Naive Question About PPFs

Question

I am a maths major, and am taking an introduction to microeconomics course this semester, and am confused by how we deduce the shape of PPF's.

For example, I was given the following problem:

Larry, Moe, and Curly all work 10 hours in a day, and can work on either mowing lawns, or washing cars. In one hour Larry can wash one car or mow one lawn, Moe can mow one lawn, or wash two cars, and Curly can wash one cars or mow two lawns.

The problem then asked me to find the various productions of each service given how much time each person spent doing one of the jobs. From there, it asked me to draw a PPF and label my points on it.

I would like to be able to go about this from a more mathematical point of view. Let $l,m,c$ be the amount of hours Larry, Moe, and Curly spend mowing lawns respectively. Then the total production of this service in terms of $l,m,c$ is given by: $$L(l,m,c)=l+c+2m$$ while the total production of washing cars is then: $$C(l,m,c)=(10-l)+2(10-c)+(10-m)$$ The PPF is then a map: $$\begin{align} PPF:\mathbb{R}^3&\longrightarrow \mathbb{R}^2\\ (l,m,c)&\longmapsto (C,L) \end{align}$$ The total production output I think is given by $L+C$, so the line we draw representing the boundary of the $PPF$ should be the maximum of $L+C$ subject to the constraint: $$\begin{align} 0\leq l+m+c\leq 30 \end{align}$$ but proceeding with Lagrange multipliers yields a set of inconsistent equations, indicating to me that my reasoning is off somewhere but I am not sure where. Furthermore, all of these equations are linear or affine so I am not sure how I'm supposed to get a bowed out boundary of the PPF. Any help on how to think about problems like this from a mathematical stand point would be very helpful.

edit: I guess it should really be a map from $[0,10]\times[0,10]\times[0,10]\rightarrow \mathbb{R}^2$.

Answer 1

Larry's $(L)$ individual PPF is $c_L+l_L=10$, where $0\leq c_L \leq 10$.

Moe's $(M)$ individual PPF is $\dfrac{c_M}{2}+l_M=10$, where $0\leq c_M \leq 20$

Curly's $(C)$ individual PPF is $c_C+\dfrac{l_C}{2}=10$, where $0\leq c_C \leq 10$

Here $c_i$ denotes number of cars washed by $i\in\{L, M, C\}$, and $l_i$ denotes number of lawns mowed by $i\in\{L, M, C\}$. To determine a point on the combined PPF, we can maximise the total number of cars washed $(=c)$, given a certain number of lawns are mowed $(=l)$ i.e. we solve: \begin{eqnarray*}\max_{c_i, l_i} & c_L + c_M + c_C = c \\ \text{s.t. } & l_L+ l_M +l_C= l \\ & c_L+l_L=10, 0\leq c_L \leq 10 \\ & \dfrac{c_M}{2}+l_M=10, 0\leq c_M \leq 20 \\ & c_C+\dfrac{l_C}{2}=10, 0\leq c_C \leq 10 \end{eqnarray*} All $(c, l)$ pairs that solves the above constitutes the PPF.

Answer 2

The PPF is then a map: PPF:R3(l,m,c)⟶R2⟼(C,L)

No. According to Wikipedia,

Graphically bounding the production set for fixed input quantities, the PPF curve shows the maximum possible production level of one commodity for any given production level of the other, given the existing state of technology.

https://en.wikipedia.org/wiki/Production%E2%80%93possibility_frontier

The PPF is the (non-axis) boundary of the region of all possible outputs. For n outputs, it's a n-1 surface. Here, there are two possible outputs, so it's a one dimensional curve.

If zero lawns are mowed, then Larry and Curly can wash at most 10 cars each, and Moe can wash 20 cars, for a total of 40 cars. 40 is the maximum number of cars that can be washed if zero lawns are mowed; the three of them could instead spend all their time cooking food, and no cars would be washed. Since we're only concerned with the boundary, however, we take the maximum 40. This gives a point (40,0).

If 2 lawns are mowed, then it could be that Curly cut them, in which case he spent one hour doing so, in which case one fewer car got washed. If Moe mowed the laws, then he spent two hours doing so, so four fewer cars were washed. If Larry mowed the lawns, then he spent two hours doing so, and two fewer cars were washed. So the maximum number of cars that were washed is 39, giving us the point (39,2).

Continuing this logic, we can have the points (38, 4), (37,6), (36,8), (35,10), (34, 12), (33, 14), (32, 16), (31, 18), (30, 20). Once we get to (30, 20), however, we're at the point where Curly is spending all his time mowing lawns, so if we want any more lawns mowed, they need to be done by Larry. This give us (29, 21), (28, 22) ... (20, 30). From there, we would have to have Moe mowing the lawns, giving us (18, 29) ... (0, 40).