Optimal Production
• Firm gets to pick both capital and labor. Y = F(L , K)
Isoquant • An isoquant is a graph of all the combinations of inputs that yield the same amount of output.
Example Q=4
Example Q=4 Q=3 Q=5
Isoquant properties • Isoquants never cross. – The same bundle of inputs can’t produce two different levels of output!
CAN’T CROSS Q = 16 Q=9
CAN’T CROSS Q = 16 Q=9
Isoquant properties • Isoquants never cross. – The same bundle of inputs can’t produce two different levels of output! • Isoquants usually bow toward the origin. – “increasing marginal rate of technical substitution” – “Replacing your 100 th machine with workers is easier than replacing your last machine. ” • Output increases as you move “northeast”
Bowed in and increasing Q = 16 Q = 25 Q=9
Isocost • Isocost is a graph of all the combinations of inputs that lead to the same cost. • Let w be the price of labor and r be the price of capital. • Suppose you wanted a cost of C
Example • Suppose: – w = 10 –r=5 – C = 100 If you only hired workers, you could hire: 100/10 = 10 units of labor
Example • Suppose: – w = 10 –r=5 – C = 100 If you only rented capital, you could rent: 100/5 = 20 units of capital
Example • Suppose: – w = 10 –r=5 – C = 100 If you fired a unit of labor, you could then rent: 10/5 = 2 units of capital
Isocost K = 100/5 w = 10 r=5 C = 100 slope=-10/5 L = 100/10
• C/w is the amount of labor you could hire if you only hired labor. • C/r is the amount of capital you could rent if you only rented capital. • If you fired one worker, you would have w to spend. • With w to spend, you could rent w/r units of capital.
Isocost K = C/r C slope=-w/r L = C/w
Isocost K = 100/5 w = 10 r=5 C = 100 slope=-10/5 L = 100/10
Suppose C fell K = 75/5 = 15 C = 100 C = 75 slope=-10/5 L = 75/10 = 7. 5 w = 10 r=5
Why is the slope the same? K = 75/5 = 15 C = 100 C = 75 slope=-10/5 L = 75/10 = 7. 5 w = 10 r=5
Why is the slope the same? K = 75/5 = 15 C = 100 C = 75 slope=-10/5 L = 75/10 = 7. 5 w = 10 r=5
Isocost K = 100/5 w = 10 r=5 C = 100 slope=-10/5 L = 100/10
Suppose the wage increased K = 100/5 = 10 C = 100 slope=-20/5 L = 100/20 = 5 w = 20 r=5
Cost increases to the “northeast” Holding input prices constant w = 10 r=5 C = 100 C = 75 C = 50
Firm problem • If a firm decides to produce q, it wants to find the lowest cost way of doing so. • Once it picks q, revenue is p*q. To maximize profit once it has picked q, it needs to minimize cost.
Example Suppose this firm decides to produce 4 units. Q=4
Example Suppose this firm decides to produce 4 units. Q=4 Suppose: w = 10 and r = 10 What are the slopes of the isocost lines?
Example Suppose this firm decides to produce 4 units. Q=4 Suppose: w = 10 and r = 10 What are the slopes of the isocost lines? -1
Example Q=4 Cost is increasing to the northeast. Want the lowest cost that will produce Q=4. Occurs where the isocost is tangent to the isoquant.
Example Q=4 Cost is increasing to the northeast. Want the lowest cost that will produce Q=4. Occurs where the isocost is tangent to the isoquant. A: no way to produce Q=4 and spend that among
Example Q=4 B: can produce 4 here, but not the cheapest Cost is increasing to the northeast. Want the lowest cost that will produce Q=4. Occurs where the isocost is tangent to the isoquant. A: no way to produce Q=4 and spend that among
Example Q=4 B: can produce 4 here, but not the cheapest Cost is increasing to the northeast. Want the lowest cost that will produce Q=4. Occurs where the isocost is tangent to the isoquant. C: the cheapest way to produce Q=4 A: no way to spend that amount and produce Q=4
Example Q=4 If this firm wants to produce 4 units of output, it should use 4 units of labor and 4 units of capital.
Rules • The cost-minimizing way to produce q is where the isocost and isoquant are tangent.
Example Q=4 If this firm wants to produce 4 units of output, it should use 4 units of labor and 4 units of capital. What is the total factor cost?
Example Q=4 If this firm wants to produce 4 units of output, it should use 4 units of labor and 4 units of capital. What is the total factor cost? 4*10 + 4*10 = 80
Example Q=4 Suppose the firm now wants to produce 3 units of output instead of 4. Obviously it can do this at a lower cost. Q=3
Example Q=4 Suppose the firm now wants to produce 3 units of output instead of 4. Obviously it can do this at a lower cost. Q=3
Example Q=4 Q=3 If the firm wants to produce 3 units of output, it should use 3 units of labor and 3 units of capital.
Example Q=4 If the firm wants to produce 3 units of output, it should use 3 units of labor and 3 units of capital. What is the total factor cost? Q=3
Example Q=4 If the firm wants to produce 3 units of output, it should use 3 units of labor and 3 units of capital. What is the total factor cost? 3*10 + 3*10 = 60 Q=3
What is the marginal cost of going from 3 units to 4 units? Q=4 Q=3
What is the marginal cost of going from 3 units to 4 units? If the firm wants to produce 3 units of output, it should use 3 units of labor and 3 units of capital. Q=4 Q=3 What is the total factor cost? 3*10 + 3*10 = 60 If this firm wants to produce 4 units of output, it should use 4 units of labor and 4 units of capital. What is the total factor cost? 4*10 + 4*10 = 80
Rules • The cost-minimizing way to produce q is where the isocost and isoquant are tangent. • Cost is increasing in q, the amount you want to produce. • Factor demand for all inputs is increasing in q.
Example Suppose the price of labor increases to 40? Q=4 What are the slopes of the new set of isocost lines?
Example Suppose the price of labor increases to 40? Q=4 What are the slopes of the new set of isocost lines? -40/10 = -4
Example Q=4 Cost is increasing to the northeast. Want the lowest cost that will produce Q=4. Occurs where the isocost is tangent to the isoquant.
Example Q=4 If this firm wants to produce 4 units of output, it should use 3 units of labor and 7 units of capital.
Example Q=4 If this firm wants to produce 4 units of output, it should use 3 units of labor and 7 units of capital. What is the total factor cost?
Example Q=4 If this firm wants to produce 4 units of output, it should use 3 units of labor and 7 units of capital. What is the total factor cost? 3*4 + 7*10 = 190
Rules • The cost-minimizing way to produce q is where the isocost and isoquant are tangent. • Cost is increasing in q, the amount you want to produce. • Factor demand for all inputs is increasing in q. • Cost is increasing in the price of your inputs. • Factor demand is decreasing in own price. • With 2 inputs, factor demand is increasing in the price of the other input.
Ag Example
Orange Industry
Orange Industry • In the United States:
Use the model • Suppose immigration reform leads to fewer migrant workers and thus higher wages for farm workers? – What happens to demand for farm labor? – What happens to demand for picking machines? – What happens to the total factor cost?
Use the model • Suppose scientists discover that consuming 3 oranges a day increases the likelihood of winning the lottery. – What happens to demand for farm labor? – What happens to demand for picking machines? – What happens to the total factor cost?
The Cost Curves • It is too difficult to derive the cost curves intuitively when there is more than one input. • If you every asked why you learned calculus in high school, this is why. • For now, trust me, they “look” the same.
• The marginal cost curve is still the firm supply curve. • The horizontal sum of firm supply curves is still the market supply curve.