A firm minimizes costs to produce exactly 500 units. If the wage rate rises while capital rental costs stay constant, what happens to the conditional demand for labor?
ALabor demand increases because the firm must compensate for the higher cost by working inputs harder
BLabor demand is unchanged because conditional demand depends only on output level, not input prices
CLabor demand decreases as the firm substitutes toward relatively cheaper capital, holding output fixed
DLabor demand decreases and output falls, as the firm can no longer afford to produce 500 units
Conditional factor demand holds output fixed (at 500 units here) and minimizes cost. When wages rise, capital becomes relatively cheaper, so the cost-minimizing input mix shifts toward capital and away from labor — substitution along an isoquant. Output does NOT change (that is the 'conditional' part). Option D confuses conditional factor demand (output fixed) with unconditional Marshallian demand (where output adjusts to maximize profit).
Question 2 Multiple Choice
Shephard's lemma states that the conditional demand for input i can be derived directly from the cost function. How?
AIt is the partial derivative of the cost function with respect to output y
BIt is the partial derivative of the cost function with respect to the input price w_i
CIt is the ratio of the cost function to the number of inputs
DIt is found by inverting the production function at the optimal input bundle
Shephard's lemma: x_i(w, y) = dc(w, y)/dw_i. This is the producer-theory analog of the Hicksian demand result in consumer theory (where compensated demand equals the derivative of the expenditure function with respect to price). The elegance is that once you have the cost function, all factor demands follow from differentiation — no need to re-solve the minimization problem for each input.
Question 3 True / False
Conditional factor demand is homogeneous of degree zero in input prices — if all input prices double, the cost-minimizing input quantities do not change.
TTrue
FFalse
Answer: True
When all input prices scale by the same factor, relative prices are unchanged, so the cost-minimizing input bundle (the tangency between the isocost line and the isoquant) is unchanged. Only the total cost doubles. This homogeneity property is a consequence of cost minimization, not an assumption — and it can be tested empirically to check whether firms behave as cost minimizers.
Question 4 True / False
Conditional factor demand and Marshallian (unconditional) input demand answer the same question from different angles, so they usually give the same input quantities at the optimum.
TTrue
FFalse
Answer: False
They answer different questions. Conditional factor demand asks: given that output is fixed at y, what inputs minimize cost? Marshallian input demand asks: given input and output prices, what inputs maximize profit, allowing output to adjust? They coincide only at the profit-maximizing output level. For any other output level, conditional demands reflect the cost-minimizing mix for that target while Marshallian demands reflect the full output-and-input optimization.
Question 5 Short Answer
Why does the matrix of own- and cross-price substitution effects for conditional factor demands have to be symmetric and negative semidefinite? What guarantees these properties?
Think about your answer, then reveal below.
Model answer: These properties are consequences of the cost function being concave in input prices, which results from cost minimization. Symmetry — the effect of w_j on x_i equals the effect of w_i on x_j — follows from Young's theorem applied to the cost function (mixed partial derivatives are equal). Negative semidefiniteness means own-price effects are non-positive: a higher price for an input can only reduce or maintain its conditional demand, never increase it at fixed output. These are mathematical implications of optimization, not behavioral assumptions, making them testable predictions about firm behavior.
The parallel to consumer theory is exact: the Slutsky matrix of compensated demand derivatives is also symmetric and negative semidefinite for the same reason (expenditure function concavity). Both results say that optimization imposes structure on how demands respond to prices.