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Microeconomic Theory, Problem Set 3

1. (1.5 points) For the following production function q = 16KL0.5 and wage, w = $10 and rental rate of capital r = $5 ﬁnd the cost minimizing combination of labor and capital used to produce 100 units of output. Also ﬁnd the total cost of production, average cost of production, and the marginal rate of technical substitution at this cost minimizing combination of labor and capital. (Hint: The numbers might not work very nicely so keep everything as fractions).

2. (1.5 points) Read this whole question before you start to sketch. You may want to draw this a couple times to make it neat. Draw the long-run total cost curve and a short-run total cost curve. Make them typically shaped (how they were shaped in class). At some quantity of output (your choice) where the short-run total cost does not equal the long-run total cost, call it q1 , draw a pair of dashed lines whose slopes represent the average long-run total cost and average short-run total costs respectively at that quantity, q1 . For another quantity of output (further away from q1 is better for this illustration) where the short-run total cost does not equal the long-run total cost (your choice), call it q2 , draw a pair of dotted lines whose slopes represent the short-run and long-run marginal costs respectively at q2 . Finally, iden- tify the output quantity where the average total cost is minimized and explain how you know.

3. (0.5 point) If the long-run marginal cost intersects the short-run marginal cost below both the long-run or short-run average cost, which of the following must be true?

a. Economies of scale b. Diseconomies of scale

c. Neither a. nor b. d. It could be either a. or b.

e. It cannot be determined with the information provided

4. (1 point) Match based on the description

1. Short-run average cost a. Always less than or equal to SR avg cost

2. Short-run total cost b. U-shaped, depends on ﬁrm size

3. Short-run marginal cost c. Always decreasing as q increases

4. Short-run variable cost d. It’s derivative is long-run marginal cost

5. Short-run ﬁxed cost e. When plotted, the ﬁxed cost is its vertical intercept

6. Long-run average cost f. Intersects SR marginal cost at the shared point on the LR and SR total cost curves

7. Long-run total cost g. Intersects the SR avg cost at SRAC’s minimum

8. Long-run marginal cost h. Is always FC less than Short Run total cost.

5. (1.5 points) Draw the long-run expansion path of a typical production function. Draw three pairs of isoquants and isocost lines. On the same graph, draw a second set of isocost lines where the wage has increased considerably. Draw these isocost lines in such a way that you are able to create a new expansion path using the same isoquants. Comment on how and why the expansion path changed.

6. (1 point) Suppose a ﬁrm’s short-run cost function is C = / 、q3 –5q2 + 15q + 100. What is the equation for marginal cost? What is the equation for average total cost? Use those answers to ﬁnd what quantity minimizes average total cost (you will need to use a calculator or a computer to ﬁnd the answer). Round your answer to the nearest tenth of a unit.

7. (1.5 points) The following are the cost functions of ﬁrms that produce 1. leather alone, 2. Hamburger alone, and 3. Leather and hamburger together. C stands for total costs.

1. CL = 5qL(0).75 + 5

2. CH = 3qH + 10

3. CLH = 5qL(0).75 + 3qH + 12

Do either hamburgers or leather producers have economies of scale? Are there economies of scope associated with producing hamburgers and leather together? Show or explain how you know.

8. (1.5 points) What is the relationship between the marginal product of labor curve and the short-run marginal cost curve? Show using graphs as well as providing a written explanation.