Economics 208 - write an equation for sams budget

Question

Answer the following questions. Be sure to explain your answers and show your work.

1.  Explain why it would be unreasonable for an individual to have two indifference curves that cross each other.

2.  Suppose that Sam's preferences are represented by the utility function U(x, y) = x²y². The prices of x and y are p_x and p_y respectively and Sam has income m.

(a) Write an equation for Sam's budget constraint.

(b) Write an equation for Sam's marginal rate of substitution between the two goods.

(c) Solve for Sam's demand functions for the two goods as functions of the prices and income.

(d) Determine what fraction of income Sam spends on good x.

3.  Consider the following market demand function for good x:x = 5m - 4p_x + 7p_y.

(a) Does the law of demand hold for this demand function?

(b) Are goods x and y substitutes or complements?

(c) Is good x a normal or an inferior good?

(d) Suppose that initially p_x=2, p_y=1, and m=8. What is the price elasticity of demand? What does the elasticity mean in words?

4. John has a utility function given by U(C, l) = C^1/2l^3/4, where C is consumption and l is leisure.  Suppose John receives a wage of w per unit of labour he supplies and the maximum number of hours he may spend working and in leisure activities is L^-.  John also receives a non-labour income of M.

(a) Graph John's budget constraint in consumption-leisure space.

(b) Derive John's labour supply function and his consumption demand function.

(c) Are leisure and consumption normal goods?

(d) Does John's labour supply curve slope upwards or downwards? What does this imply about the strength of the substitution effect relative to the income effect?

(e) How does a change in non-labour income affect John's consumption and labour supply?  Explain why a change in income affects consumption and labour supply the way that it does.

5. Sam lives for only two periods.  He earns M₁ in the first period and M₂ in the second period.  His utility function is U(C₁, C₂). Suppose that Sam can borrow and lend at the interest rate r.

(a) Determine the equations for consumption in both periods and savings that maximize Sam's utility.

(b) How does savings respond to a decrease in the interest rate? Illustrate this case when Sam is a saver initially. Is Sam better off after this change? Is it possible for Sam to become a borrower after this change?

6. An exchange economy contains just two consumers, Astrid and Birger, and two commodities, salmon and cheese.  Astrid's initial endowment is 4 units of salmon and 1 unit of cheese.  Birger's initial endowment has no salmon and 7 units of cheese.  Astrid and Birger have identical utility functions given by U(S, C) = S^1/2C^1/2.

(a) Draw an Edgeworth box diagram to illustrate the initial situation.

(b) Explain why the initial endowment is not Pareto efficient.

(c) Write an equation that characterizes the Pareto efficient allocations of cheese and salmon.

(d) Suppose we now introduce a competitive market for salmon and cheese and Astrid and Birger can buy and sell cheese at the prices (p_s, p_c) = (4,2). Illustrate and explain the disequilibrium in the market for salmon and for cheese.What change will occur to generate an equilibrium?

(e) Illustrate the competitive equilibrium in the Edgeworth box diagram.  Is it Pareto efficient?

(f) Suppose inflation occurs and the prices of cheese and salmon both double.  What happens to the competitive equilibrium?