Final answer:
The output of an economy is determined by its production function, and a Cobb-Douglas production function where capital earns one-fourth of total income would have capital input (K) with an exponent of 0.25, meaning Y = A * K^0.25 * L^0.75.
Step-by-step explanation:
What determines the amount of output an economy produces is a fundamental question in economics, typically answered by looking at a production function. This function describes the output based on several inputs, commonly labor and capital. To write a Cobb-Douglas production function where capital earns one-fourth of total income, we need to assign exponents to capital (K) and labor (L) that represent their respective shares in income generation.
In the traditional Cobb-Douglas production function, Y = A * K^α * L^β, where Y represents total production/output, A is total factor productivity, K is the capital input, L is the labor input, α is the output elasticity of capital, and β is the output elasticity of labor.
Since we are given that capital earns one-fourth of total income, α would be 0.25. Assuming constant returns to scale where α + β = 1, β would therefore be 0.75. The production function would be expressed as Y = A * K^0.25 * L^0.75, assuming A represents a constant level of technology.