Differentiation, Curve Sketching, and Cost Functions

1. Differentiation, Curve Sketching, and Cost Functions

2. Assumptions • When applying calculus to economics we are making approximations. True functions in economics jump from value to value. However, because the jumps are small, we can approximate with continuous functions. • We assume that the approximations are close to the actual functions. • At some point the cost functions will reach optimal values for total output. • We also assume that all economic factors remain constant. (e.g. inflation, price of supplies, demand for product, etc…)

3. Assumptions • Desired output level is attainable. • There are both fixed and variable cost. • There is a positive fixed cost. • Total cost increases as total quantity produced increases. • At low levels of production the marginal cost decreases as production as increases. • At high levels of production, the marginal cost increases as the production increases. • The cost of production for a quantity, q, can be approximated by a twice continuously differentiable function: C(q). • Quantity, q, is always positive. • The objective of the firm is to maximize profits.

4. Cost of Production • Cost is defined as the value of everything a seller must give up to produce a good. • For a given level of production, q, the cost function is C(q). This cost function will take into consideration both the variable and fixed costs. • In regards to the cost function, C(q), • C(q) has a positive y-intercept • C(q) is increasing • For an interval 0 < q < q*, C(q) is concave down. • For the interval q* < q < , C(q) is concave up.

5. Marginal Cost of Production • Marginal cost is defined as the increase in total cost that arises from an extra unit of production. • Marginal Cost, MC(q), can be found by calculating the derivative of the cost function, C’(q). MC(q) = C’(q) • C(q) always has a positive slope, therefore Marginal Cost is always positive. • Marginal cost is at a minimum where it’s derivative is 0. MC’(q) = 0 • The graph of the Marginal Cost function first decreases, reaches a minimum, then increases. • The minimum of the Marginal Cost function occurs at the same production quantity, q, as the inflection point of C(q).

6. Average Cost of Production • Average Cost, AC(q), is defined as the total cost divided by total quantity, C(q) / q. • AC(q) = C(q) / q • AC(q) will always be positive. • AC(q) approaches infinity as q approaches 0. • AC(q) has a minimum at • AC’(q) = 0 • which occurs when AC(q) = MC(q). This is the only point at which these lines intersect.

7. Demand and Revenue • There is a given demand function, D(q), which determines the price the firm can charge for it’s goods. • The graph of demand function, D(q), will show the relationship between the price of a good and the quantity demanded. • Revenue, R(q), is determined by multiplying D(q) by the production quantity, q. • R(q) = D(q) * q

8. Marginal Revenue and Profit • Marginal revenue is defined as the change in total revenue from an additional unit sold. • Marginal Revenue, MR(q), can be found by taking the derivative of the Revenue function. • MR(q) = R’(q) • The quantity, q, at which profit is maximized is found when the Marginal Revenue is equal to the Marginal Cost. • MR(q) = MC(q). • Profit is determined by subtracting the total cost from the total revenue. • Profit = R(q) – C(q)

9. ModelInterpretations • This model is built on the assumptions that our firm is in a economic vacuum. In reality economic factors would be constantly change. • Mathematically, the model provides accurate results given the Cost function, C(q), and Demand function, D(q), are correct.

10. Questions?