Questions: Isoquants and Factor Substitution

Question 1

A production manager says 'Our isoquants are perfectly straight lines.' What does this reveal about the firm's technology?

• A. Capital and labor exhibit diminishing marginal returns along every isoquant
• B. Capital and labor are perfect substitutes — one can replace the other at a constant rate without limit
• C. The firm uses fixed-proportion (Leontief) technology where inputs must be combined in rigid ratios
• D. The MRTS increases as more labor is substituted for capital, reflecting increasing returns

Answer: (B)

Straight-line isoquants indicate a constant MRTS — the rate at which labor substitutes for capital is the same regardless of how much of each input you use. This is the definition of perfect substitutability. If you can always replace one unit of capital with two units of labor (for instance) no matter where you are on the isoquant, the two inputs are perfect substitutes. Fixed-proportion (Leontief) technology is the opposite extreme: right-angle isoquants, not straight lines. Diminishing MRTS produces the typical bowed-inward isoquant — the curvature reflects the changing substitution rate, not straight lines.

Question 2

A firm's current input mix has MRTS = MP_L/MP_K = 3, while the wage-to-rental ratio w/r = 2. What adjustment should the firm make to minimize cost?

• A. Hire more capital and less labor, moving up the isoquant toward a higher capital-labor ratio
• B. Hire more labor and less capital, since the marginal product per dollar is higher for labor than for capital
• C. Keep the current input mix — the MRTS and w/r are close enough to approximate optimality
• D. Adjust wages and rental rates by negotiating with input markets until they equal the MRTS

Answer: (B)

Cost minimization requires MRTS = w/r. Here MRTS (3) > w/r (2), which means MP_L/MP_K > w/r, or equivalently MP_L/w > MP_K/r — the last dollar spent on labor buys more output than the last dollar spent on capital. The firm should reallocate spending toward labor (hire more workers) and away from capital (use less). As more labor is substituted for capital along the isoquant, diminishing marginal product reduces MP_L and the scarcer capital becomes more productive, so the MRTS falls. The firm keeps substituting labor for capital until MRTS = w/r = 2, at which point the last dollar on each input yields equal marginal product.

Question 3

The inward-bowing (convex) shape of a typical isoquant reflects the principle of diminishing MRTS: as more labor is substituted for capital, each additional unit of labor can replace less and less capital.

• T. True
• F. False

Answer: True

Moving along an isoquant from upper-left (lots of capital, little labor) to lower-right (lots of labor, little capital), the MRTS decreases. This is because diminishing marginal product applies to both inputs: as you add more labor, each additional worker contributes less to output (MP_L falls); as you remove capital, each remaining unit becomes more scarce and more productive (MP_K rises). Since MRTS = MP_L/MP_K, both forces push MRTS downward. The result is an isoquant that gets flatter as you move right — a convex curve bowing toward the origin.

Question 4

If the MRTS is constant at most point along an isoquant, this indicates the technology uses fixed input proportions — a Leontief production function.

• T. True
• F. False

Answer: False

Constant MRTS indicates perfect substitutes, not fixed proportions. A constant MRTS produces straight-line isoquants: the substitution rate between inputs never changes. Leontief (fixed-proportion) technology is the opposite extreme, producing right-angle isoquants: the MRTS is zero along horizontal segments (adding labor does nothing) and undefined/infinite along vertical segments (adding capital does nothing). These are two distinct extreme cases. Leontief isoquants have a kink, not a constant slope.

Question 5

What does it mean for an isoquant to bow inward (be convex to the origin), and why does this shape arise naturally from diminishing marginal products of both inputs?

Model answer: A convex isoquant means the MRTS diminishes as you substitute more labor for capital along the curve. Geometrically, the isoquant gets flatter as you move right (more labor, less capital). This shape arises because diminishing marginal product acts on both inputs simultaneously. As you add more labor along the isoquant, each additional worker produces less (diminishing MP_L), so you get progressively less output from each unit of additional labor. At the same time, as capital becomes scarcer, each remaining unit of capital is more productive (rising MP_K due to scarcity). Since MRTS = MP_L/MP_K, falling MP_L and rising MP_K both drive MRTS downward as you move down the isoquant — producing the characteristic concave-to-origin curvature.

The convexity of isoquants is not an assumption arbitrarily imposed — it is a consequence of diminishing marginal products. This is why economists use it to characterize 'normal' production technologies. At the extremes, if marginal products don't diminish, you get straight-line isoquants (perfect substitutes). If adding one input without the other does literally nothing, you get right-angle isoquants (perfect complements). The convex isoquant sits between: substitution is possible, but at an increasingly costly rate.