Suppose the production possibility frontier for cheeseburgers ( \(C\) ) and milkshakes (M) is given by \\[ C+2 M=600 \\] a. Graph this function. b. Asuming that people prefer to eat two cheeseburgers with every milkshake, how much of each product will be produced? Indicate this point on your graph. c. Given that this fast-food economy is operating efficiently, what price ratio \(\left(P_{c} / P_{u}\right)\) must prevail?

To plot the PPF, first, solve the equation \(C + 2M = 600\) for C and M in terms of the other variable. To solve for C, \(\displaystyle C = 600 - 2M\). To solve for M, \(\displaystyle M = \frac{1}{2}(600 - C)\). Now, we can plot the PPF on a graph with C on the x-axis and M on the y-axis. You'll see a straight line descending from the point \((C, M) = (600, 0)\) to the point \((C, M) = (0, 300)\).

Next, we need to find the amount of cheeseburgers and milkshakes under the condition that people prefer to eat 2 cheeseburgers for every 1 milkshake. To represent this, we can write the relationship as: \\[\frac{C}{M} = \frac{2}{1}\\] We can write C as \(2M\), and substitute this into the PPF equation: \\[(2M) + 2M = 600\\] Solving for M: \\[4M = 600\\] \\[M = 150\\] Now that we have M, we can find the corresponding C: \\[C = 2M = 2(150)\\] \\[C = 300\\] Thus, the preferred allocation is \((C, M) = (300, 150)\). Mark this point on the PPF graph.

Finally, we need to find the price ratio \(P_c / P_u\) for the economy to operate efficiently. To do this, we consider the trade-off between cheeseburgers and milkshakes that occurs along the PPF. The slope of the PPF represents the trade-off between the goods (the opportunity cost). In this case, the slope of the PPF, as found in Step 1, is: \\[\frac{\Delta M}{\Delta C} = -\frac{1}{2}\\] The meaningful part here is the value "2", which represents the amount of cheeseburgers that must be given up to produce one more milkshake. This negative sign shows that there's a trade-off between producing cheeseburgers and milkshakes. The price ratio must equal the negative of the PPF slope (the opportunity cost), which implies: \\[\frac{P_{c}}{P_{u}} = -\frac{\Delta M}{\Delta C} = -(-\frac{1}{2})=\frac{1}{2}\\] Therefore, for the economy to operate efficiently, the price ratio must be \(P_c / P_u = 1/2\).