explain all

1. (55 points) Egbert consumes only axes (X) and mead (Y ). His utility function is U = XY , his income is I and the prices of axes and mead are pXand pY respectively. (a) What is Egbert's demand for for axes and mead? (Show all the steps.) (b) What is Egbert's indirect utility function? (c) What is Egbert's compensated demand for axes and mead? (d) What are the numerical values for (b),(c) and (d) if I = 24, pX= 1, pY= 2? (e) Now the price of mead rises to pY= 3. What income would Egbert need to be able to reach his old utility level at the new prices? (f) If he had this income, what quantities of each good would he consume at the new prices? (g) Given his actual income, how much of each good will Egbert consume at the new prices? (h) What is the income effect of this price change? (i) What is the substitution effect of this price change? (j) Show the Slutsky equation holds for this price change. (k) What loss in welfare (in $) does Egbert suffer due to the price change? Give three different possibilities.

2. (5 points) The effect on an isoquant of Hicks-neutral technological progress - redraw this diagram. Show (either on the same diagram or a second one) the effect of labor-saving technological progress. Make clear (on the diagram and with some text) what the difference is between the effects of the two types of technological progress.

1. A firm's analysts estimate that the firm can manufacture a product according to the production function :

Q = F (K ,L) = K3/4 L1/4

a. Calculate the average product of labour, APL, when the level of capital is fixed at 81 units and the firm uses 16 units of labour. How does the average product of labour change when the firm uses 256 units of labour?

b. Find an expression for the marginal product of labour, MPL, when the amount of capital is fixed at 81 units. Then, illustrate that the marginal product of labour depends on the amount of labour hired by calculating the marginal product of labour for 16 and 81 units of labour.

c. Suppose capital is fixed at 81 units. If the firm can sell its output at a price of $200 per unit of output and can hire labour at $50 per unit of labour, how many units of labour should the firm hire in order to maximize profits?

2. A firm's product sells for $4 per unit in a highly competitive market. The firm produces output using capital (which it rents at $25 per hour) and labour (which is paid a wage of $30 per hour under a contract for 20 hours of labour services). Complete the following table and use that information to answer these questions.

a. Identify the fixed and variable inputs. b. What are the firm's fixed costs? b. What is the variable cost of producing 475 units of output? c. How many units of the variable input should be used to maximize profits? d. What are the maximum profits this firm can earn? e. Over what range of the variable input usage do increasing marginal returns exist? f. Over what range of the variable input usage do decreasing marginal returns exist? g. Over what range of input usage do negative marginal returns exist?

Table: K L Q. MPk APk APL VMPK 0 20 0 1 20 50 2 20 150 3 20 300 4 20 400 5 20 450 6 20 475 7 20 475 8 20 450 9 20 400 10 20 300 11 20 150

3. Explain the difference between the law of diminishing marginal returns and the law of diminishing marginal rate of technical substitution.

4. An economist estimated that the cost function of a single-product firm is:

C (Q) = 90 + 35Q + 25Q2 + 10Q3

Based on this information, determine: a. The fixed cost of producing 10 units of output. b. The variable cost of producing 10 units of output. c. The total cost of producing 10 units of output. d. The average fixed cost of producing 10 units of output. e. The average variable cost of producing 10 units of output. f. The average total cost of producing 10 units of output. g. The marginal cost when Q=10.

6. A manager hires labour and rents capital equipment in a very competitive market. Currently, the wage rate is $9 per hour and capital is rented at $10 per hour. If the marginal product of labour is 50 units of output per hour and the marginal product of capital are 60 units of output per hour is the firm using the cost-minimizing combination of labour and capital? If not, should the firm increase or decrease the amount of capital used in its production process?

7. A firm produces output according to a production function: Q= F(K,L) = min{4K,4L}

a. How much output is produced when K=2 and L=3? b. If the wage rate is $60 per hour and the rental rate on capital is $40 per hour, what is the cost-minimizing input mix for producing 8 units of output? c. How does your answer to part b change if the wage rate decreases to $40 per hour but the rental rate on capital remains at $40 per hour?

There are 10 potential entrants into the market for tomatoes. All tomato growers are identical. They face a fixed cost of 500 if they choose to enter the market and then have a marginal cost of 10 per bushel of tomatoes produced. Those who enter the market produce a homogeneous tomato with demand P = 150 2Q, where Q is the number of bushels. The entering firms compete according to Cournot competition. Before choosing quantities firms sequentially announce whether they are entering and pay their fixed cost if entering.

1. Suppose N firms enter the market. Since this is a symmetric Cournot model, we've shown earlier in the class that in Nash Equilibrium each firm will produce, qN = (1/(N + 1))*((A c)/B) . So for this model, qN = 1/(N + 1) *(150 10)/2 = 70/(N + 1) . Given this, what is the price of tomatoes and the total quantity of tomatoes produced as a function of N? 2. What is the profit of the entering firms as a function of N? 3. What is consumer surplus as a function of N? 4. In a free entry equilibrium, how many firms will enter the market? 5. What is the profit of the first firm to announce it will enter the market? 6. Suppose a social planner were to choose the number of firms to enter the market to maximize total welfare, how many firms would enter?

. (70 points total) ) Homer Simpson, does not abide by the life cycle theory of consumption. Homer has a "let's live life like it's our last day" mentality and thus, he prefers to consume more today, relative to the future. In particular, Homer prefers to consume exactly twice as much today (c), relative to consumption next period (cf). Homer's current income = $300K and his future expected income = $200K. He has no wealth (neither current nor expected) since he lives like today is his last. Given that Homer faces a real interest rate of 10% ( 0.10). Please answer the following questions.

1.(5 points) Calculate Homer's optimal consumption bundle showing all work. Then draw a completely labeled graph (the two period consumption model) depicting this initial optimal consumption bundle as point C*A.

2.(5 points) Ben Bernanke and the Fed are not happy with the state of the economy (we are in the midst of the Great Recession!) and worry about an impending recession. As a result, the Fed lowers rates so that the new real rate is zero (0%) Recalculate the optimal bundle for Homer and add this point to your graph and label as point C*B.

3.Marge is worried about saving for the kids' college education and does some research investigating whether most consumers have spending patterns like Homer. She encounters the life cycle theory of consumption and finds that this popular economic theory suggests that the way to maximize life time utility is to perfectly smooth consumption through time. Marge discusses this new development with Homer and convinces him to change his preferences to that of a perfect smoother, like our friend Dagwood. Resolve for Homers optimal basket. Real rates are zero still.

4. Ben Bernanke and the Fed are still not happy with the state of the economy (we are in the midst of the Great Recession!) and worry about an impending recession. As a result, the Fed lowers rates again so that the new real rate is negative 5%. Recalculate the optimal bundle for Homer (given his new preferences) and add this point to your graph and label as point C*D .

5.Did the Fed policy work as in stimulating the economy as measured by the change in Homer's current consumption? Why or why not? Explain using the income and substitution effects. Do these income and substitution effects work in the same or opposite directions? Explain.

6.) In the space below, draw two savings functions for Homer. The first savings function is for the initial conditions, before Homer changes his preferences. Label as point A where r = .10, consistent with the initial conditions. Label as point B, where r = zero.

The second savings function is after Homer changes his preferences. Label as point C, when real rates are zero and as point D, when real rates are negative 5% (-.05).