Economics 4 Questions: You Are The Manager Of A Monopoly

Economics 4 Questions1 You Are The Manager Of A Monopoly That Faces

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1) Determine the maximum profits of a monopoly with demand curve P = 230 - 20Q and costs C = 5 + 30Q.

2) For a product with demand QXd = P, find the inverse demand curve, and analyze consumer surplus at prices of $45 and $30; explain what happens to consumer surplus as the price falls.

3) Given the own price elasticity of demand for gasoline is -0.8, the Rothschild index is 0.3, and sales are $2,700,000, find the price elasticity of demand for a representative gasoline retailer's product.

4) If a firm’s revenues are $20,000 from product X and $90,000 from product Y, with the own price elasticity of demand for X at -3 and cross-price elasticity between Y and X at -1.3, determine how total revenues change if the price of X increases by 1%.

Paper For Above instruction

The following paper provides an in-depth analysis and solution to the four economic questions posed, demonstrating mastery of microeconomic concepts such as monopoly profit maximization, demand curve analysis, consumer surplus, elasticity calculations, and revenue impact analysis due to price changes.

1. Maximizing Monopoly Profits

A monopoly faces the demand curve P = 230 - 20Q, with cost function C = 5 + 30Q. To find the firm's maximum profits, we need to identify the profit-maximizing quantity and price. The profit function is defined as total revenue minus total cost:

π(Q) = TR - TC = P × Q - C

Substituting the demand curve into total revenue:

TR(Q) = (230 - 20Q) Q = 230Q - 20Q²

Cost function is C = 5 + 30Q. Therefore, the profit function becomes:

π(Q) = 230Q - 20Q² - 5 - 30Q = (230Q - 30Q) - 20Q² - 5 = 200Q - 20Q² - 5

To maximize profit, differentiate with respect to Q and set to zero:

dπ/dQ = 200 - 40Q = 0

Solving for Q:

Q* = 200 / 40 = 5

Plugging Q* back into the demand equation to find the profit-maximizing price:

P* = 230 - 20(5) = 230 - 100 = 130

Maximum profit is then:

π(5) = 200(5) - 20(5)^2 - 5 = 1000 - 500 - 5 = 495

Thus, the firm's maximum profits are __$495__ (without rounding).

2. Consumer Surplus and Demand Analysis

Given the demand curve QXd = P, which implies Q = P. Inverting this demand function gives the inverse demand curve:

P = Q

a. When Px = $45, consumer surplus (CS) is calculated as the area of the triangle between the demand curve and the price line up to the quantity demanded:

CS = 0.5 × (Q at Px=45) × (Difference between maximum willingness to pay and Px)

At Px=45, the quantity demanded is Q = 45. The maximum willingness to pay is where demand drops to zero, which in this linear case is at Q=0, so maximum price is theoretically unlimited. But since demand is P=Q, the highest willingness to pay at Q=45 is $45, so CS is:

CS = 0.5 × 45 × (Maximum price - Px) = 0.5 × 45 × (45 - 45) = 0 (since at this point, the consumer's maximum willingness is exactly $45). However, more precisely, consumer surplus at Px=45 is the area of the triangle between the demand curve and the price line from Q=0 to Q=45, with maximum willingness to pay at Q=Qmax, which theoretically goes to infinity. Given this context, assuming the demand extends to Q=Qmax where P=Qmax, consumer surplus is better visualized at the specific price points.

b. When Px = $30, the quantity demanded Q=30. The consumer surplus is:

CS = 0.5 × Q × (Maximum willingness to pay - Px) = 0.5 × 30 × (Q at Q=0) — but as above, since demand is P=Q, consumer surplus simplifies to:

CS = 0.5 × Q × (Q - Px) = 0.5 × 30 × (30 - 30) = 0

In practice, consumer surplus reduces as the price approaches the maximum willingness to pay, and at the exact demand point, the surplus is the area of the triangle between the demand curve and the price line; since the demand curve is P=Q and the maximum willingness is as Q approaches infinity, detailed calculations would involve more market-specific parameters.

d. The general effect of a fall in the price of a good is an increase in consumer surplus as consumers are willing to purchase more of the good at a lower price, increasing the area under the demand curve above the market price, which directly indicates increased consumer benefit.

3. Elasticity of Demand for Gasoline Retailer

The own price elasticity of market demand for gasoline is given as -0.8. The Rothschild index measures the degree of market concentration and is 0.3, with a retailer's sales volume at $2,700,000. The elasticity of demand faced by a typical retailer is calculated as:

Elasticity for individual retailer = (Market elasticity) × (Rothschild index)

Therefore,

Elasticity = -0.8 × 0.3 = -0.24

Since elasticity values are typically expressed as positive magnitudes, the elasticity is 0.24, indicating inelastic demand at the retailer level.

4. Revenue Changes Due to Price Increase

The firm receives revenues of $20,000 from product X and $90,000 from Y, with the own price elasticity of demand for X at -3 and the cross-price elasticity between Y and X at -1.3. When the price of X increases by 1%, the change in total revenue can be estimated as:

Revenues from X:

ΔR_X = Revenue_X × Price elasticity of demand_X × (% change in price)

ΔR_X = $20,000 × (-3) × 0.01 = -$600

Indicating a decrease of $600 in revenue from X.

Revenues from Y are affected via cross-price elasticity:

ΔR_Y = Revenue_Y × cross-price elasticity × (% change in P_X)

ΔR_Y = $90,000 × (-1.3) × 0.01 = -$1,170

Overall change in total revenue:

ΔTotal Revenue = ΔR_X + ΔR_Y = -$600 - $1,170 = -$1,770

Hence, the firm's total revenues will decrease by $1,770 if the price of product X increases by 1%.

Conclusion

Through comprehensive analysis, we find that the monopoly maximizes profits at a Q of 5 units and a price of $130, with maximum profits of $495. Consumer surplus varies with price changes, increasing as prices drop, consistent with microeconomic theory. The elasticities indicate inelastic demand for gasoline at the retail level, and a 1% increase in the price of product X leads to a $1,770 decrease in total revenues for the firm, illustrating the significance of elasticity considerations in pricing strategies.

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