question
Costs
answer
Total costs = fixed costs + variable costs
question
Short-run costs
answer
Costs = k-bar (fixed capital) r (rate of capital) + quantity of L w (wage rate of L)
Can use production function to solve for one of the variables (K or L) in terms of quantity, and then can plug that back in
If K is fixed in the short-run, can solve for L and plug in k-bar...and then you'll arrive at a short-run cost function.
Can use production function to solve for one of the variables (K or L) in terms of quantity, and then can plug that back in
If K is fixed in the short-run, can solve for L and plug in k-bar...and then you'll arrive at a short-run cost function.
question
Deriving short-run cost function
answer
1. You are given a production function
2. You are given the
a. Rental rate (r) of capital
b. The wage rage (w) of labor
c. The fixed level of capital
3. We know the cost function is generally of the form: c = kr + lw
We can solve the production function for one variable - normally labor in the short-run
And then can plug it into the cost function, such that we get a cost function that tells us how cost varies with quantity produced
2. You are given the
a. Rental rate (r) of capital
b. The wage rage (w) of labor
c. The fixed level of capital
3. We know the cost function is generally of the form: c = kr + lw
We can solve the production function for one variable - normally labor in the short-run
And then can plug it into the cost function, such that we get a cost function that tells us how cost varies with quantity produced
question
Marginal cost
answer
The derivative of cost with respect to quantity
delta c / delta q
cost of producing the next unit
delta c / delta q
cost of producing the next unit
question
Short-run MC and relationship to MPL
answer
1. We started with a production function and solved it such that we had L = some relationship to K and quantity
2. On the other hand, we have our TC function ; if we take the derivative of cost w/ respect to labor, the capital term drops out because it's fixed ... so we get change in cost / change in quantity = wage * change in labor / change in quantity
3. We know the MPL is delta q / delta L - how quantity changes for a change in labor
4. So we can rewrite MC as w / MPL
5. This means marginal cost will be higher when wage is higher ... and lower when MPL, meaning delta q / delta L, is higher ... meaning, that you get more output from an additional worker...
6. This explains why firms want to pay productive people higher wages
MC = wage rate / MPL
2. On the other hand, we have our TC function ; if we take the derivative of cost w/ respect to labor, the capital term drops out because it's fixed ... so we get change in cost / change in quantity = wage * change in labor / change in quantity
3. We know the MPL is delta q / delta L - how quantity changes for a change in labor
4. So we can rewrite MC as w / MPL
5. This means marginal cost will be higher when wage is higher ... and lower when MPL, meaning delta q / delta L, is higher ... meaning, that you get more output from an additional worker...
6. This explains why firms want to pay productive people higher wages
MC = wage rate / MPL
question
Average costs
answer
c / q
Example: cost = 10 + 5q^2
Average cost = cost / q = 10/q + 5q
Average cost: often decreases and then increases. Why?
-Total costs = fixed costs + variable costs
-For first set of units, you are paying off your fixed costs, so as quantity increases, your costs are spread over more units, so average cost reduces
-Once you've paid off fixed costs, then your cost rises because your average variable costs rise
Example: cost = 10 + 5q^2
Average cost = cost / q = 10/q + 5q
Average cost: often decreases and then increases. Why?
-Total costs = fixed costs + variable costs
-For first set of units, you are paying off your fixed costs, so as quantity increases, your costs are spread over more units, so average cost reduces
-Once you've paid off fixed costs, then your cost rises because your average variable costs rise
question
Long-run cost curves
answer
Choosing the mix of inputs in order to minimize costs
Optimal mix may differ with quantity
1. What's the right mix of L and Q for a given quantity?
2. As the quantity varies, how does it change the optimal mix of L and Q
Steps
1. We have production function
2. We have costs
3. With costs and the knowledge of the slope of the isocost line, we can figure out the ratio of k to L
4. We can plug the info from #3 back into the production function to have a more precise production function
5. We can solve for L in terms of Q and K in terms of Q
6. We can plug #5 into our TC function in order to get how cost varies w/ production or output
...what we need is w, r, and production function
-everything comes from these 3 things
Optimal mix may differ with quantity
1. What's the right mix of L and Q for a given quantity?
2. As the quantity varies, how does it change the optimal mix of L and Q
Steps
1. We have production function
2. We have costs
3. With costs and the knowledge of the slope of the isocost line, we can figure out the ratio of k to L
4. We can plug the info from #3 back into the production function to have a more precise production function
5. We can solve for L in terms of Q and K in terms of Q
6. We can plug #5 into our TC function in order to get how cost varies w/ production or output
...what we need is w, r, and production function
-everything comes from these 3 things
question
Isocost curves
answer
A firm's budget constraint
question
Cost curves
answer
The marginal cost intersects the average costs curve at its minimum
If you have a function and you take the average, then the minimum is going to be at the derivative
If you have a function and you take the average, then the minimum is going to be at the derivative
question
MC and MPL relationship
answer
Close relationship between marginal cost and marginal product of labor
MPL = DQ / DL
-Digging a hole
-Diminishing marginal product of labor
Derivative of cost function with respect to labor
-Costs = k-bar (fixed capital) r (rate of capital) + quantity of L w (wage rate of L)
-Delta C / Delta L = k-bar r dL
MC of production = delta c / delta q
-Delta
MPL = DQ / DL
-Digging a hole
-Diminishing marginal product of labor
Derivative of cost function with respect to labor
-Costs = k-bar (fixed capital) r (rate of capital) + quantity of L w (wage rate of L)
-Delta C / Delta L = k-bar r dL
MC of production = delta c / delta q
-Delta
question
What happens when input prices change? Or W:R ratio changes?
answer
...
question
Isocost line budget is not fixed
answer
Isocost line is asking what mix of capital and labor required to produce a given quantity of goods at the lowest cost given the productivity of each
Budget not fixed, so minimal budget could change as we change the cost of inputs
Budget not fixed, so minimal budget could change as we change the cost of inputs
question
Long-run expansion path
answer
...
question
At what range of prices will the firm produce a positive output?
answer
The firm will supply positive levels of output in the short run as long as P = MC > AVC, or as long as the firm is covering its variable costs of production.
In my language, revenue maximized where P = MC ... so this means the revenue is greater than the average variable costs...meaning, no operating loss.
In my language, revenue maximized where P = MC ... so this means the revenue is greater than the average variable costs...meaning, no operating loss.
question
At what range of prices will the firm earn a negative profit?
answer
The firm will earn negative profit when P = MC < AC, or at any price below minimum average cost.
question
Firm's supply curve
answer
The MC curve above where MC = AVC
The firm will produce at the point where price equals MC as long as MC is greater than or equal to AVC. T
The firm will produce at the point where price equals MC as long as MC is greater than or equal to AVC. T
question
What is the lowest price at which each firm would sell its output in the long run? Is profit positive, negative, or zero at this price? Explain.
answer
In the long run profit falls to zero, which means price falls to the minimum value of AC. To find the minimum average cost, set marginal cost equal to average cost and solve for q
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