Production & Costs Continued… Agenda: Consumer and Producer Theory: similarities and differences Isoquants & The Marginal Rate of Technical Substitution Diminishing vs. Decreasing Returns Isocosts Putting it together: Optimal Production & Examples Optimal Quantity Long Run and Short Run Cost Curves

Consumer Utility Maximization Indifference curves Budget constraint Utility curves Different notation Same meaning!

The Production Mountain Quantity per unit of labor holding capital constant Quantity per unit of capital holding labor constant Isoquants: Combinations of capital and labor that produce a given quantity Long Term we can vary both capital and labor

The Marginal Rate of Technical Substitution If we change the amount of capital we use, how much do we need to change the amount of labor to make the same quantity? Isoquants Airplane Game Isoquants When you have a lot of capital, it will have a low marginal product, so you can give up a unit and not have to add a lot of labor; but when you have a little capital, giving up one unit will require a lot of labor to make the same amount. Looking from the labor perspective: when you have a little labor, you have a high marginal product, so giving up a unit requires a lot of additional capital. However, when you have a lot of labor you have a low marginal product, so you can give up a lot and only need a little more capital to produce the same amount. paper labor 1 X Y 1

Isoquants vs. Indifference Curves What are the similarities and differences between Isoquants and Indifference curves? Diminishing MRTS: If you have a lot of labor, you have a low rate of Marginal product. If you give up a little, you only need a little more capital to produce the same quanitity. Convexity from diminishing marginal rate of technical substitution More is better Quantity is a cardinal measure Can only change both capital and labor in the long run. Convexity from preference assumption More is better Utility is an ordinal measure Individuals make trade-offs both at one time and over time

Diminishing vs. Decreasing returns All Isoquants are convex and slope down: diminishing MRTS Quantity increases at a decreasing rate as all inputs increase: decreasing returns +20 +40 +60 +60 +60 +90 +60

C=PKK + PLL Isocost Lines What is an equation to represent the total cost of production? C=PKK + PLL Can we re-arrange this to fit the equation for a line in (L,K) space?

What is the optimal input combination GIVEN cost or quantity? “No matter what the structure of industry may be… (for profit or not for profit) … the objective of most producers is to produce any given level and quality of output at the lowest possible cost. Equivalently, the producer wants to produce as much output as possible from a given expenditure on inputs.” (Frank p. 233) Duality w = cost of labor Maximize Q given C Minimize C given Q Note, ratio of w/r is absolute value – slope of isocost line is negative. r = cost of capital Marginal products per dollar

How should the firm adjust its mix of capital and labor? Example: If the MRTS between capital and labor is 1/2, the interest rate is 5% (use 5) and the wage rate is $10 per hour, is the firm maximizing production? How should the firm adjust its mix of capital and labor? The firm could be making more for the same cost! Use LESS labor, MORE capital The firm is spending more than it has to! 2 1/2

(that said… change term structure, reduce benefits, training, perks…) Example: If the marginal product of labor is 5 and the marginal product of capital is 2, the price of labor is $20 and the cost of capital is 4%, is the firm optimizing production? To increase the marginal product of labor, reduce labor. To decrease the marginal product of capital, increase capital. You can NOT control interest rates or reduce wages in perfect capital or labor markets. (that said… change term structure, reduce benefits, training, perks…)

But how do we know how much to produce in the first place? Supply Curves are MC curves ABOVE the minimum AVC! Short-run Individual firm supply curve Why not here?? In other words…where do supply curves come from? What if price is here? Economic loss! Shutdown! P<AVC Test yourself: Does the supply curve have anything to do with fixed costs?

Short Run vs Long Run Costs Key: long run total costs are those associated with the optimal inputs for a given quantity Short run cost curves Long run cost curves No fixed or variable costs in the long run! Inflection point Total costs can be zero in the long run Tangency point Key – the total costs are the cost associated with the optimal inputs for a given quantity The change in total costs reflects a change in optimal inputs to reach a new quantity isoquant What happens when we combine the short and long run curves…

The long-run average cost curve is the envelope of the short run curves. Viner told his draftsman PK Wong, a mathematician, to draw the envelope curve so that it intersected at the minimum of the short run curves. Wong told him that was impossible. The debate/confusion raged for decades. Some tried to explain the discrepancy using a time dimension, but Silberberg gives a nice intuitive proof of why this must be the case. Why doesn’t the LAC curve intersect with the minimum points of the short run cost curves? “…it is the occasional errors of geniuses like Viner…which seminally advance the body of science.” Paul Samuelson (1972) JPE 72:5 – 11. Jacob Viner

And the marginal cost curves must have the same slope Why doesn’t the LAC curve intersect with the minimum points of the short run cost curves? When the short run cost curve is tangent to the long run curve the average cost has the same slope And the marginal cost curves must have the same slope The minimum point on the short run curve will be where the slope of the average equals the marginal This will be everywhere above the long run cost curve except where the long and short run curves are tangent at the long run minimum. Why isn’t that slope always the minimum slope of the short run curves? Emphasize the difference between long-run unconstrained and short run constrained – short run curves must lie above the long run ones. Silbergerg on the Envelop Theorem http://www.ems.bbk.ac.uk/faculty/phdStudents/kim/me_env.pdf Eugene Silberberg

This mix of labor & capital is optimal to make Q2 Short run cost curves Capital is fixed At high Q too much labor At low Q too much capital This mix of labor & capital is optimal to make Q2 Since everything is variable in the long run, we are going to compare the short-run average variable cost to the long run average cost The only think that is varying in the short-run graph is labor GIVEN the production process with fixed capital in the short run graph, it is optimal to produce Q3 where MC = ATC. HOWEVER, with the given costs of capital and labor, and the level of capital given in the short run curve, the optimal (least cost) output is 2. The short-run minimum cost to produce Q3 will have more capital and less labor. Long run cost curves: everything is variable