1) Marry is at the supermarket buying her bi-weekly groceries. As she arrived at the store right before closing, not much is left on the shelves; so her purchases are limited to beef, B, and an assortment of organic vegetables, V. Beef costs $50/kg and vegetables $10/kg. Her utility function is expressed as ????(????, ????) = ????ˆ0.6 ????ˆ03. She has $150 to spend. Please answer the following questions using this information:
- Given the vegetables cost less than beef, explain why Marry would not only buy vegetables.
- Write Marry’s constrained maximization problem (objective function subject to her budget constraint).
- Using the Lagrangian technique, find how much beef and vegetables would Marry buy?
- Show that for the preferences represented as above, demand for beef is a function of only its own price, ????B, and income, ????, but not the price of vegetables, ????V. Note: Rather than using the given prices and income, use ????B, ????V and ????. Find and comment on the comparative statics, i.e., how do changes in income and prices affect the demand for beef?
- What is the degree of homogeneity of the beef demand function you found in part (4)?
- Suppose there is a promotion for beef and the price falls to $40/kg. Calculate the compensated income, ????’, for Marry to consume the same amount of beef and vegetables under the new prices. Under the promotional price, how much beef does Marry buy? Decompose the change in her demand for beef into substitution and income effects using Slutsky decomposition. Comment on signs of these effects, i.e., is beef a normal or an inferior good for Marry?
2) Suppose the total cost of a representative perfectly competitive apple producer is given as ???????? = 12 + 6???? + ????(. All apple producers in the market are assumed to be identical. Suppose further that the demand for apples is estimated as ????) = 18,000 − 500???? and market supply is ????’ = 2,000 + 500????.
- Find the equilibrium market price and total supply of apples in the market.
- What is the profit maximizing quantity of apples each company would produce? Find the total revenue, total cost and profits associated with the profit maximizing quantity.
- Comment on whether this is an equilibrium in the short-run or in the long-run. Which assumption of perfectly competitive markets do you base your response on?
- What is the short-run supply function of this apple producer?
- What is the number of companies in the market in the short run?
- Using the assumptions of the perfectly competitive model, comment on what will happen in the market in the long run. What will be the new equilibrium price? What will be the number of companies? Assume input prices will remain the same, no matter what, regardless of the number of apple producers in the market.
3) A college professor is planning for his retirement years. His utility function is ????(???? , ???? ) = 3????tˆ0.5, + 2crˆ0.5, where ????t represents his consumption today (period 1), his active years of teaching, and ????r represents his consumption in his retirement years (period 2). During his active years of teaching, he makes a total of $3 million, while in his retirement years his total income is $1 million. He can borrow or lend at an interest rate of 25% between the two periods.
- Write an equation that describes the professor’s budget assuming he will spend all his income during his lifetime.
- If the professor chooses neither to borrow nor to lend during his active years, what will be his marginal rate of substitution between his consumption today and his retirement years?
- If the professor aims at maximizing his utility, how much does he consume in each period (use the Lagrangian method)? Does he save for his retirement years? If so, how much?
- At what interest rate would the professor choose to consume the same amount in his working years as in his retirement years? What would be his consumption in each period?
4) The production function of a firm is as follows:
1 ????(????,????) = 500????????ˆ0.5 ,
where K is capital measured in machine-hours, L is labor in person-hours, and y is the yearly output. The hourly wage rate, w, is $10, and the hourly rental rate of capital, r, is $20.
- Show what type of returns to scale the technology of this firm displays (constant, increasing, or decreasing).
- Calculate the marginal products of labor and capital.
- Derive the firm’s short-run cost function (as a function of output given the factor prices) if the capital is fixed at 10,000 machine hours. Start with setting the short-run cost minimization problem of the firm subject to the technology constraint.
- Derive the firm’s short-run marginal cost function and the short-run average cost function. At what level of output do these curves intersect? Show that at this point, the average cost function is at its minimum.
5)For the demand function
????d(????,????) = 1???? / 2????
- Calculate the price elasticity. Provide an economic interpretation (elastic or inelastic).
- Calculate the income elasticity. Provide an economic interpretation.
6)The long-run cost function of a firm producing widgets in a competitive market is given by
????ˆ2+10 for ????>0
0 for ???? = 0
where y is the quantity of widgets.
- Find the lowest price at which this firm will supply a positive number of widgets
in the long run. What is the output of the firm at this price?
- For the price you found in part (a), if there are 1,000 identical firms operating in this market, what would be the equilibrium market demand for widgets?