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In Chapter 6 of the text and in lecture we are introduced to the concept of technology and production. Technology is the way things (goods and services) and put together and this technology embodies all of the wisdom, tools, and know-how that goes into the making of things. For example, the technology of today’s dental care (a service) includes high speed drills, pain killers, and years of study in graduate school.
The production function tells us that for a given technology, what combinations of factors of production yield us output. We say that:
X = F ( K , L ) which means that there is a functional relationship between capital, labor, and the output of “X”.
Ultimately, we will need the production function and its underlying technology to help us define the cost curves for the firm. After all, cost will be the use of factors by the firm to produce a quantity of “X” multiplied by the price paid for these factors.
From our derivation of cost structures that come form the Production Function, we can derive the supply curve for the firm – the price each firm will need to cover costs at different output levels.
The production function tells us the amounts of factors we need to produce any given quantity of a good or service. It also tells us the cheapest and most efficient way to do this. (That is, the production function tells us how X is being produced with factors of production and the best available technology.)
We use isoquants to depict the production function in a two dimensional world. Between the use of capital (K) and labor (L), the isoquant tells us at what rate the two factors must be substituted against each other to keep production at certain level. Just like the indifference curves that describe an individual’s utility function, isoquants can be seen as contour lines on a map whose position and height in the two-dimensions is determined by the underlying technology of production.
We will eventually see that in the SHORT RUN, at least one factor is fixed for the firm and its quantity used in production cannot be changed. In the LONGRUN, all factors can be changed.
Isoquants can tell us many things: they can tell the degree to which factors of production are complements, substitutes, and how for a given level of production, they can be traded at the rate of MRTS – marginal rate of technical substitution (also referred to in the textbook as the RTS). The MRTS, or slope of the isoquant is negative, it falls as you substitute one factor for another, and it is the ratio between the marginal physical
products of the individual factors.
That is: MRTS between capital (vertical axis) and labor (horizontal axis) = MPP(labor)/MPP(capital). This also can be written as MPL/MPK.
We can also show with isoquants whether or not a production function exhibits constant, increasing, or decreasing returns to scale.
So, let’s get to the homework:
(1) Is it possible for a firm to have two isoquants that cross over each other? Could a firm’s isoquant ever slope upward? Why or why not?
(2) A firm has a production function of the form Q = K (1/2)*L (1/2) where the (1/2s) are the exponents. What is the output if the firm uses 9 units of labor and 4 units of capital? If the firm doubles its input levels, what happens to the level of output? Does this firm’s production function exhibit constant, increasing, or decreasing returns to scale?
(3) Let’s assume that your GOAL is to maximize your “output” that you define as getting an “A” course grade in ECN 100 at UCD. How would you go about constructing a production function (the inputs) that would help you achieve this result? How would each of the factors relate to the desired output? How would you go about assessing the marginal productivity of each of the factors of “production” in this case?
Okay – now chose one of your factors of production that is an input to the getting of the a grade as THE MOST IMPORTANT FACTOR, and state how the available level of technology would affect its role as an input to the desire output, X = grade.
(4) Do you realize that you have your OWN PERSONAL production function?
The value or ability of whatever you produce is a function of the various inputs that you put to productive use working with your body and mind. Now, think of an important output goal you would like to achieve (like getting a Law Degree or becoming a bicycle racer, or whatever) and devise an “input” strategy to reach this goal in an effective manner. Once you decide on these inputs, how do you think the current level of Technology helps determine how these inputs will work in your efforts to produce an output result? This is an important problem and we will also discuss it in lecture.
(5) What might be the three most important inputs to the following production activities?
The manufacture of Honda Civics
The growing of organic lettuce
Nordstrom Department Store
Your ability to do math problems
Running for political office
A Jamaba Juice® franchise.
Now choose two inputs to the production of a Jamba Juice smoothie, and draw an isoquant map. Show how a technological change would alter the map so that the Jamba Juice Corp. uses more of one of your chosen inputs relative to the other input. What would you expect to happen to the MRTS (MPL/MPK) after this change? Show how technological change would alter the isoquant map so that more smoothies could be
produced with less of all inputs.
(6) What does it mean when we say that the marginal rate of technical substitution (RTS) has a “negative” sign? If the RTS = MPL/MPK, and the ratio of MPL/MPK is rising along along an isoquant, are we moving up or down the isoquant? Given that we are either moving up or down the isoquant, what is the significance of the change in the ratio of the marginal products of the two inputs, labor and capital?
(7) A firm faces an input technology that does NOT allow substitution between the two inputs, labor and capital. Draw a possible isoquant map for this firm. Okay, now technology changes in a way that still does NOT allow substitution between the units at any level of production, but uses more of one of the inputs relative to the other. How does that change the isoquant map? Now, suddenly a new technology arises whereby all of the output is produced with capital without labor (say, completely robotic production).
How does that change in isoquant map?
(8) Let’s say that you want to go into Frozen Yoghurt Shop that offers a huge assortment of yoghurt flavors, serving sizes, fat contents, and toppings. The output of the shop is the variety of yoghurt products and the pleasant environment of eating the ice cream. Okay, name FIVE important inputs to the production function of this ice cream shop. Which of these inputs would you classify generally as capital input and which as a labor input?
Here you see that a production function in the real world can have many, many inputs. However, when we simplify our production function into a two-dimensional space and two major input-types, labor and capital, we do not stray far from the reality of business. Our theory is sound!
(9) This question might take some thought - How do you think that the Microsoft Word® word processing software has improved the productivity of individuals who have to write documents. How would you depict this productivity and its change with an isoquant map with two main inputs (you choose the inputs). What does your map say about the relative marginal products of your inputs and how might these marginal products change with the invention and use of the Word program?
(10) A technology is invented that makes labor and capital perfect substitutes in the production of candy. Draw an isoquant map for candy. Now, from your original drawing of the isoquant map, assume that the marginal product of labor suddenly falls relative to capital, but the inputs in candy production remain as perfect substitutes. How does that change your isoquant map?
(11) Here is a “dynamic technology” question – When it comes to automobile production it is an accepted fact that American automakers did not readily adopt and use Asian and European technologies in the production of cars. The result was a loss of global markets for the American producers. So the USA government imposed tariffs and other restrictions on the import of foreign cars. In response the European and Asians constructed production plants in the USA in low-cost, non-union states like Tennessee
and Kentucky. As a result the Americans (especially in the Detroit factories) then begin to integrate the Asian and European technologies into the production to compete with the new arrivals. Part of the new technology was to substitute capital (adding robots) for labor in the production of cars.
Using isoquant maps, show the typical American car maker production function prior to the new arrivals. How might the maps change after the new arrivals arrived? What might be different between the two isoquant maps with respect to the height of the maps and the expected RTSs on the maps?
