Question

a. On a given evening \(\mathrm{J}\). P. enjoys the consumption of cigars \((C)\) and brandy \((B)\) according to the function $$U(C, B)=20 C-C^{2}+18 B-3 B^{2}$$ How many cigars and glasses of brandy does he consume during an evening? (cost is no object to \(\mathrm{J}\). P.) b. Lately, however, J. P. has been advised by his doctors that he should limit the sum of brandy and cigars consumed to \(5 .\) How many glasses of brandy and cigars will he consume under these circumstances?

Short answer

Answer: J.P.'s optimal consumption under the constraint is 19/4 cigars and 5/4 glasses of brandy.

Step-by-step solution

Compute partial derivatives of the utility function

Take the partial derivative with respect to cigars \((C)\) and brandy \((B)\). $$\frac{\partial U}{\partial C}=20-2C$$ $$\frac{\partial U}{\partial B}=18-6B$$

Solve first-order conditions

Find the values of \(C\) and \(B\) that maximize utility by setting the partial derivatives equal to zero. $$20-2C=0$$ $$2C=20$$ $$C=10$$ $$18-6B=0$$ $$6B=18$$ $$B=3$$ His optimal consumption is 10 cigars and 3 glasses of brandy. b. Utility maximization under the constraint

Create the Lagrange function

Formulate the Lagrange function by adding the constraint using a Lagrange multiplier (\(\lambda\)). $$L(C,B,\lambda)=20C-C^2+18B-3B^2+\lambda(5-C-B)$$

Compute partial derivatives

Take the partial derivative with respect to cigars \((C)\), brandy \((B)\), and the Lagrange multiplier (\(\lambda\)). $$\frac{\partial L}{\partial C}=20-2C-\lambda$$ $$\frac{\partial L}{\partial B}=18-6B-\lambda$$ $$\frac{\partial L}{\partial \lambda}=5-C-B$$

Solve first-order conditions

First, we solve for \(\lambda\) by equating the first two partial derivatives, then plugging the result into the third partial derivative equation. $$20-2C-\lambda = 18-6B-\lambda$$ $$2C=6B+2$$ $$C=3B+1$$ Substitute this into the third partial derivative equation: $$5-(3B+1)-B = 5-4B = 0$$ $$4B=5$$ $$B=\frac{5}{4}$$ Now plug the value of \(B\) back into the previous equation to get the value of \(C\): $$C=3(\frac{5}{4})+1$$ $$C=\frac{15}{4}+1$$ $$C=\frac{19}{4}$$ Under the constraint, J.P. will consume \(\frac{19}{4}\) cigars and \(\frac{5}{4}\) glasses of brandy.

Key concepts

Partial Derivatives

When analyzing utility functions in microeconomics, partial derivatives are a fundamental concept that allows us to understand how the utility changes with respect to one good while holding the quantity of other goods constant. To find the rate of change of utility caused by a minor change in the quantity of cigars or brandy, we use partial derivatives.

For example, the partial derivative of the utility function with respect to cigars \( C \) is calculated by treating brandy \( B \) as a constant. This gives us a formula \( \frac{\partial U}{\partial C} = 20 - 2C \), which tells us by how much utility changes for each additional cigar. Similarly, we calculate the partial derivative with respect to brandy to find out the change in utility for each additional glass.

Lagrange Multiplier

The Lagrange multiplier technique is a powerful tool used to find the maximum or minimum of a function subject to constraints. It combines an objective function—in this case, the utility function—with the constraint into a single equation known as the Lagrangian.

When J.P.'s doctor advises him to limit his cigar and brandy consumption, we introduce a consumption constraint and employ a Lagrange multiplier \( \lambda \) to account for it. The new Lagrangian function is \( L(C,B,\lambda)=20C-C^2+18B-3B^2+\lambda(5-C-B) \). The presence of \( \lambda \) enforces the constraint that the sum of cigars and brandy must not exceed five.

First-Order Conditions

The first-order conditions are obtained by setting the partial derivatives of the Lagrangian to zero. These conditions are necessary for finding local maxima or minima of a function, and in the context of microeconomics, they help locate the point of maximum utility.

We solve the system of equations given by these first-order conditions to determine the values of cigars, brandy, and the Lagrange multiplier that satisfy both the utility maximization and the constraint imposed. In J.P.'s scenario, equating the partial derivatives from the utility and the constraint equation enables us to solve for the optimal consumption of cigars and brandy when following his doctor's advice.

Consumption Constraint

A consumption constraint limits the quantity of goods that can be consumed. This is often due to budget limitations, health advice, or other real-world constraints. In our exercise, the constraint is an external condition set by J.P.'s doctors, limiting the total consumption of cigars and brandy to five.

To apply this constraint mathematically, we incorporate it into the Lagrange function, thereby allowing us to use the method of Lagrange multipliers to adjust the utility maximization problem accordingly. The outcome illustrates the new optimal consumption levels under the constraint, which are distinct from the unconstrained scenario.