Question

Each day Paul, who is in third grade, eats lunch at school. He likes only Twinkies \(\left(7^{\prime \prime}\right)\) and Orange Slice \((5),\) and these provide him a utility of $$\text { utility }=U\\{T, S)=\mathbf{V} 7 \mathbf{X}$$ a. If Twinkies cost \(\$ .10\) each and Slice costs \(\$ .25\) per cup, how should Paul spend the \(\$ 1\) his mother gives him in order to maximize his utility? b. If the school tries to discourage Twinkie consumption by raising the price to \(\$ .40,\) by how much will Paul's mother have to increase his lunch allowance to provide him with the same level of utility he received in part (a) \(?\) How many Twinkies and cups of Slice will he buy now (assuming that it is possible to purchase fractional amounts of both of these goods)?

Short answer

Additionally, when the price of Twinkies increases to 40 cents each, how much more should Paul's mother give as an allowance, and what new quantities of Twinkies and Orange Slice will Paul buy? Answer: With the initial prices of 10 cents for Twinkies and 25 cents for Orange Slice, Paul should buy 5 Twinkies and 2 cups of Orange Slice to maximize his utility. When the price of Twinkies increases to 40 cents, Paul's mother should increase his allowance by $0.50, and Paul will buy 2.5 Twinkies and 4 cups of Orange Slice.

Step-by-step solution

Set up the budget constraint

: The budget constraint can be expressed as the sum of the cost of Twinkies and the cost of Orange Slice, which should be equal to the total budget ($1): \(0.10T + 0.25S = 1\)

Derive the demand functions for Twinkies and Orange Slice

: We can rearrange the budget constraint equation from step 1 above so we have the cost of Twinkies in terms of Orange Slice: \(T=(1-0.25S)/0.10=10-2.5S\) Now we can plug this into the Utility function: U(S)=√((10-2.5S)*S)

Determine the optimal combination of Twinkies and Orange Slice to maximize utility in part a

: To find the optimal combination to maximize utility, we will take the derivative of the utility function with respect to Orange Slice and set it equal to zero: \(\frac{dU(S)}{dS} = 0\) The derivative of the utility function with respect to Orange Slice is: \(\frac{dU(S)}{dS}= \frac{1}{2\sqrt{(10-2.5S)S}}\left(\frac{d}{dS}((10-2.5S)S)\right)\) Taking the derivative and simplifying, we get: \(\frac{dU(S)}{dS}= \frac{-2.5S^2+10S}{\sqrt{(10-2.5S)S}}\) Now, set the derivative equal to zero and solve for S: \(0=frac{-2.5S^2+10S}{sqrt{(10-2.5S)S}}\) Solving the quadratic equation, we get S = 2. To find the optimal number of Twinkies, we plug in the value of S into the expression we found in step 2 for T: \(T=10-2.5(2)=5\) So, Paul should buy 5 Twinkies and 2 cups of Orange Slice to maximize his utility.

Calculate the increase in allowance and new quantities of Twinkies and Orange Slice in part b

: First, we need to recalculate the budget constraint considering the new price of Twinkies: New budget constraint: \(0.40T+0.25S=1+\Delta x\), where \(\Delta x\) is the increase in allowance. Next, we must maintain the original utility level computed in part a: \(U(5,2)=√(5 \times 2)=√10\) Now, we need to find the new optimal combination of T and S that maximizes utility, given the new prices and budget constraint. Substitute the new Twinkie price into the budget constraint: \(T= (1-0.25S+ \Delta x)/0.40\) Replace T in the utility function: \(U(S)=√((1-0.25S+\Delta x)/0.40 \times S)\) Set the utility equal to the original level: \(√10=√((1-0.25S+\Delta x)/0.40 \times S)\) Square both sides to get rid of the square root: \(10=(1-0.25S+\Delta x)/0.40 \times S\) Now, we must solve for the unknowns, \(\Delta x\) and S. Solving for S, we have: \(S=\frac{20}{3-0.625\Delta x}\) Solving for the increase in allowance, \(\Delta x\) = 0.5. Now that we have found the increase in allowance, we can determine the new quantities of Twinkies (T) and Orange Slice (S) that Paul will buy. Plugging the increase in allowance back into the expression for S, we find: \(S=\frac{20}{3-0.625(0.5)}=4\) Then, find the new quantity of Twinkies (T) that Paul will buy, based on the increase in allowance, using the new budget constraint: \(T=(1.5-0.25(4))/0.40=2.5\) Thus, Paul's mother must increase his lunch allowance by $0.50, and he will now buy 2.5 Twinkies and 4 cups of Orange Slice.

Key concepts

Budget Constraint

The concept of a budget constraint plays a crucial role in economics, especially in consumer choice theory. In layman’s terms, it represents the combinations of goods and services that a consumer can purchase given their limited income. In our exercise, Paul's budget constraint is defined by the amount of money he has to spend on Twinkies and Orange Slice for his lunch, which is $1.

When we express this mathematically, we see it as an equation where the sum of the cost of Twinkies and Orange Slice cannot exceed Paul’s budget. This constraint helps to narrow down the feasible options Paul has for his lunch combination and is an essential step in solving his utility maximization problem.

Demand Functions

In the context of the utility maximization problem, demand functions reflect the relationship between the quantity of goods a consumer wants to buy and the price of those goods, given the consumer’s budget constraint. These functions are derived from the consumer's utility function, which in Paul’s case is solely dependent on the amounts of Twinkies and Orange Slices he consumes.

To maximize utility, we manipulate the budget constraint to express the quantity of one good in terms of the other, helping us to understand Paul’s preferences and how he would adjust his consumption when prices change. In the textbook solution, we deduced that given his budget and the cost of each item, Paul would ideally want to purchase 5 Twinkies and 2 Orange Slices.

Marginal Utility

The term marginal utility describes the additional satisfaction, or utility, a person derives from consuming one more unit of a product. Economists assume that individuals make purchases with the goal of maximizing their overall utility. In mathematical terms, marginal utility is represented by the derivative of the utility function with respect to a good.

In Paul’s lunch choice, to find the optimal mix of Twinkies and Orange Slice, we calculated the marginal utility of consuming these items. The solution showed the process of equating marginal utility to zero to find the point where Paul’s utility is maximized with no further gain from extra consumption.

Price Elasticity of Demand

The price elasticity of demand measures how sensitive the quantity demanded of a good is to a change in its price. It’s a crucial concept when discussing market economics, as it helps to understand how consumers will react to price changes. The demand is said to be elastic if a small change in price leads to a large change in the quantity demanded, and inelastic if the quantity demanded is relatively insensitive to a change in price.

In part b of the exercise, we assessed the impact of a price change of Twinkies on Paul’s consumption. The higher cost meant that Paul would need an increase in his budget to maintain the same level of utility. The textbook solution shows the shift in demand due to a price change, reflecting the concept of price elasticity.