In the two-period consumption savings model (in which the representative consumer has no control over his real labor income y1 and y2), suppose the representative consumer's utility function is u(c1, Bc2) , where, as usual, c1 denotes consumption in period 1, c2 denotes consumption in period 2, and B is a preference parameter. Suppose that all variables are real and the interest rate is r.

please only solve part d)

(a) Using a Lagrangian and assuming the utility function is 1-0 u u= -1 (Bc2)1-0 - 1 + 1-0 1-0 show how the representative consumer's optimal choices of consumption over time depends on B. Provide an economic interpretation for parameter B.

b) Illustrate the effect of an increase in B on private savings in period 1. Provide economic interpretation for the result you find

(c) Suppose that during the current pandemic, consumers who did not experience a change in income decreased their consumption and increased their savings. Is an increase in B and the effects you analyzed in parts (a) and (b) above consistent with the idea that consumption fell and savings increased because of the pandemic (the term B can be interpreted as a measure of consumer confidence)?

d) The Permanent Income Hypothesis states that individuals consider their future lifetime earnings when making their current consumption decision.Discuss briefly how the multi-period models we are considering here ( regardless of two-period, three-period, n-period or infinite -period) ar consistent with the Permanent Income Hypothesis. Give an example using the solution to part (a).