# Computing the competitive equilibrium from the edgeworth box

Consider the following Edgeworth economy. There are two consumers $$i \in {1,2}$$ and two goods x and y. Consumer $$i$$ consumes $$(x_i,y_i)$$, where $$x_i ≥0$$ and $$y_i ≥0$$. Endowments are $$ω_1 =(a,0)$$ and $$ω_2 =(0,b)$$,where a>0 and b > 0. That is, consumer 1 is endowed with good x and consumer 2 is endowed with good y. We are given $$u_1=\min\{x,y\}$$ and $$u_2=\min\{x,2y\}$$. I need to find the competitive equilibrium and draw the offer curves. What I got so far based on MGW ch.15,pg:515-525 is find the offer curves for both consumers and my results are:

$$OC_1=(\frac{P_1a}{P1+P_2},\frac{P_1a}{P1+P_2})$$

$$OC_2=(\frac{2P_2b}{2P1+P_2},\frac{P_2b}{2P1+P_2})$$ (if I haven't made any mistakes in calculations).

From that we determine the walrasian eq. prices by equaling $$\frac{P_1a}{P1+P_2}+\frac{2P_2b}{2P1+P_2}=a(=\omega_{11}+(\omega_{12}))$$ and get that $$\frac{P_1^*}{P_2^*}=\frac{a-2b}{2(b-a)}$$ again if I haven't done any miscalculations.

Is this the correct way of getting the competitive equilibrium?

The picture below presents the competitive equilibrium in different situations:

• This is literally a set of formulas and image shared without source or attribution, close to zero knowledge is imparted... Commented May 27 at 13:29
• Any job that requires you to solve problems according to formulas can (and will) be performed by a program. Are you preparing people for these jobs? Or are you a firm believer in the unfairness of the testing system, and think that all students should get a passing grade? Commented May 27 at 13:31
• I created this picture. Do I also need to mention that when I create the picture? Carefully going through the picture explains why three cases are needed. This is an old question with no useful answer. I posted it because I believe that this is a useful answer.
– Amit
Commented May 27 at 13:37
• It's nice that you created the picture; my point was that if it came from a textbook or something one could at least go the source to read the surrounding text and get the theory, not just the formula. Missing a theoretical explanation I find the statement "Carefully going through the picture explains why three cases are needed." astonishing. Commented May 28 at 2:59
• You are welcome to write a complete theoretical answer of your own in a new answer. In my opinion, my answer has sufficient graphical explanation - "A picture is worth a thousand words" Thanks for your inputs/comments/suggestions.
– Amit
Commented May 28 at 3:29