DSE 2013 Question 33: Budget constraint and consumer equilibrium

An individual lives for two periods, consuming only rice. In period 2 she costlessly produces 1 unit of rice; in period 1 she produces nothing but can borrow rice at interest rate $r>0$ (borrowing $z$ in period 1 means repaying $z(1+r)$ in period 2). Consumption is $x_1,x_2\ge 0$ with utility $U=x_1+\beta x_2$, $0<\beta<1$. Rice is available only from own production and borrowing.

Now suppose there are $N$ agents in this two-period economy, identical except for their discount factors, with $\beta$ uniformly distributed on $[\tfrac12,1]$. Assuming $r\le 1$, the demand function for rice in period 1 will be: