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Suppose there are two persons, 1 and 2, who consume one good in the quantities x1 and x2, respectively, whereby x1+ x2 = X = 10. The utility functions of 1 and 2 are respectively:

u1= 4x1

u2= 4 (lnx2)

  1. In numbers, what is the maximum possible utility person 1 could achieve ie., if 1 gets all X?
  2. If x1= 10 - e2, what is the maximum value of person 2's utility?
  3. Find the equation of the UPF. Express this equation with u2 on the left and everything else including u1 on the right (this expression will contain ln).
  4. Write the equation of the utilitarian social welfare function, W, in terms of x2
  5. According to utilitarianism, what is the social welfare maximizing allocation to person 2, x2*?
  6. According to utilitarianism, what is the social welfare maximizing allocation to person 1, x1*?
  7. What is the maximum numerical value of social welfare, W*?

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