Economics 11
Caltech Spring 2010
Problem Set 4
Homework Policy: Study You can study the homework on your own or with a group of fellow students. You should feel free to consult notes, text books and so forth.
The quiz will be available Wednesday at 5pm. Following the Honor code, you should find 20 minutes and do the quiz, by yourself and without using any notes. Paper and pen should be all you need. Then turn it in by Thursday 5pm. (drop off in box in front of Baxter 133). It will include one question from each section
The answers to the whole homework will be available Friday at 2pm.
Definitions Please explain each term in three lines or less!
• Shadow value The shadow value of capital refers to the value associated with a constraint, i.e. when capital cannot be adjusted in the short‐run it creates a constraint on the profit. • Marginal product The additional output that can be produced by one more unit of a particular input while holding all other inputs constant. It is usually assumed that an input’s marginal productivity diminishes as additional units of the input are put into use while holding other inputs fixed. If , , • Marginal cost The cost of producing an additional unit of output. • Short‐run marginal cost The cost of producing an additional unit of output without changing anything you cannot change quickly (such as building new factories).
In the short run, some inputs (e.g., capital) are fixed, i.e., cannot be adjusted to change output levels. In such case (e.g., assuming that capital is not adjustable in the short run), the marginal cost given the amount of capital used is the derivative of the total costs with respect to the quantity of output. In other words, | , , • Short‐run average cost of production The short‐run average cost is the short run total cost of production divided by quantity, where one factor of production is fixed.
It is just the short‐run total cost of productions (i.e., holding K fixed) divided by the output, i.e., | the average of the total cost in the short run per unit of output:
• Revealed preference The preference of consumers can be revealed by their purchasing habits. • Fixed costs Costs that do not change as the level of output changes in the short run. Fixed costs are in many respects irrelevant to the theory of short run price determination. • Total Factor Productivity A variable which accounts for effects in total output not caused by inputs. For example, a year with unusually good weather will tend to have higher output, because bad weather hinders agricultural product. A variable like weather does not directly relate to unit inputs, so weather is considered a total‐factor productivity variable (Wiki).
Word problems Please explain each question in a few sentences.
• Consider fixed capital, what are the implications of the first order condition of the profit maximization problem? What does the second order condition imply? Given a fixed level of capital K, the firm chooses L to maximize profit. From the firm’s (short‐run) profits , , we obtain the first order condition:
, , 0 ,
where the left‐hand side of the last equality if the value of the marginal product of labor, and the right hand‐side is the wage rate. This implies that the firm will add workers to the production process up to the point where a worker just pays his salary, in the sense that the value of the marginal product for that worker equals the cost of hiring him. The second‐order condition, in turn, can be written as:
, . This expression has to be negative (or non‐positive) since, at a maximum, the slope goes from positive (when the function is increasing up to the maximum) to zero (a the maximum) to a negative number (as the variable rises past the maximum).
• Why do ships and taxis go at slower speeds as a short run adjustment to recessions? Given that capital equipment – and thus, the cost of capital – is fixed in the short run, taxis and ships adjust to recessions by lowering variable costs. By going at slower speeds, the cost of gas decrease, thus lowering variable costs.
• Sketch the average total cost (ATC), average variable cost (AVC) and marginal cost (MC) of a firm (dollars vs. quantity). Why does the MC curve necessarily intersect at the minimum of the AVC and ATC curves?
Maximize: 0 Multiply through by : So at the minimum of , , follow similar steps to get intersects at the minimum of .
• Why does a profit‐maximizing firm produce the quantity , where price equals marginal cost, given price is as large as minimum average variable cost? Why does a profit‐maximizing firm shut down if the price falls below the minimum average variable cost of production?
The profit maximizing firm maximizes profit (shaded blue) when it produces quantity where price equals marginal cost. The profit‐maximizing firm shuts down if price falls below the minimum AVC because the firm will start losing money.
The firm’s (short‐run) profits are given by: | , and thus it maximizes profits by
choosing the quantity satisfying
a) 0 | .
This is a good strategy only if producing a positive quantity is desirable, i.e.
if | 0, , that is, if
| , b)
| , where is the average cost ignoring the investment in capital equipment (recall that the cost of capital is fixed in the short run, and thus 0, 0). Hence, the profit‐maximizing
firm produces the quantity where price equals marginal cost, provided that p is as large as minimum average variable cost. If price falls below minimum average variable cost, the firm shuts down, since it is more convenient for the firm to face the fixed costs of producing 0 units than to produce a positive quantity (i.e., it loses less by producing 0).
• What does 0 and 0, where , is the production function and and are capital and labor, imply about labor and capital? Provide an example of each case and draw the respective isoquants. 0 implies that and are substitutes—increase in makes an increase in boost production less.
0 implies that and are complements—increase in makes an increase in boost production more.
• Suppose we have a profit‐maximizing firm and is the fixed optimum price required. What does a decline in output price imply about cost of production? What does an increase in wages imply about labor productivity?
