(5) 由哪一個衡量值可顯示：投資組合 P 的績效超越或低於市場投資組合 M 多少？ (2 分) （10 分）

(a) Crystal lives only with two goods, coffee (x1) and chocolate (x2). Suppose that Crystal has the following quasi-linear utility function:u (x1,x2) = α1lh x1 + α2x2.Each day, Crystal spends w dollars on coffee and chocolate. Suppose that the prices of coffee and chocolate are respectively pi and pz. 'That is, the budget constraint of Crystal can be written as:p1x1+p2x2 = w.Suppose that there is an inner solution. Derive x1 (p1, p2, w) and x2 (p1, p2, w), the demand functions of coffee and chocolate of Crystal. That is, solve the following problem:and write x1 and x2 as functions of p1, p2, and I.

(b) u(p1, p2,w), the indirect utility function, gives Crystal's maximal attain-able utility when faced with (Pi, P2), the vector of goods prices, and w, the amount of income:(p1,p2,W) =u[x1(p1,p2,W),x2(p1,p2,w)].Write down the indirect utility function of Crystal. That is, plug in your answers of (a) into the quasi-linear utility function.

(c) Roy's identity is a major result in microeconomics having applications in consumer choice and the theory of the firm. The lemma relates the demand function to the derivatives of the indirect utility function. Specifcally, demand function for good i can be calculated asCalculate

(a) Suppose that Loretta's instantaneous utility function takes the CRRAform. Compute her coefficient of relative risk aversion:where u' (C) and u" (C) are respectively the first and second derivatives of u (C).

(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). 

4. In the Dixit-Stiglitz model, the representative consumer's utility function canwhere 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). DetermineStephen's utility. That is, determine

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.

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