1. Unconstrained Optimization

2. Unconstrained Optimization The goal of unconstrained optimization is to maximize the net benefit of an activity; a common example of net benefit is profit. Net benefit NB = Total Benefit – Total Cost = TB-TC Ideally, net benefit is maximized where Marginal Benefit = Marginal Cost ∆NB/∆A = MB – MC = 0 at maximum Δ means “change in” Marginal benefit MB = (Δtotal benefit)/(Δ activity) = ΔTB/ΔA Marginal cost MC = (Δ total cost)/(Δ activity) = ΔTC/ΔA

3. Marginal Benefit Marginal benefit (MB) is the change in total benefit for an additional unit of an “activity” (often an input or output). Calculating marginal benefit for an additional input shows the marginal productivity of that input. The terms input and output can be thought of in economic terms. An input or output could be a ton of steel, additional coding in a computer program, or an hour of labor. →The activity (A) is a more generic term MB = ∆TB/∆A

4. Marginal Cost Marginal cost (MC) is the change in total cost after engaging in one more unit of an activity (or producing one more unit of output). Marginal cost shows the costs that were incurred to produce that one additional unit of output or activity. MC = ∆TC/∆A

5. Optimization Table • Creating an optimization table is an easy way to see changes as inputs or activities (X) change • An example of an optimization table follows, along with how to find missing information • Some basic algebra along with the three formulas presented in this section allows us to solve for the entries for the blank cells NB = TB − TC ∆NB/∆A = MB – MC MB = ∆TB/∆A MC = ∆TC/∆A

6. Example: Start with this information

7. What Does This Show Us? • Ideally, a firm produces where MB – MC = 0 • In this example, MB-MC does not ever equal 0 for any level of X (output), but any profit maximizing firm will produce so long as MB > MC. • This firm produces until X=3. • When X=3, the firm obtains the highest NB with MB > MC. • At X=4, MB < MC and NB declines.

8. The cost of a stolen MP3 Player is $50.00. Guards are paid $300. How many guards should the firm hire?