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Consider the problem of maximizing profits for the production function of the form f(x) =xa where...
A monopolist has demand function Q(P)-ap-ε (with lel > 1) and total cost function TC(Q)-cQ (a)...
A monopolist has demand function Q(P)-ap-ε (with lel > 1) and total cost function TC(Q)-cQ (a) Show that the demand elasticity is -e (b) Find the firm's optimal price as a function of c and ε. (c) What happens to price as є ічі.e. є approaches 1 from the right side of the number line)? (d) What is the monopoly's profit-maximizing output?
The probability mass function of a random variable X is given by Px(n)r n- (a) Find...
The probability mass function of a random variable X is given by Px(n)r n- (a) Find c (Hint: use the relationship that Ση_0 n-e) (b) Now assume λ = 2, find P(X = 0) (c) Find P(X>3)
Exercise 2.3. Consider the objective function cr (a) Let p be an ascent direction for cx....
Exercise 2.3. Consider the objective function cr (a) Let p be an ascent direction for cx. Show that the objective at a +op for every a 0 is larger than the objective at r. (b) Let p be a descent direction for cTr. Show that the objective at +op for every a >0 is smaller than the objective at r.
Question 3. Micro Review. Suppose that a firm has a production function Q = kalb, where...
Question 3. Micro Review. Suppose that a firm has a production function Q = kalb, where a>0 and b>0. K is capital and L is labor. Assume the firm is a price taker and takes the prices of inputs, (r and w) as given. 1) Write down the firm's cost minimization problem using a Lagrangean. 2) Solve for the optimal choses of L and K for given factor prices and output Q. 3) Now use these optimal choices in the...
3. A monopolist chooses output (x) to maximize profit (T) where r(x) = p(x)x-c(x) In this...
3. A monopolist chooses output (x) to maximize profit (T) where r(x) = p(x)x-c(x) In this equation, p(x) denotes the price of x and c(x) is the cost of producing x. Since demand curves are downward sloping, we assume that p,(x) <0. In addition, we will assume that marginal cost is positive and non-decreasing; that is, c'(x) > 0 and c"(x) 2 0. Derive a condition on the demand curve such that marginal revenue is downward sloping. Derive the first...
Q1. Assume that X is a continuous and nonnegative random variable with the cumulative distribution function...
Q1. Assume that X is a continuous and nonnegative random variable with the cumulative distribution function F and density f. Let b>0. (a) Write the forinula for E(X b)+1. (b) Apply the general formula from (a) to exponential distribution with parameter λ > 0.
Problem 9: 10 points Suppose that X, Y are two independent identically distributed random variables with...
Problem 9: 10 points Suppose that X, Y are two independent identically distributed random variables with the density function f(x)= λ exp (-Az), for >0. Consider T- and find its cumulative distribution function and density function.
7. Let X1,... , Xn be iid based on f(x; 6) -22e-z?/e where x > 0....
7. Let X1,... , Xn be iid based on f(x; 6) -22e-z?/e where x > 0. Show that θ=-yx? is efficient
Find all values x= a where the function is discontinuous. 7 if x <4 f(x) =...
Find all values x= a where the function is discontinuous. 7 if x <4 f(x) = x- 9 if 4 sxs7 7 if x>7 O A. a=7 O B. a=9 OC. a=4 OD. Nowhere
5. A firm has costs C(v) 18+, for y >0 and C(0) - 0 and faces...
5. A firm has costs C(v) 18+, for y >0 and C(0) - 0 and faces a demand curve of P(y) -9-y and faces a demand curve of Ply a) Solve for the monopoly price, quantity and profit. b) Solve for firm price, output, and profits under marginal cost pricing (i.e., a first-best allocation). c) Show that this is one of those cases where it is possible to have a product which cannot be profitably sustained in the market, but...
A monopolist has demand function Q(P)-ap-ε (with lel > 1) and total cost function TC(Q)-cQ (a) Show that the demand elasticity is -e (b) Find the firm's optimal price as a function of c and ε. (c) What happens to price as є ічі.e. є approaches 1 from the right side of the number line)? (d) What is the monopoly's profit-maximizing output?
The probability mass function of a random variable X is given by Px(n)r n- (a) Find c (Hint: use the relationship that Ση_0 n-e) (b) Now assume λ = 2, find P(X = 0) (c) Find P(X>3)
Exercise 2.3. Consider the objective function cr (a) Let p be an ascent direction for cx. Show that the objective at a +op for every a 0 is larger than the objective at r. (b) Let p be a descent direction for cTr. Show that the objective at +op for every a >0 is smaller than the objective at r.
Question 3. Micro Review. Suppose that a firm has a production function Q = kalb, where a>0 and b>0. K is capital and L is labor. Assume the firm is a price taker and takes the prices of inputs, (r and w) as given. 1) Write down the firm's cost minimization problem using a Lagrangean. 2) Solve for the optimal choses of L and K for given factor prices and output Q. 3) Now use these optimal choices in the...
3. A monopolist chooses output (x) to maximize profit (T) where r(x) = p(x)x-c(x) In this equation, p(x) denotes the price of x and c(x) is the cost of producing x. Since demand curves are downward sloping, we assume that p,(x) <0. In addition, we will assume that marginal cost is positive and non-decreasing; that is, c'(x) > 0 and c"(x) 2 0. Derive a condition on the demand curve such that marginal revenue is downward sloping. Derive the first...
Q1. Assume that X is a continuous and nonnegative random variable with the cumulative distribution function F and density f. Let b>0. (a) Write the forinula for E(X b)+1. (b) Apply the general formula from (a) to exponential distribution with parameter λ > 0.
Problem 9: 10 points Suppose that X, Y are two independent identically distributed random variables with the density function f(x)= λ exp (-Az), for >0. Consider T- and find its cumulative distribution function and density function.
7. Let X1,... , Xn be iid based on f(x; 6) -22e-z?/e where x > 0. Show that θ=-yx? is efficient
Find all values x= a where the function is discontinuous. 7 if x <4 f(x) = x- 9 if 4 sxs7 7 if x>7 O A. a=7 O B. a=9 OC. a=4 OD. Nowhere
5. A firm has costs C(v) 18+, for y >0 and C(0) - 0 and faces a demand curve of P(y) -9-y and faces a demand curve of Ply a) Solve for the monopoly price, quantity and profit. b) Solve for firm price, output, and profits under marginal cost pricing (i.e., a first-best allocation). c) Show that this is one of those cases where it is possible to have a product which cannot be profitably sustained in the market, but...
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