In this question, we consider an "economy" with two consumers and one firm. There is a single physical good in the economy; call it "bread" for simplicity. Consumer i starts with ei 0 units of the good and zi > 0 units of time available for work, for i = 1 and 2. If the consumers spend a total time z working, f(z) units of bread will be produced where f is a known function. The consumers can trade bread. For example, if Consumer 1 starts with two units of bread and Consumer 2 starts with one unit, it is physically possible for Consumer 1 to consume one unit and Consumer 2 to consume two units without any production; whether Consumer 1 is willing to accept this arrangement is a story for another day. An allocation is a pair (x; z) where x 2 R2 and z 2 R2, where xi is the units of bread consumed by Consumer i and zi is the units of time worked by Consumer i. Consumer i's utility from an allocation (x; z) is ui(xi; zi). We make the following assumptions throughout the question

• f is twice continuously differentiable, strictly increasing, weakly concave, and f(0) = 0.
• ui is well-defined on the entire R2, twice continuously differentiable, strictly increasing in its first argument (bread) and strictly decreasing in its second argument (working time), and strictly quasiconcave.

To characterize all the Pareto optimal allocations (whose definition is good to know but not needed for this question), we solve the following problem: for any given v 2 R, we seek for all feasible allocations that maximize Consumer 1's utility subject to the constraint that Consumer 2's utility is at least v. Daniel formulates the aforementioned problem as follows.

max x2R2,z2R2 u1(x1; z1); s:t: x1 0; x2 0; z1 0; z2 0; u2(x2; z2) v; x1 + x2 e1 + e2 + f(z1 + z2); z1 + z2 z1+z2:

Jose read this and has two critiques. First, he argues that zi should be less than or equal to zi for every i, which is a stronger condition than Daniel's final constraint. Secondly, he argues that there is no sense of leaving some bread unconsumed, so the second last constraint should be equality. 1. Formulate the correct optimization problem. This can be done by affirming Daniel's formulation, correcting any mistakes he made or starting from scratch. 2. Does the problem always have a solution? If so, prove your assertion; if not, find conditions under which the problem has a solution or prove that the problem never has a solution.