (p) * Cantidad (q). Pero necesitamos expresar la cantidad en términos de precio usando la ecuación de la demanda. Como p = Va, podemos resolver para q: q = p/V * TR = p * (p/V) = p²/V * **Costo total (CT):** Costo unitario ($3) * Cantidad (q) * TC = 3q = 3(p/V) * **Ganancia:** * Beneficio = TR - TC = (p²/V) - (3p/V) **3. Encontrar el precio óptimo** Para maximizar las ganancias, un monopolista producirá donde el ingreso marginal (MR) sea igual al costo marginal (MC). Encontremos esos: * **Ingreso Marginal (MR):** La derivada de TR con respecto a p: MR = (2p/V) * **Costo Marginal (MC):** La derivada de TC con respecto a p: MC = (3 / V) **4. Ajuste MR = MC** * (2p/V) = (3/V) * 2p = 3 * p = 3/2 = $1.50 **5. Considerando la ecuación de la demanda** La ecuación de la demanda (p = Va) implica que 'V' es una constante que representa la pendiente de la curva de demanda. Necesitaremos el valor de 'V' para calcular un precio específico. Si V es muy pequeño, incluso el precio óptimo de 1,50 dólares puede no dar como resultado una ganancia positiva. **En resumen** * El precio de 1,50 dólares dará las mayores ganancias al monopolista, pero sólo si la curva de demanda permite una ganancia positiva dada la estructura de costos. Necesitará un valor para 'V' en la ecuación de demanda para obtener una respuesta más definitiva.

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