In: Economics
For Theresa, one hamburger is a perfect substitute for two hot dogs. Theresa's weekly hamburger/hot dog
budget is $12. If the price of a hamburger is $2 and the price of a hot dog is $1.50, how many hamburgers and
hot dogs does Theresa consume every week?
SOLUTION:
The question is a very interesting analysis of how can we find the equilibrium points when the ICs are actually indicating Perfect Substitute Goods.
For this analysis, we must first discuss very basic but important points.
When we have intersecting points of Perfect Substitute Indifference curves (ICs) and the Budget Line (BL), we should always remember that we DO NOT have Tangency Point Solutions, but rather have Corner Point Solutions. This is because, as Perfect Substitute goods indicate straight-line IC curves as given in the question, we do not get any such point where we can have tangency between the Budget Line and the Indifference curve.
So, how do we go about solving the equation?
We first consider the given information.
It is given that Theresa is willing to give up 2 Hot-Dogs for 1 Hamburger. This simple sentence actually gives us a lot of information to form our utility function. However, a strong understanding of the concept of Marginal Utility is necessary to understand this idea.
Thus, from the first sentence, we know that our Marginal Rate of Substitution MRS = - 2.
Thus, we can say that MRS = - MUx/MUy = - 2
What this basically means is that Theresa could give up 2 units of Hot-Dogs, and get 1 unit of Hamburger and be equally as well off as she was before. Thus, She is willing to give up 2 Hog Dogs for 1 Hamburger, and considers them as Perfect Substitutes.
Thus, we can have a very simplified IC as U = 2x + y, where x = Hamburger, and y = Hot-Dogs, and the MRS = - 2. - - - - -- (1)
(MUx = dU/dx = 2, and MUy = dU/dy = 1, thus MRS = - 2)
Also, given that her budget is $12, and the Price of Hamburger (Px) is $2, and the Price of Hot-Dog (Py) is $1.5, we can write our Budget Line.
As M = x*Px + y*Py, we have M = 12 (Total Money), Px = 2, and Py = 1.5
So, we have our Budget Line as, 12 = 2x + 1.5y - - - -- (2)
Thus, now we have both our IC, as well as our BL.
Plotting the Utility Function (1) using any arbitrary number indicates to us that the slope of the IC will always be constant at MRS = - 2, indicating that the ICs are for Perfectly Substitute Goods.
Thus, we can write that, Slope of IC = MRS = 2, and
Slope of BL = Px/Py = 2/1.5 = 1.33
Thus, the Slope of IC > Slope of BL
This point indicates to us that we can have 2 corner point solutions. Once at the x-axis, when the IC intersects with the BL, and once at the y-axis.
To make these calculations easier, take any random number for Utility (Between 1 and 20) and put into the utility function and solve the IC and BL simultaneously.
We get our solutions for U = 8, and U = 12 (Numbers found using hit and trial method)
Consider the diagram having plotted equation 1 and 2,
Thus Blue lines indicate the two ICs and the Red is the BL, where at,
U = 8, we have x (Hamburger) = 0, and y (Hot-Dog) = 8, and Theresa Exhausts her budget, as Py*y = 1.5 * 8 = 12 (Budget Exhausted for the Week), and
For U = 12, we have x (Hamburger) = 6, and y (Hot-Dog) = 0, and Theresa Exhausts her budget, as Px*x = 2* 6 = 12 (Budget Exhausted for the Week).
Thus, to answer the question we can have 2 sets of possible consumption every week, where Theresa can have either 8 Hot-Dogs and 0 (zero) Hamburgers or 0 (zero) Hot-Dogs and 6 Hamburgers.