L9: Costs

Suppose 𝑞=sqrt(K) = sqrt(L), 𝑟=5, and 𝑤=50. How much labor will the firm use to produce 𝑞=110, if it minimizes costs?

100 (Exercise 9.4.1)

Under what conditions is cost minimized for a fixed level of production?

A tangency between the isoquant and an isocost Cost is minimized for a fixed level of production when the isoquant associated with that level of production is tangent to an isocost line. Any isocost line that is shifted inward from that point will never intersect the isoquant. Any isocost line that is shifted outward from that point is higher than the minimum cost. This is analogous to finding the consumer's bundle of greatest utility by finding the tangency between the consumer's indifference curves and the budget constraint. Although firms do seek to maximize profit (rather than minimizing costs), this is not what the question asks.

In the short run, how are marginal cost and the marginal product of inputs related?

MC = w/MP_L 𝑀𝑃_𝐿=Δ𝑞/Δ𝐿

What costs are "fixed" even in the long run?

Sunk costs Sunk costs remain the same (i.e. are "fixed") under any level of production, in the short or the long term. Ordinary fixed costs, like capital, are still variable in the long run.

What's the correct order of steps for solving a cost function? a. Write down the production function, 𝑞 in terms of 𝐾 and 𝐿 b. Solve for 𝐾 in terms of 𝑞 c. Solve for 𝑞 in terms of 𝐿 d. Plug in values for 𝑤 and 𝑟, and solve for 𝐶 in terms of 𝑞 e. Set 𝑀𝑅𝑇𝑆 equal to price ratio, and solve for 𝐾 in terms of 𝐿 f. Solve for 𝐿 in terms of 𝑞 g. Write down the cost function, 𝐶 in terms of 𝑟, 𝑤, and 𝑞

This sequence is a, e, c, f, b, g, d. The steps are delineated in the video: 1. Write down the production function, 𝑞 in terms of 𝐾 and 𝐿 2. Set 𝑀𝑅𝑇𝑆 equal to price ratio, and solve for 𝐾 in terms of 𝐿 3. Solve for 𝑞 in terms of 𝐿 4. Solve for 𝐿 in terms of 𝑞 5. Solve for 𝐾 in terms of 𝑞 6. Write down the cost function, 𝐶 in terms of 𝑟, 𝑤, and 𝑞 7. Plug in values for 𝑤 and 𝑟, and solve for 𝐶 in terms of 𝑞

When is the average total cost minimized?

When the average total cost is equal to the marginal cost The average total cost is minimized when the average total cost is equal to the marginal cost. When the marginal cost is below the average total cost, producing one more unit will lower the average total cost. When the marginal cost is above the average total cost, producing one more unit will raise the average total cost. So to minimize average total cost, produce additional units until the marginal cost of the unit stops decreasing the average total cost.

Which statements about cost are true? A. Total cost is the product of average total cost and the quantity produced B. Total cost is the sum of each short run marginal cost of all units produced C. Average total cost is the sum of average fixed cost and average variable cost D. Average fixed cost is always declining with quantity produced in the short run E. Total cost is always declining with quantity produced

A, C, D. (You put C, D.) Total cost is the product of average total cost and the quantity produced, because the average total cost is simply total cost divided by quantity produced Total cost is the sum of each short run marginal cost of all units produced plus the fixed cost. Average total cost is the sum of average fixed cost and average variable cost, by definition. Average fixed cost is always declining with quantity produced. As more units are produced, the average fixed cost is spread among all of them. Total cost is always increasing with quantity produced. Total cost will never drop by producing an additional unit.

What is true about the long-run expansion path? A. It intersects a series of tangency points between isocost lines and isoquant lines B. It lies tangent to a series of isocost lines C. It is linear if the MRTS is linear D. It represents the long-run cost curve

A, C, D. As can be seen in Figures 9-5, the long-run expansion path intersects a series of tangency points between isocost lines and isoquant lines, because those lines represent the cost-minimizing points at each level of quantity produced. It is never tangent to any isocost lines; rather it intersects them. As pointed out in lecture, the long-run expansion path is linear if the MRTS is linear, and it is alternately called the long-run cost curve.

The LRAC curve is a lower bound of multiple SRAC curves. What differentiates these SRAC curves?

Each SRAC curve is for a different fixed level (or "amount") of capital In the short run, capital is fixed by definition. Each SRAC curve is optimized for a certain fixed level of capital. In the long run, capital is flexible, and the LRAC can "choose" between SRACs.