110年 - 110 國立清華大學碩士班考試入學試題_經濟學系:微積分與統計#105269

(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.

(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 as

Calculate

(a) Suppose that Loretta's instantaneous utility function takes the CRRA
form. 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:

(c) If the true data generating process contains a constant term such
that

where u: is a random error term. Is the estimator derived in part (a) unbiased?
Show your work and explain carefully.

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

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, let

Determine , the ridge estimator for B.

X, denoted as E(X), and variance, denoted as Var(X).

(a)Derive the OLS estimator .

(b) Calculate the variance of .

(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.