Problem 8.1. Exercise 8.1 Family Economics, I in Ljungqvist \& Sargent (4th edition, p. 304). There is one consumption good and one input, labor. A family has two members named 1 and 2. The family is run by person 1 . His welfare function is 1logc1+2[logc2n2] where 1 and 2 are positive Pareto weights, c1,c2 are consumption of person 1 and person 2 , respectively, and n2 is labor supplied by person 2. Person 1 has no labor but is endowed with s units of the consumption good, where s>0. Feasible allocations satisfy c1+c2n2+s a. Formulate the Pareto problem as a Lagrangian. b. Solve the Pareto problem for an optimal allocation and Lagrange multiplier. c. Interpret the Lagrange multiplier as a "shadow price", and tell the object of which it is the shadow price. d. Describe how the family could be reorganized as a competitive economy. e. Compute a competitive equilibrium of the economy that you identified in part d. f. Please tell how n2 would respond to different values of s. Problem 8.1. Exercise 8.1 Family Economics, I in Ljungqvist \& Sargent (4th edition, p. 304). There is one consumption good and one input, labor. A family has two members named 1 and 2. The family is run by person 1 . His welfare function is 1logc1+2[logc2n2] where 1 and 2 are positive Pareto weights, c1,c2 are consumption of person 1 and person 2 , respectively, and n2 is labor supplied by person 2. Person 1 has no labor but is endowed with s units of the consumption good, where s>0. Feasible allocations satisfy c1+c2n2+s a. Formulate the Pareto problem as a Lagrangian. b. Solve the Pareto problem for an optimal allocation and Lagrange multiplier. c. Interpret the Lagrange multiplier as a "shadow price", and tell the object of which it is the shadow price. d. Describe how the family could be reorganized as a competitive economy. e. Compute a competitive equilibrium of the economy that you identified in part d. f. Please tell how n2 would respond to different values of s