Question

Connie has a monthly income of \(\$ 200\) that she allocates between two goods: meat and potatoes. a. Suppose meat costs \(\$ 4\) per pound and potatoes \(\$ 2\) per pound. Draw her budget constraint. b. Suppose also that her utility function is given by the equation \(U(M, P)=2 M+P .\) What combination of meat and potatoes should she buy to maximize her utility? (Hint: Meat and potatoes are perfect substitutes.) c. Connie's supermarket has a special promotion. If she buys 20 pounds of potatoes (at \(\$ 2\) per pound), she gets the next 10 pounds for free. This offer applies only to the first 20 pounds she buys. All potatoes in excess of the first 20 pounds (excluding bonus potatoes are still \(\$ 2\) per pound. Draw her budget constraint. d. An outbreak of potato rot raises the price of potatoes to \(\$ 4\) per pound. The supermarket ends its promotion. What does her budget constraint look like now? What combination of meat and potatoes maximizes her utility?

Short answer

a. The budget constraint will be a line starting from 50 (pounds of meat Connie can afford if she spends all her money on meat) to 100 (pounds of potatoes she can afford if she only buys potatoes). b. She can choose any combination of meat and potatoes that lies on her budget line, because both goods give her the same utility per dollar. c. The budget line will be broken at 20 and 30 pounds of potatoes, representing the promotional offer. d. The budget line will be steeper and start from 50 units of meat and end at 50 units of potatoes. As utility per dollar for meat surpasses potatoes, she will spend her entire income on meat.

Step-by-step solution

- Plot the Budget Constraints (Part a)

Plot the budget constraint by setting up the equation using the formula: \(P_M \cdot M + P_P \cdot P = I\), where \(P_M\), \(P_P\), \(M\), and \(P\) represent the price of meat, price of potatoes, quantity of meat, and quantity of potatoes, respectively, and \(I\) represents the income. Plug the given values into the equation and rearrange the equation to \(P = (I - P_M \cdot M) / P_P\). This equation represents the budget line, which is a downward sloping line starting from the point \(M = I / P_M\) and ending at \(P = I / P_P\).

- Find the Optimal Consumption Bundle (Part b)

Calculate the optimal combination using the utility function \(U(M, P) = 2M + P\). Since meat and potatoes are perfect substitutes for Connie, she will spend all her income on the good that provides the highest utility per dollar. Calculate utility per dollar for both goods, \(U_M / P_M = 2 / 4 = 0.5\) and \(U_P / P_P = 1 / 2 = 0.5\). Since both are equal, she can buy any combination of meat and potatoes within her budget constraint.

- Draw the New Budget Constraint (Part c)

Modify the budget line to reflect the promotional offer. The equation for the first 20 pounds of potatoes remains the same but for quantities greater than 20, the line is steeper upwards for the next 10 pounds (free potatoes), and beyond this quantity, the slope of the budget line is again the same as the original. The new budget line starts at \(M = I / P_M\), goes to \(P = 20 + 10 = 30\) (including free potatoes) at \(M = (I - 20 \cdot P_P) / P_M\), and then runs parallel to the original line.

- Plot the Final Budget Constraint and Determine the Optimal Combination (Part d)

Take the new prices and draw a new budget line, which is steeper than the original due to an increase in potato prices. Find the new optimal combination using the utility per dollar (like in Step 2). If the utility per dollar for meat is greater than potatoes, all income is spent on meat, and if utility per dollar for potatoes is more than meat, spend all income on potatoes. Recalculate the utility per dollar for each, with the new potato price. If the utility per dollar of any goods changes, she will buy more of the one with higher utility per dollar.