# Question on General Equilibrium: how to write offer curves?

QUESTION:

Consider simple two-person, two-good economy in which agents’ utility functions are given by

$$U_1(x_{11}, x_{21}) = min\{x_{11}, x_{21}\}$$, and $$U_2(x_{12}, x_{22}) = min\{4x_{12}, x_{22}\}$$.

Endowments are w1 =(30,0) and w2 =(0,20).

If neither agents can have negative consumption of either good, what is Walrasian equilibrium?

SOLUTION:

With Leontiev preferences, the indifference curves of both agents are right-angles. For agent 1 their vertices lie on the line $$x_{11} = x_{21}$$, whereas, for agent 2, they lie on the line $$x_{22} = 4x_{12}$$. These lines are respective offer curves of both agents when both prices are strictly positive. For the case when one of the prices is zero, we have following offer curves:

————

$$OC_1(P_1, P_2)=$$

if $$(P_1, P_2) \in$$ {0} $$\times R_+$$ then $$OC_1(P_1, P_2)= \{(x_{11}, w_{21}): x_{11} > w_{21} \}$$,

if $$(P_1, P_2) \in$$ $$R_+ \times$$ {0} then $$OC_1(P_1, P_2)= \{(w_{11}, x_{21}) : x_{21} > w_{11} \}$$,

$$OC_2(P_1, P_2)=$$

if $$(P_1, P_2) \in$$ {0} $$\times R_+$$ then $$OC_2(P_1, P_2)= \{(x_{11}, w_{21}) : x_{11} > (1/4) w_{21} \}$$,

if $$(P_1, P_2) \in$$ $$R_+ \times$$ {0} then $$OC_2(P_1, P_2)= \{(w_{11}, x_{21}) : x_{21} > 4 w_{11} \}$$,

my question is how to write this offer curve? I don't ask to solve this question. I dont understand the point where the offer curves are written. Especially I find to be difficult to write the OC2 for the second consumer. I mean writing OC_2 is especially difficult for me. If you show how to write offer curves, I will really glad. Thank you.

$$EO_2BO_1$$ is 2's offer curve and $$EO_1A$$ is 1's offer curve. Set of competitive equilibrium allocations is given by the intersection of the two offer curves that is the set of allocations on the line segment $$O_1B$$.