Questions: Production with Two Variable Inputs: Isoquants

Question 1

A firm moves along an isoquant, substituting more labor for capital while keeping output constant. What happens to the marginal rate of technical substitution (MRTS) as the firm uses progressively more labor?

• A. MRTS rises, because more labor makes each unit of labor more productive
• B. MRTS stays constant, because total output is unchanged
• C. MRTS falls, because labor becomes less productive at the margin and capital more productive as their proportions change
• D. MRTS rises and then falls, following a U-shaped pattern

Answer: (C)

Diminishing marginal product is the key. As the firm uses more labor relative to capital, the marginal product of labor (MP_L) falls while the marginal product of capital (MP_K) rises. Since MRTS = MP_L/MP_K, the numerator falls and the denominator rises — so MRTS declines. This is exactly why the isoquant is convex: moving right along it, each additional unit of labor released requires surrendering less and less capital to maintain the same output. The declining MRTS is not a separate assumption — it follows from diminishing marginal product.

Question 2

If a car manufacturer finds that robots and assembly workers are perfect substitutes (one robot always replaces exactly three workers), what is the shape of their isoquants?

• A. Convex, bowing toward the origin, because diminishing returns still apply
• B. L-shaped right angles, because the two inputs must be used in fixed proportions
• C. Straight lines with constant slope, because the substitution rate never changes
• D. Concave curves, because the firm can substitute at increasing rates

Answer: (C)

Perfect substitutes have isoquants that are straight lines with constant (and negative) slope. The MRTS is constant — one robot always replaces exactly three workers regardless of how many of each are employed. This is the polar opposite of convex isoquants (where MRTS changes with the ratio) and L-shaped isoquants (perfect complements, where no substitution is possible). Real production processes almost never have perfect substitutes; this case sets an extreme benchmark.

Question 3

Isoquants slope downward because using more of one input reduces output, which is expected to be offset by using more of the other input.

• T. True
• F. False

Answer: False

Isoquants do slope downward, but not because using more of one input reduces output. Both labor and capital are productive — using more of either, holding the other constant, generally increases output. Isoquants slope downward because we are constraining output to be constant: if you gain more of one productive input, you can release some of the other and still produce the same amount. The downward slope reflects a trade-off between inputs, not a productivity penalty.

Question 4

The marginal rate of technical substitution at any point on an isoquant equals the ratio of the marginal products of the two inputs: MRTS = MP_L / MP_K.

• T. True
• F. False

Answer: True

This equality follows directly from the condition that we move along the isoquant — output is constant. If the firm gains dL units of labor, output rises by MP_L · dL. To keep output constant, the firm must reduce capital by dK such that MP_K · dK = MP_L · dL, giving dK/dL = MP_L/MP_K. Since MRTS is defined as −dK/dL (the absolute value of the isoquant slope), MRTS = MP_L/MP_K. Understanding this equality connects the geometric property (isoquant slope) to the underlying production technology.

Question 5

Why are typical isoquants convex (bowing toward the origin) rather than straight lines, and what property of production technology causes this shape?

Model answer: Isoquants are convex because of diminishing marginal product. When a firm uses a lot of capital and little labor, labor is scarce relative to capital — labor's marginal product is high and capital's is low, so MRTS = MP_L/MP_K is large. As the firm substitutes more labor for capital (moving along the isoquant), labor's marginal product falls and capital's rises due to diminishing returns to each input. The MRTS therefore declines, meaning each successive unit of labor added allows the firm to release less and less capital. This declining MRTS produces the convex (bowed-inward) shape — straight lines would require constant MRTS (perfect substitutes), which would require no diminishing returns.

The connection to diminishing marginal product is the key: the convex shape is not assumed separately but follows from the production technology. If inputs had increasing marginal products, isoquants would be concave — but such production functions are economically implausible for most real processes.