At the optimum, the profit‐maximizing firm produces the quantity where the price equals the marginal cost. Ceteris paribus, a decline in price thus will require a corresponding decrease in the marginal cost of productions – and hence a decline in the cost of production. Ceteris paribus, an increase in wages has to be matched by an increase in labor productivity, since profit , maximization requires that .
Technical problems • Suppose a company has total cost given by where capital is fixed in the short‐run. o What is the short‐run average total cost and marginal cost? Plot these curves. 2
o For a given quantity , what level of capital minimizes total cost? What is the minimum average total cost of ? Minimize total cost wrt : 0 2 2 √2
Minimize average total cost wrt : 0 2 2 √2
o Suppose capital can be adjusted in the long‐run. Does this company have an increasing return to scale, decreasing returns to scale or constant returns to scale? So we optimize the total cost with respect to and plug in: √2 2√2 1 √2 2√2 So the average total cost is constant independent of so constant return to scale.
• Professor Smith and Professor Jones are going to write a textbook together. Their production function for the book is / / , where is the number of pages in the finished book, is the number of working hours spent by Smith, and is the number of hours spent working by Jones. Smith values his labor as $3 per working hour. He has spent 900 hours preparing for the first draft. Jones, whose labor is valued at $12 per working hour, will revise Smith’s draft to complete the book. o How many hours will Jones have to spend to produce a finished book of 150 pages? Of 300 pages? Of 450 pages?
3 12 Plug in 12 3 Minimize total cost with respect to : 12 3 0 2 2 But we also need to stipulate that 900. If 150, then Smith has already worked more than the equilibrium amount so we just need the smallest TC even though it won’t be a minimum. For 900, 12 3 3 12/36 0 So 900 150 25 900 Similarly, if 300, and 900 12 3 3 12/9 0 So 900 300 100 900
And if 450 and 900, then 12 12 3 3 0 4 So 900 450 225 900
o What is the marginal cost of the 150th pages of the finished book? Of the 300th page? Of the 450th page? In the previous part we showed that Smith doesn’t put in any more hours for 450 so 30 900 12 12 2700 2700 900 24 900 If 150, 4. If 300, 8. If 450, then 12. • A lawn mowing company uses two sizes of mowers to cut lawns. The smaller mowers have a 24‐ inch blade and are used on lawns with many trees and obstacles. The larger mowers are exactly twice as big as the smaller mowers and are used on open lawns where maneuverability is not so difficult. The two production functions available to the company are1:
Output Capital input Labor input (square feet) (# of 24’’ mowers) Larger mowers 8000 2 1 Small mowers 5000 1 1 o Graph the 40000 square feet isoquant for the first production function. How much and would be used if these factors were combined without waste?
The isoquant curves for large mowers look like: 8000 min , Set 40000 8000 min , , we get 10, 5 o Answer the previous part for the second function.
1 Nicholson’s Microeconomic Theory
The isoquant curves for large mowers look like: 5000 min , Set 40000 5000 min , , we get 8, 8 o How much and would be used without waste if half of the 40000‐square‐foot lawn were cut by the method of the first production function and half by the method of the second? How much and would be used if 3/4 of the lawn were cut by the first method and 1/4 were cut by the second? What does it mean to speak of fractions of and ? Assuming we can only use integer numbers of lawn mowers: 1) Half by first method, half by second: We would cut 24000 with the first mower and 20000 with the second mower. First mower: 3 min , 6, 3; second mower: 4 min , 4, 4. There would be a waste of 4000 square feet mowed. 2) by first method, by second: We would cut 32000 with the first mower and 10000 with the second mower. First mower: 4 min , 8, 4; second mower: 2 min , 2, 2. There would be a waste of 2000 square feet mowed. o On the basis of your observations in the previous part, draw a 40000 isoquant for the combined production functions.
Assuming and are indivisible, to draw the 40000 isoquant we would need to enumerate integer combinations of large and small mowers that cover at least 40000 square feet.
If and could be divisible we can graph the isoquant as follows:
We can get an expression for the combined and : Let be the fraction of 40000 cut by mower 1. 10 8 1 8 2 5 8 1 8 3 We can get an equation of in terms of and by substituting from the second equation into the first equation: 8 2 8 2 .
o Suppose a gardener is going to go into business. She has access to both technologies what is the optimal number of mower of each type that she should purchase in order to minimize costs? When she hires workers or rents mowers she had to do it in whole hours the wage is $10 per hour; the rental cost of a unit of capital is $5 per hour. If we assume only integer and the cost function can be plotted by making a table of combinations of small and large mowers that will minimize cost for different intervals of area mowed:
(square feet) Small mower Large mower Cost(q) 1 0 15 0 1 20 2 0 30 1 1 35 … … … …
In fact past 13K the best solution is the interger value of K/8k large mower and for the residual a small mower if it is less than 5K and a large on if it larger than 5K.
However, if we assume and are divisible we can derive the following closed form of the cost function: Cost function: 10 5 Following steps similar to the previous part: 4 20000 4 20000 20000 4 8 3 8 3 3 40000 40000 2000 2 Given is a fixed constant , we want to minimize the cost, so substituting the above equation into the cost function: 3 10 5 10 15 2000 2 200 15 Increasing capital lowers cost, so the gardener should only use large mowers.