# budget line for ration quota

In this situation there is a particular commodity, like rice, is both available at a subsidized rate from a fair price shop (ration shop) and at a higher price from the open market. Suppose a consumer can buy a certain (fixed) quantity of rice at a lower price from the ration shop (that is, there is a ration quota). In addition, he can buy more of rice (assume a uniform quality of rice) from the open market at a higher price. Assuming that consumers preferences are represented by standard downward sloping, smooth, convex indifference curves.

I'm trying to answer 2 problems here.

(a). Graphically depict the consumer’s equilibrium (assuming he exhausts the ration quota and in addition buys from the open market).

(b). Suppose rice is a normal good. What will happen to the quantity purchased in the open market (over and above the ration quota) if the subsidized price (price at which the ration quota rice could be bought) is increased (but is still lower than the open market price)? Will your conclusion change if rice is an inferior good? Briefly explain.

My approach

(a). Suppose $$p_r$$ and $$p$$ are the prices of the commodity in ration shop and open market respectively. $$q$$ is the ration quota for the rice, $$M$$ is the income of the consumer and $$r$$ is the quantity of rice that the consumer will buy from the open market.

Then the Budget line of the consumer will be like $$B(p_s , p, r) = max{(q.p_r + (r-q).p \leq M)}$$

Since $$q\leq r$$ always, therefore there is going to be a kink in the budget line at $$q$$ and $$r^*$$ (the optimal qnty of rice consumed) is either going to be at $$q$$ or after $$q$$.

(b). As rice is a normal good, so Increase in $$p_r$$ will increase the slope of rationed portioned of the consumer's budget line, but since the $$p_r$$ is still $$\leq$$ $$p$$ therefore the kink is going to persist and the optimal $$r$$ will be reduced because of the substitution effect dominating the income effect, $$i.e.$$ there is going to be a reduction in the purchasing power of the consumer.

Please let me know if I am approaching this in the correct manner.