Technology

What a firm can make

A firm’s technology is the recipe turning inputs into output. With two inputs we write the production function \(y = f(x_1,x_2)\) — the most output obtainable from \(x_1\) units of input 1 and \(x_2\) of input 2.

The input-space analogue of an indifference curve is an isoquant: all input combinations that produce the same output. Its slope is the technical rate of substitution (TRS) — how much of input 2 you can drop if you add one unit of input 1 and hold output fixed: \[ \text{TRS} = -\frac{MP_1}{MP_2}. \]

With Cobb–Douglas technology \(f = x_1^{a}x_2^{b}\) the marginal products are \(MP_1 = a\,x_1^{a-1}x_2^{b}\) and \(MP_2 = b\,x_1^{a}x_2^{b-1}\), so \(\text{TRS} = -(a/b)(x_2/x_1)\) — diminishing, which is why isoquants bow toward the origin.

The exponents also fix returns to scale — what happens to output when you scale both inputs by \(t\): \[ f(tx_1,tx_2)=t^{\,a+b}f(x_1,x_2). \] So \(a+b<1\) is decreasing returns, \(a+b=1\) constant, \(a+b>1\) increasing.

Try it. Slide the exponents a and b. The isoquant map reshapes, and the panel reports returns to scale from \(a+b\). Larger exponents bow the isoquants and steepen the trade-off between inputs.

Exponent on input 1 (a): 0.50 Exponent on input 2 (b): 0.50

What to notice

• Isoquants farther from the origin are higher output — the exact analogue of higher indifference curves meaning more utility.
• Set a + b = 1 (e.g. 0.5 and 0.5) for constant returns: doubling both inputs doubles output. Push the sum above 1 for increasing returns, below for decreasing.
• A larger exponent on an input makes its marginal product matter more, tilting the isoquants and steepening the input trade-off (TRS).