1. **Problem statement:** We analyze the returns to scale from the given table of output (Q) outputs for different combinations of labour (L) and capital (K). 2. **Returns to scale:** We examine how output changes when inputs are scaled up proportionally. - For example, at (1L, 1K), $Q=100$. - Doubling inputs to (2L, 2K), $Q=170$ which is less than double (2*100=200). - Tripling inputs to (3L, 3K), $Q=200$ which is less than triple (3*100=300). This implies **decreasing returns to scale** because output increases less than proportionally when input is increased. 3. **Isoquants:** An isoquant connects input combinations producing the same output. - Select outputs from the table to write all input combinations corresponding to these isoquants: - $Q=190$ isoquant: (2L, 4K), (3L, 3K), (4L, 2K) - $Q=220$ isoquant: (3L, 4K), (4L, 3K) - $Q=150$ isoquant: (1L, 4K), (2L, 2K) - These sets form the isoquants which can be graphed with L on x-axis and K on y-axis. 4. **Profit maximization for competitive firm:** Given $$ C = wL + rK $$ $$ Q = L^{\alpha} K^{\beta} $$ Maximize profit $\pi = pQ - C$, where $p$ is output price. 5. **Lagrangian:** $$ \mathcal{L} = pL^{\alpha} K^{\beta} - wL - rK $$ Set partial derivatives to zero: $$ \frac{\partial \mathcal{L}}{\partial L} = p\alpha L^{\alpha-1}K^{\beta} - w = 0 $$ $$ \frac{\partial \mathcal{L}}{\partial K} = p\beta L^{\alpha}K^{\beta-1} - r = 0 $$ 6. **Rearranging:** $$ p\alpha L^{\alpha-1} K^{\beta} = w $$ $$ p\beta L^{\alpha} K^{\beta-1} = r $$ Divide equations to eliminate $p$: $$ \frac{p\alpha L^{\alpha-1}K^{\beta}}{p\beta L^{\alpha}K^{\beta-1}} = \frac{w}{r} $$ Simplify: $$ \frac{\alpha}{\beta} \frac{1}{L} K = \frac{w}{r} $$ Rewrite: $$ \frac{K}{L} = \frac{\beta w}{\alpha r} $$ 7. **Express $K$ in terms of $L$:** $$ K = \frac{\beta w}{\alpha r} L $$ 8. **Substitute into production:** $$ Q = L^{\alpha} \left( \frac{\beta w}{\alpha r} L \right)^{\beta} = L^{\alpha + \beta} \left( \frac{\beta w}{\alpha r} \right)^{\beta} $$ Solve for $L$: $$ L = \left( \frac{Q}{\left( \frac{\beta w}{\alpha r} \right)^{\beta}} \right)^{\frac{1}{\alpha + \beta}} $$ 9. **Substitute back to get $K$:** $$ K = \frac{\beta w}{\alpha r} L $$ Thus, $L$ and $K$ chosen maximize profit given prices and production parameters.