Marginal Analysis 1. 2. Suppose a competitive firm has as its total cost function: TC=29+2q2TC=29+2q2 Suppose the firm's output can be sold (in integer units) at $77 per unit. Using calculus and...

Marginal Analysis
1.
2.
Suppose a competitive firm has as its total cost function:
TC=29+2q2TC=29+2q2
Suppose the firm's output can be sold (in integer units) at $77 per unit.
Using calculus and formulas (don't just build a table in a spreadsheet as in the previous lesson), what is the total profit at the optimal integer output level?
Please specify your answer as an integer. In the case of equal profit from rounding up and down for a non-integer initial solution quantity, proceed with the higher quantity.
Hint 1: The first derivative of the total cost function, which is cumulative, is the marginal cost function, which is incremental. The na
ated lecture and formula summary explain how to compute the derivative.
Set the marginal cost equal to the marginal revenue (price in this case) to define an equation for the optimal quantity q.
Hint 2: When computing the total cost component of total profit for a candidate quantity, use the total cost function provided in the exercise statement (rather than summing the marginal costs using the marginal cost function).
3.
Assume that a monopolist faces a demand curve for its product given by:
p=100−1qp=100-1q
Further assume that the firm's cost function is:
TC=470+9qTC=470+9q
Using calculus and formulas (don't just build a table in a spreadsheet as in the previous lesson) to find a solution, what is the profit (rounded to the nearest integer) for the firm at the optimal price and quantity?
Round the optimal quantity to the nearest hundredth before computing the optimal price, which you should then round to the nearest cent. Note: Non-integer quantities may make sense when each unit of q represents a bundle of many individual items.
Hint 1: Define a formula for Total Revenue using the demand curve equation. Then take the derivative of the Total Revenue and Total Cost formulas to compute the Marginal Revenue and Marginal Cost formulas, respectively. Use these Marginal Revenue and Marginal Cost formulas to perform a marginal analysis.
Hint 2: When computing the total revenue component of total profit for each candidate quantity, use the total revenue function computed from the demand curve equation (rather than summing the marginal revenues using the marginal revenue function).
Supply and Demand
1.
Assume that the demand curve D(p) given below is the market demand for apples:
Q=D(p)=250−15pQ=D(p)=250-15p, p > 0
Let the market supply of apples be given by:
Q=S(p)=42+10pQ=S(p)=42+10p, p > 0
where p is the price (in dollars) and Q is the quantity. The functions D(p) and S(p) give the number of bushels demanded and supplied.
What is the equili
ium quantity in this market?
Round the equili
ium price to the nearest cent, use that rounded price to compute the equili
ium quantity, and round the equili
ium quantity DOWN to its integer part.
2.
The demand curve for tickets at an amusement park is:
Q=D(p)=1300−46pQ=D(p)=1300-46p, p > 0
All customers pay the same ticket price. The marginal cost of serving a customer is $13.
Using calculus and formulas (don't just build a table in a spreadsheet as in the Marginal Analysis I lesson) to find a solution, what is the total contribution margin (revenue less variable costs) to the amusement park?
Round the equili
ium quantity DOWN to its integer part and round the equili
ium price to the nearest cent.
Hint: The first derivative of the total revenue function, which is cumulative, is the marginal revenue function, which is incremental. The formula summary explains how to compute the derivative.