Unconstrained Optimization: Single Variable Basic Math for Economics – Refresher 1 Introduction  In both micro and macroeconomic contexts, optimization is a frequently relied upon tool.  We  use it to maximize profits, minimize costs, etc. We break optimization down into two types:  Unconstrained Optimization deals with situations where we seek to either maximize or minimize one or several variables without any restrictions on the values they can take.  Constrained Optimization, on the other hand, imposes limits on the values that our variables can take. Things like budget constraints, production functions, etc. 2 Unconstrained Optimization  Let’s review unconstrained optimization by looking at the profit maximization function.  Suppose a monopolist served a market that faced the inverse demand function of 𝑝 = 250 − 2𝑞 and a constant marginal cost of production of 𝑐 = 50. What value of 𝑞 maximizes the monopolist’s profits? What is the corresponding price and profit level?  We already know that the monopolist can set his marginal revenue equal to his marginal costs and determine his optimal quantity that way. This time, let’s set it up using an optimization problem. 3 Unconstrained Optimization  To properly set up an optimization problem, we need a few elements:  First, we need to define the problem. This is a maximization problem, so let’s start by writing that. max  Next, we need to list the choice variables; the ones that we are optimizing for. In this case, there is only one, 𝑞. We list these variables under our optimization condition. max 𝑞 4 Unconstrained Optimization  Lastly, we list our objective function. This is the goal of the problem. In this case, the monopolist is maximizing his profits, max 𝜋 𝑞  We can expand on this, however, since 𝜋 = 𝑇𝑅 − 𝑇𝐶. Substituting, max 𝑇𝑅 − 𝑇𝐶 𝑞  Once more, we can substitute 𝑇𝑅 = 𝑝𝑞 (and use the inverse demand function for 𝑝) and 𝑇𝐶 = 50𝑞 (since the marginal cost of production is constant) to further expand on this, obtaining, max 250 − 2𝑞 𝑞 − 50𝑞 𝑞 5 Unconstrained Optimization max 𝑞 250 − 2𝑞 𝑞 − 50𝑞  Once here, we are ready to calculate our first-order (and second-order, if need be) conditions.  Being able to set up a problem this way will keep you organized and reduce the likelihood of making mistakes going forward.  As a tip, the first thing you should do when facing an optimization problem is to clearly define the problem. Make sure you consider all the variables of interest as well as properly defining the objective function. 6 Unconstrained Optimization max 𝑞  250 − 2𝑞 𝑞 − 50𝑞 When we calculate first-order conditions, we take the derivative of the objective function for each of our choice variables, then set it equal to zero (since that is where a maximum or minimum occurs).  Thus, we calculate the derivative of our profit function with respect to 𝑞, since that is our only choice variable, then we set it equal to zero. 𝑑𝜋 = 250 − 4𝑞 − 50 = 0 𝑑𝑞  Taking a closer look at this, notice that it’s the same condition as marginal revenue equals marginal cost. 250 − 4𝑞 − 5ด0 = 0 𝑀𝑅 𝑀𝐶 7 Unconstrained Optimization 𝑑𝜋 = 250 − 4𝑞 − 50 = 0 𝑑𝑞    From here, we simply solve this equation for 𝑞 to find our equilibrium.  Rearranging terms, 4𝑞 = 200 Lastly, dividing both sides by 4, we obtain our equilibrium quantity, 𝑞 ∗ = 50 If we wanted to, we could go back and calculate our equilibrium price and profit level from this information. Let’s make sure we have a maximum however, first. 8 Unconstrained Optimization 𝑑𝜋 = 250 − 4𝑞 − 50 = 0 𝑑𝑞  Recall that for this value of 𝑞 to be a maximum (rather than a minimum), the second derivative must be negative.  We can calculate the second derivative by just differentiating this function one more time, 𝑑2 𝜋 = −4 < 0 2 𝑑𝑞  Since this function is negative for every possible value of 𝑞, we know that we have a maximum. 9 Unconstrained Optimization 𝑞 ∗ = 50  Now we can calculate our equilibrium price and profit level.  We find our equilibrium price by plugging the equilibrium quantity back into the inverse demand function, 𝑝∗ = 250 − 2𝑞 ∗ = 250 − 2 50 = 150  Lastly, we obtain our profit level by plugging the equilibrium quantity back into the profit function, 𝜋 ∗ = 250 − 2𝑞 ∗ 𝑞∗ − 50𝑞 ∗ = 5,000 10

Unconstrained Optimization Single Variable

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