4. In the Dixit-Stiglitz model, the representative consumer's utility function can

where q (w) is consumption of variety w, n is the mass of varieties available to con-
sumers, and p is is a measure of substitutability. O ＜ p ＜ 1.
Suppose that for any w  (0,1), Stephen consumes g(W) = exp(W). Determine
Stephen's utility. That is, determine

(b) Following (a), suppose that Loretta's instantaneous consumption is at the steady state. That is, C(t) = . Determine Loretta's lifetime utility:

(c) Following (a), suppose that Loretta's instantaneous utility function takes the CRRA form with e = 1. Determine the instantaneous utility of Loretta. That is, determine 

(a) Eric would like to find an expression for R(t) by differentiating the equation with respect to t. Take the derivative of the left hand side:

(b) Following (a), take the derivative of the right hand side and find the expression for R(t). 

5. In Econometrics, nowaday ridge, lasso, and elastic net are widely used. In general, for the following estimating equation:an elastic net regression can be written as:where λ1 ≥ 0 and λ2 ≥ 0. When λ1 = 0, the elastic net reduces to a ridge； when λ2 = 0, the elastic net reduces to a lasso.For simplicity, Ray consider a one-dimensional ridge. That is, for 1 ＞ 0, letDetermine , the ridge estimator for B.

(a)Find the value of θ.

(a) Are these three estimators unbiased? Explain and show your work.

(c)  Are these three estimators consistent? Explain and show your work.

(b)  What is the expected GPA of male students living in rural areas?What is the expected GPA of female students living in rural areas? Express in terms of B's and show your work.

(a)Another researcher asks why this equation does not include both a female indicator (female = 1 if female and - O if not), and a male indicator (male = 1 if male and = 0 if not). Explain why or why not.