1 Consider a two period consumption investment model studied in class Letc 1 andc 2 denote the consumption levels andy 1 andy 2 denote the income levels in periods 1 and 2 Consumer can borrow and lend in period 1 at the interest rater Consumer's utility function is given by(c 1 ) (c 2 ) (a)Intuitively, what assumptions should we impose on the first and second derivative of()to make the model economically meaningful (b)Write down the equation of the wealth constraint Graph the wealth constraint (c)State the consumption investment problem in the context of this model (i e what is this consumer trying to achieve ) (d)We know that at the optimum the slope of the highest attainable indifference curve for this consumer must be equal to the slope of the wealth constraint Use this fact to derive the optimality condition characterising the optimal consumption levels in the two periods (e)Interpret the condition derived in (d) (f)Suppose that the period utility function is logarithmic, that is ( 1 ) ( 2 ) ( 1 ) ( 2 ) Use the Lagrangian method to solve for the optimal consumption levels in the two periods (g)Suppose thaty 1 100andy 2 50and 0 1 Compute the utility gain this consumer obtains from the access to the capital market (Hint compare the utility levels with and without an opportunity to borrow or save) 2 Consider a two period consumption investment model studied in class Suppose thaty 1 2000,y 2 1296, 0 08and(c 1 , c 2 ) c 1 c 2 Solve the consumption investment problem Does this consumer benefit from the access to the capital market Explain 3 Consider and entrepreneur whose initial capital is equal to $100 This capital can be invested at the market interest rate of 8 In period 1, she also has access to a private investment venture which generates return according to the following transformation function K 10x 0 5 , whereis the funds generated in period 2 andis the amount invested in period 1 What is the optimal allocation of funds between these two investment opportunities in period 1

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1. Consider a two-period consumption-investment model studied in class. Letc 1 andc 2 denote the consumption levels andy 1 andy 2 denote the income levels

1. Consider a two-period consumption-investment model studied in class. Letc₁andc₂denote the consumption levels andy₁andy₂denote the income levels in periods 1 and 2. Consumer can borrow and lend in period 1 at the interest rater. Consumer's utility function is given by(c₁) + (c₂).

(a)Intuitively, what assumptions should we impose on the first and second derivative of()to make the model economically meaningful?

(b)Write down the equation of the wealth constraint. Graph the wealth constraint.

(c)State the consumption-investment problem in the context of this model (i.e. what is

this consumer trying to achieve?)

(d)We know that at the optimum the slope of the highest attainable indifference curve for

this consumer must be equal to the slope of the wealth constraint. Use this fact to derive the optimality condition characterising the optimal consumption levels in the two periods.

(e)Interpret the condition derived in (d).

(f)Suppose that the period utility function is logarithmic, that is

(₁) + (₂) = (₁) + (₂)

Use the Lagrangian method to solve for the optimal consumption levels in the two periods.

(g)Suppose thaty₁= 100andy₂= 50and = 0.1. Compute the utility gain this consumer obtains from the access to the capital market. (Hint: compare the utility levels with and without an opportunity to borrow or save).

2. Consider a two-period consumption-investment model studied in class. Suppose thaty₁= 2000,y₂= 1296, = 0.08and(c₁, c₂) = c₁c₂. Solve the consumption-investment problem. Does this consumer benefit from the access to the capital market? Explain.

3. Consider and entrepreneur whose initial capital is equal to $100. This capital can be

invested at the market interest rate of 8%. In period 1, she also has access to a private

investment venture which generates return according to the following transformation function: K = 10x^0.5

, whereis the funds generated in period 2 andis the amount invested in period 1. What is the optimal allocation of funds between these two investment opportunities in period 1?