Assume that the social welfare of an economy at any time is measured by the utility...

  

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Assume that the social welfare of an economy at any time is measured by the utility from consumption C. U = U(C), with Uc > 0 and Uce <0. Let's denote K as capital, / as investment (KI), Y = f(K), where f is the production function (fx > 0 and fix < 0), and Y is the output. Given capital K, and consumption C, investment is given by I=K' f(K) C. Note that C' K'f'(K) - K". 1. Write the society's utility as a function of K and K'. [5 points] 2. We assume that the discounts the future at a positive discount rate of r. The society's goal is to maximize social welfare in a finite time horizon T, with T > 0. (a) Write the integrand function, F, for the society's objective functional form. [5 points] (b) With the boundary conditions K(0) Ko and K(T) = 0, write the the dynamic optimization problem of the society. [5 points] (c) Write the Euler equation of the above problem and show that it yields: UCC (K)- 10 points) U'(C) (d) Does the integrand F(t, K, K') a concave function in (K, K')? Explain. [5 points] 3. Define the (negative) elasticity of marginal utility as (C) - (a) Using 2.(c), show that P(C)= f'(K)-r. [5 points] (b) Provide an economic interpretation of (1). [5 points] 4. In this question, we assume that U(C) InC and f(K) K. (a) Determine (C). [5 points] U(C) (b) Using (1), find the expression of C as a function of t. [5 points] (1) (c) Since K' f(K)-C, use 4.(b) and the boundary conditions to find K as a function of t. Then, find the expression of C(t). [10 points] Assume that the social welfare of an economy at any time is measured by the utility from consumption C. U = U(C), with Uc > 0 and Uce <0. Let's denote K as capital, / as investment (KI), Y = f(K), where f is the production function (fx > 0 and fix < 0), and Y is the output. Given capital K, and consumption C, investment is given by I=K' f(K) C. Note that C' K'f'(K) - K". 1. Write the society's utility as a function of K and K'. [5 points] 2. We assume that the discounts the future at a positive discount rate of r. The society's goal is to maximize social welfare in a finite time horizon T, with T > 0. (a) Write the integrand function, F, for the society's objective functional form. [5 points] (b) With the boundary conditions K(0) Ko and K(T) = 0, write the the dynamic optimization problem of the society. [5 points] (c) Write the Euler equation of the above problem and show that it yields: UCC (K)- 10 points) U'(C) (d) Does the integrand F(t, K, K') a concave function in (K, K')? Explain. [5 points] 3. Define the (negative) elasticity of marginal utility as (C) - (a) Using 2.(c), show that P(C)= f'(K)-r. [5 points] (b) Provide an economic interpretation of (1). [5 points] 4. In this question, we assume that U(C) InC and f(K) K. (a) Determine (C). [5 points] U(C) (b) Using (1), find the expression of C as a function of t. [5 points] (1) (c) Since K' f(K)-C, use 4.(b) and the boundary conditions to find K as a function of t. Then, find the expression of C(t). [10 points]