4. Ariel's T-shirt company hires labor at a rate of $8 per hour and rents capital (sewing machines) at a cost of $128 per machine per day. Throughout this question, you can think of Ariel renting fractions of sewing machines if necessary. Her production function is given by q = 6L3K3 where q measures the number of T-shirts produced each day, L measures the total number of hours worked by all employees in a day and K measures the number of sewing machines. (a) Using the Lagrangian technique, find the cost-minimizing combination of labor and capital for the production of q T-shirts per day. (b) Write down an equation for Ariel's long-run expansion path. (c) Find Ariel's long-run total, average, and marginal cost as a function of the number of T-shirts she produces each day. (d) Sketch Arieľ's long-run total expansion path in a diagram. Sketch at least two isoquants with the associated isocost lines. Using your diagram, determine how Ariel's long-run expansion path will shift if the cost of renting a sewing machine for a day decreases. Explain why this change occurs.

Suppose that Tim runs his own lawn care service and that his firm is competitive in the lawn care industry. He has a fixed number of tools that can be used to maintain a yard, but he needs to determine the productivity gained from hiring various numbers of workers per week. Which of the following statements best represents the idea behind the marginal product of labor ( MPL )? When Tim hires more workers, he can mow more lawns per week, but at a decreasing rate. If Tim hires twice as many workers per week, he can mow twice as many lawns per week. If Tim buys one more lawn mower, he can mow 13 more lawns per week. If Tim hires an additional worker per week, he can mow 11 more lawns per week. The following graph portrays the weekly total product curve for lawn mowing given a fixed amount of capital and varying labor inputs. (Note: Each black point (plus symbol) represents a point on the total product curve.) 100 90 во 70 60 50 40 30 20 10 xxxx_ Total Product ° 0 1 2 4 5 7 8 10 LABOR…

Suppose that John is a shrubber. As a master shrubber, he hires apprentices, L, at wage w and purchases capital, K, (electric shrubbery trimmers, for example) to create his high-end shrubberies. Capital is expensive, so he generally borrows money to purchase the equipment. The cost of capital is r. John can produce shrubberies according to the production function q = 10 · K^1/4 L^1/4. We can assume John is a price taker. If wages are $10/hour, the cost of capital is $40/hour, and shrubberies sell for p = $160, how many shrubberies will John produce and sell? How much labor and capital will he purchase to produce that output? What are his profits?

3. Elsa's handicrafts has the production function f(x1,x2) = (min{x1 ,2x2})2, where x, is the amount of plastic used, x, is the amount of labor used, and f(x1,x2) is the number of lawn ornaments produced. Let w, be the price per unit of plastic and w, be the wage per unit of labor. а. What is Elsa's cost function c(w1,w2,y)? b. If wi = w2 = 1, then what is Elsa's marginal cost of producing y units of output? What is the number of units of output that she would supply at price p? At these factor prices, what is her average cost per unit of output? c. If the competitive price of the lawn ornaments she sells is p = 48, and w, = w2 = 1, how many will she produce? How much profit will she make? d. More generally, at factor prices w, and w2, what is her marginal cost? At these factor prices and an output price of p, what is the number of units she will choose to supply?

In Example 6.4, wheat is produced according to the production function: q=100 (K0.7L0.3). Beginning with a capital input of 4 and a labor input of 49, show that the marginal product of labor and the marginal product of capital are both decreasing. (Round your responses to two decimal places.) The MP at 5 units of capital is The MP at 6 units of capital is The MP at 50 units of labor is. The MP at 51 units of labor is Does this production function exhibit increasing, decreasing, or constant returns to scale? OA. Decreasing returns to scale because the inputs exhibit diminishing marginal returns. OB. Constant returns to scale because a proportionate increase in all inputs results in the same proportionate increase in output. OC. Increasing returns to scale because the inputs exhibit diminishing marginal returns.

1. Assume that a firm has the following production function: q = L0.5K0.6 d. Determine, both, the marginal product of labour and capital functions (ignore c.). e. Calculate, both, the marginal product of labour and capital when L = 64, K = 32 and interpret your answers. f. Calculate the marginal rate of technical substitution when L = 64, K = 32 and interpret your answer.

Using the information from the isoquant/isocost diagram to the right and assuming that PL = P = $2, complete the table below. (Enter your responses as integers.) Output Units 100 200 300 Total Cost of Output $100 $ $300 Units of Labor Demanded 25 50 Units of Capital Demanded 4 50 75 Units of capital (K) 200 175- 150- 125- 8100- 75- 50- 25+ 0+ 0 25 50 9=100 9 200 100 125 150 q=300 75 Units of labor (L) 175 200 Q Q

4. A bakeshop has the following production function for cupcakes: 1) 2) 3) Q = 4K 0.5 L0.5, (MPK = 2L0.5/K0.5, MP₁ = 2K 0.5/L0.5), where K is the quantity of capital, L is the quantity of worker hired. The cost of capital is r = $1000, and the cost of labor is w = $40 per worker. Does the production function exhibit diminishing marginal return of labor? Explain. Find the cost function of the bakeshop, C(Q). (Hint: Use the conditions MRTS =MPL/MPK = w/r and Q = 4K 0.5 L0.5, solve the optimal input choices K(Q) and L(Q). Then the cost function is given by C(Q) = r K(Q) + w L(Q)). Does the bakeshop experience economies of scale, diseconomies of scale, or neither? Explain with the calculation of scale economies index.

Suppose that the price of labor (the wage rate) is $100 a day and that the rental price of capital is $50 a day. Currently the Jedie Weapons Corporation is using ten units of capital and five units of labor to produce 500 lightsabers a day. Each lightsaber sells for $2. At the current input use levels, the marginal product of capital is 40 lightsabers per day and the marginal product of labor is 60 lightsabers a day. Is the firm making an economic profit at its current levels of operation?     2. Could the firm increase its profits at the current level of output by adjusting its use of labor and capital? If so, how should input use be adjusted?

(2.2) Emily's coffee shop has a production function with L-shaped isoquants Capital (units) 12 8 4 3 6 q3=30 Labor (units) What's the optimal combination of capital and labor to efficiently produce q=30 units of coffee? 9 q2=25 q1=20