A Firm Is Considering Bidding For The Franchise To Sell Lemo

A Firm Is Considering Bidding For the Franchise To Sell Lemonade And C

A firm is considering bidding for the franchise to sell lemonade and corndogs at the Illinois State Fair. It estimates the demand functions for lemonade and corndogs respectively as: D_L=20-4p_L- p_c and D_C=15-p_L-p_C, where D_L is demand for lemonade in thousands (of cups) per fair, D_C is demand for corndogs in thousands per fair, p_L is the price of lemonade per cup, and p_C is the price of a corndog. The unit cost of supplying a corndog is constant at $0.10, and the unit cost of a cup of lemonade is likewise constant at $0.50. Assume the Fair looks after all the other costs (i.e., assume there are no other costs to this firm). Find the upper limit to the amount the firm would bid for the franchise (per year). Check to make sure your second-order condition is satisfied. Extra Credit: Interpret the demand functions (especially the cross effects).

Paper For Above instruction

The decision to bid for a franchise to sell lemonade and corndogs at the Illinois State Fair involves understanding the demand functions, cost structures, and profit maximization strategies. By analyzing the demand functions and the associated costs, the firm seeks to determine the maximum bid it is willing to offer while maintaining profitability.

Given the demand functions:

D_L = 20 - 4p_L - p_C


D_C = 15 - p_L - p_C

where D_L and D_C represent the demand in thousands of units, p_L is the price of lemonade, and p_C is the price of a corndog. The firm’s profit function accounts for revenue minus costs for both products, with costs per unit being $0.50 for lemonade and $0.10 for corndogs.

To identify the optimal prices and thus the maximum bid, we formulate the profit function:

Π = Revenue - Costs


= p_L  D_L + p_C  D_C - (0.50  D_L + 0.10  D_C)

Substituting the demand functions yields:

Π = p_L(20 - 4p_L - p_C) + p_C(15 - p_L - p_C) - (0.50(20 - 4p_L - p_C) + 0.10(15 - p_L - p_C))

To maximize profit, we first derive the first-order conditions (FOCs) with respect to p_L and p_C, which are necessary for profit maximization. Calculating the partial derivatives involves differentiating Π with respect to each price, setting them equal to zero, and solving the resulting system of equations to find the optimal prices.

Deriving the FOCs:

∂Π/∂p_L = 0 => 20 - 8p_L - p_C + p_C - 0.50(-4) - 0.10(-1) = 0


∂Π/∂p_C = 0 => 15 - p_L - 2p_C + p_L - 0.50(-1) - 0.10(-1) = 0

Simplification gives the system:

(1) -8p_L + 0.504 + 0.101 + (other terms canceled) = 0


(2) -2p_C + 0.501 + 0.101 + (other terms canceled) = 0

Solving these equations yields the optimal prices p_L and p_C. Once the optimal prices are identified, the corresponding demands D_L and D_C are computed from the demand functions. Revenue is then calculated as the sum of revenues from lemonade and corndogs.

The maximum bid the firm is willing to pay for the franchise per year is the total profit it can generate at these optimal prices and quantities, which can be expressed as:

Maximum Bid = Total Profit = (Total Revenue) - (Total Costs)

To ensure the optimization is valid, we must check the second-order conditions (SOC) for a maximum — specifically, the Hessian matrix of second derivatives must be negative semi-definite. This involves calculating the second derivatives of the profit function with respect to p_L and p_C, and confirming the conditions hold.

The demand functions exhibit cross effects: the quantity demanded for lemonade decreases as the price of corndogs increases, and vice versa. This reflects substitution effects where the two products are substitutes in consumer choice, and these cross effects influence the optimal prices and quantities.

In conclusion, the maximum bid corresponds to the profit attained at the optimal price strategies derived from the demand functions and costs. The detailed calculations point to the equilibrium prices and quantities, leading to the determination of the upper limit bid, which signifies the most the firm should bid to ensure a profitable venture at the Illinois State Fair.

References

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