macroeconomics 2

Consider the Solow-Swan growth model, with a savings rate, s, a depreciation rate, , and a population growth rate, n. The production function is given by Y = AK + BKL1, with (0,1), where A and B are positive constants. 1. Does this production function exhibit constant returns to scale? Explain why. 2. Does it exhibit diminishing returns to capital? Explain why. 3. Express output per person, y = Y/L, as a function of capital per person, k = K/L 4. Write down an expression for y/k as a function of k and graph. (Hint: as k goes to infinity, does the ratio y/k approach zero?) 5. Use the production function in per capita terms to write the fundamental equation of the Solow- Swan model. 6. Suppose first that sA < +n Draw the savings curve and the depreciation curve. What number does the savings curve approach as k goes to zero? As k goes to infinity, the savings curve approaches a number: what number is that? Is it zero? 7. Under these parameters, will there be positive growth in the long run? (Remember that A and B are constants). Why? 8. Imagine that we have two countries with the same parameters (same A, B,s,, and n). One of them is rich and the other is poor. Which one of the two will grow faster? Why? Does this model predict convergence? 9. Suppose now that sA > +n. Draw the savings and depreciation curves. Under these circumstances, will there be positive growth in the long run? Why? 10. If s = 0.3, A = 2, B = 2, = .3 and n = 0.03, the growth rate converges to some value as time goes to infinity. What is this value?

An insurance company offered drivers auto insurance. Assume that claims by safe drivers cost the insurer $1,000 over the term of the policy and claims by reckless drivers costs $5,000. Drivers know whether they are safe or reckless, but the insurer only know that 10% of drivers are reckless.

a. What is the expected cost of losses to the insurance company? b. How much does the insurance company have to charge for auto insurance to break-even? Why?

Suppose that there are three states of the world, a, b, and c. The probabilities of the three states are 1 = 0.25, 2 = 0.5, and 3 = 0.25. Let A, B, and C denote the Arrow-Debreu securities that pay $1 in states a, b, and c, respectively. That is, A = (1,0,0), B = (0,1,0) and C = (0,0,1). Let pA = 0.4, pB = 0.5 and pC = 0.2 denote the prices of A, B, and C.

Consider a security X which is worth $2 in state a, $3 in state b, and $1 in state c. If there are liquid markets for A, B, C and X, what is the price of X?

Suppose that there are two food stores in town. La Boulangerie sells bread and La Fromagerie sells cheese. It costs $1 to make a loaf of bread and $2 to make a pound of cheese. If La Boulangerie's price is PB dollars per loaf of bread and La Fromagerie's price is PF dollars per pound of cheese, their respective weekly sales, QB thousand loaves of bread and QF thousand pounds of cheese, are given by the following equations:

QB = 14 - PB - 0.5PF

QF = 19 - 0.5PB - PF

a.) What are profit functions for La Boulangerie, B, and La Fromagerie, F?

b.) What are the best response functions in terms of setting the prices for La Boulangerie and La Fromagerie?

c.) Graph the respective best-response functions. Plot La Boulangerie's price on the vertical axis and La Fromagerie's price on the horizontal axis.

d.) Calculate the Nash equilibrium

Rather than setting their prices independently, suppose now that the two stores collude to set prices jointly so as to maximize the sum of their profits. Over the next few questions, we are going to find the joint profit maximizing prices for the stores.

e.) Start by identifying the profit function that captures the sum of the profits for the two stores i.e. J = B + F.

f.) What price should the two stores set for La Boulangerie? In other words, what PB maximizes the joint profit function? This looks a little trickier because the joint profit function, if you've calculated it correctly, has two squared terms in it instead of one. Think about it this way, though. Recall that the generic quadratic equation looks like y = ax2 + bx + c. In this part of the question, the x we're interested in is PB. Any term in our joint profit function that doesn't have some kind of PB in it is just part of the constant, 'c'. The 'x' that maximizes 'y' is still -b/2a. With this in mind, what is the best response function for the price set by La Boulangerie, PB?

Suppose that there are two food stores in town. La Boulangerie sells bread and La Fromagerie sells cheese. It costs $1 to make a loaf of bread and $2 to make a pound of cheese. If La Boulangerie's price is PB dollars per loaf of bread and La Fromagerie's price is PF dollars per pound of cheese, their respective weekly sales, QB thousand loaves of bread and QF thousand pounds of cheese, are given by the following equations:

QB = 14 - PB - 0.5PF

QF = 19 - 0.5PB - PF

a.) What are profit functions for La Boulangerie, B, and La Fromagerie, F?

b.) What are the best response functions in terms of setting the prices for La Boulangerie and La Fromagerie?

c.) Graph the respective best-response functions. Plot La Boulangerie's price on the vertical axis and La Fromagerie's price on the horizontal axis.

d.) Calculate the Nash equilibrium

Rather than setting their prices independently, suppose now that the two stores collude to set prices jointly so as to maximize the sum of their profits. Over the next few questions, we are going to find the joint profit maximizing prices for the stores.

e.) Start by identifying the profit function that captures the sum of the profits for the two stores i.e. J = B + F.

f.) What price should the two stores set for La Boulangerie? In other words, what PB maximizes the joint profit function? This looks a little trickier because the joint profit function, if you've calculated it correctly, has two squared terms in it instead of one. Think about it this way, though. Recall that the generic quadratic equation looks like y = ax2 + bx + c. In this part of the question, the x we're interested in is PB. Any term in our joint profit function that doesn't have some kind of PB in it is just part of the constant, 'c'. The 'x' that maximizes 'y' is still -b/2a. With this in mind, what is the best response function for the price set by La Boulangerie, PB?