the di?erent kinds of returns to scale?
18.3. If a + b = 1, we have constant returns to scale, a + b<1 gives decreasing returns to scale, and a + b>1 gives increasing returns to scale.
4. The technical rate of substitution between factors x2 and x1 is ?4. If you desire to produce the same amount of output but cut your use of x1 by 3 units, how many more units of x2 will you need? 18.4. 4×3 = 12 units.
5. True or false? If the law of diminishing marginal product did not hold, the world’s food supply could be grown in a ?owerpot. 18.5. True.
6. In a production process is it possible to have decreasing marginal product in an input and yet increasing returns to scale? 18.6. Yes.
19 Pro?t Maximization
1. In the short run, if the price of the ?xed factor is increased, what will happen to pro?ts? 19.1. Pro?ts will decrease.
2. If a ?rm had everywhere increasing returns to scale, what would
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happen to its pro?ts if prices remained ?xed and if it doubled its scale of operation?
19.2. Pro?t would increase, since output would go up more than the cost of the inputs.
3. If a ?rm had decreasing returns to scale at all levels of output and it divided up into two equal-size smaller ?rms, what would happen to its overall pro?ts?
19.3. If the ?rm really had decreasing returns to scale, dividing the scale of all inputs by 2 would produce more than half as much output. Thus the subdivided ?rm would make more pro?ts than the big ?rm. This is one argument why having everywhere decreasing returns to scale is implausible.
4. A gardener exclaims: “For only $1 in seeds I’ve grown over $20 in pro- duce!” Besides the fact that most of the produce is in the form of zucchini, what other observations would a cynical economist make about this situation?
19.4. The gardener has ignored opportunity costs. In order to accurately account for the true costs, the gardener must include the cost of her own time used in the production of the crop, even if no explicit wage was paid.
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5. Is maximizing a ?rm’s pro?ts always identical to maximizing the ?rm’s stock market value?
19.5. Not in general. For example, consider the case of uncertainty.
6. If pMP1 >w 1, then should the ?rm increase or decrease the amount of factor 1 in order to increase pro?ts? 19.6. Increase.
7. Suppose a ?rm is maximizing pro?ts in the short run with variable factor x1 and ?xed factor x2. If the price of x2 goes down, what happens to the ?rm’s use of x1? What happens to the ?rm’s level of pro?ts?
19.7. The use of x1 does not change, and pro?ts will increase.
8. A pro?t-maximizing competitive ?rm that is making positive pro?ts in long-run equilibrium (may/may not) have a technology with constant returns to scale. 19.8. May not.
20 Cost Minimization
1. Prove that a pro?t-maximizing ?rm will always minimize costs. 20.1. Since pro?t is equal to total revenue minus total costs, if a ?rm is
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not minimizing costs then there exists a way for the ?rm to increase pro?ts; however, this contradicts the fact that the ?rm is a pro?t maximizer.
2. If a ?rm is producing where MP1/w1 > MP 2/w2, what can it do to reduce costs but maintain the same output?
20.2. Increase the use of factor 1 and decrease the use of factor 2.
3. Suppose that a cost-minimizing ?rm uses two inputs that are perfect substitutes. If the two inputs are priced the same, what do the conditional factor demands look like for the inputs?
20.3. Since the inputs are identically priced perfect substitutes, the ?rm will be indi?erent between which of the inputs it uses. Thus the ?rm will use any amounts of the two inputs such that x1 + x2 = y.
4. The price of paper used by a cost-minimizing ?rm increases. The ?rm responds to this price change by changing its demand for certain inputs, but it keeps its output constant. What happens to the ?rm’s use of paper?
20.4. The demand for paper either goes down or stays constant.
5. If a ?rm uses n inputs (n>2), what inequality does the theory of
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revealed cost minimization imply about changes in factor prices (Δwi) and the changes in factor demands (Δxi) for a given level of output? 20.5. It implies that
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