chapter 6
decreasing returns to scale
A production function for which changing all inputs by the same proportion changes output less than proportionately.
increasing returns to scale
A production function for which changing all inputs by the same proportion changes output more than proportionately.
constant returns to scale
A production function for which changing all inputs by the same proportion changes the quantity of output by the same proportion.
MRTS:
W/R = MPL/MPK MPK/R = MPL/W
or each of the following production functions, determine if they exhibit constant, decreasing, or increasing returns to scale. Q = 2K + 15L Q = min(3K, 4L) Q = 15K0.5L0.4
直接带数进去。 L = 1 , K =1 。 Q= ? L = 2 , K =2。 Q= ? 如果刚好等于一倍的话就是constant 大于1 就是increasing 小于1就是decreasing The easiest way to determine the returns to scale for a production function is to simply plug in values for L and K, calculate Q, and then double the input levels to see what happens to output. If output exactly doubles, the production function exhibits constant returns to scale. If output rises by less than double, there are decreasing returns to scale. If output more than doubles, the production function has increasing returns to scale. So, for each of these production functions, we will start with K = L = 1 and calculate Q and then perform the same exercise for K = L = 2. Note that K and L do not have to be equal for this method to work, but it does simplify the solution a bit. If L = 1 and K = 1: Q = 2K + 15L = 2(1) + 15(1) = 2 + 15 = 17. If L = 2 and K = 2: Q = 2K + 15L = 2(2) + 15(2) = 4 + 30 = 34. Since output exactly doubles when inputs are doubled, the production function exhibits constant returns to scale. If L = 1 and K = 1: Q = min(3K, 4L) = Q = min(3(1), 4(1)) = min(3, 4) = 3. If L = 2 and K = 2: Q = min(3K, 4L) = Q = min(3(2), 4(2)) = min(6, 8) = 6. Because output exactly doubles when inputs are doubled, the production function exhibits constant returns to scale. If L = 1 and K = 1: Q = 15K0.5L0.4 = Q = 15(1)0.5(1)0.4 = 15(1)(1) = 15. If L = 2 and K = 2: Q = 15K0.5L0.4 = Q = 15(2)0.5(2)0.4 = 15(1.414)(1.320) = 27.99. Because output less than doubles when inputs are doubled, the production function exhibits decreasing returns to scale.
ISOQUANT 比较直的时候。
(a) Relatively straight isoquants indicate that MRTSLK does not vary much along the curve. Therefore, labor and capital are close substitutes for each other. L AND K很容易替换。
ISOQUANT 比较弯的时候
(b) Relatively curved isoquants indicate that MRTSLK varies greatly L AND K不容易替换。
Simplifying Assumptions about Firms' Production Behavior
1. The firm produces a single good. 2. The firm has already chosen which product to produce. 3. For whatever quantity it makes, the firm's goal is to minimize the cost of producing it. 4. The firm uses only two inputs to make its product: capital and labor. 5. In the short run, a firm can choose to employ as much or as little labor as it wants, but it cannot rapidly change how much capital it uses. In the long run, the firm can freely choose the amounts of both labor and of capital it employs. 短期，人力可变，capital不变。 6. The more inputs the firm uses, the more output it makes. 7. A firm's production exhibits diminishing marginal returns to labor and capital.每增加一个人或资源，每个人的额外效率都会减小一些。 8. The firm can buy as many capital or labor inputs as it wants at fixed prices. 9. If there is a well-functioning capital market (e.g., banks and investors), the firm does not have a budget constraint
final good
A good that is bought by a consumer.
perfect complements
Cabs K and drivers L are perfect complements. The isoquants are L-shaped, and the optimal quantity (K, L) for each output Q is the corner of the isoquant. In this case, 1 cab with 1 driver produces Q = 1, while 2 cabs with 2 drivers produce Q = 2. 直角。
long run
In production economics, the period of time during which all inputs into production are fully adjustable.
short run
In production economics, the period of time during which one or more inputs into production cannot be changed.
Production functions
Q = 10K + 5L
Cobb-Douglas production function.
Q = K0.5L0.5
Production Functions
Q = f(K, L)， can have varied form.
perfect substitute
Robots K and labor L are perfect substitutes. The isoquants are straight lines, and the MRTSLK does not change along the isoquant. In this case, two humans can substitute for one robot. 一条直线。
MRTSLK
the amount of capital needed to hold output constant if the quantity of labor used by the firm changes. 如果labour 改变了 要改变多少k才能保持同样的产量。
Q = Af(K, L)
where A is the level of total factor productivity, a parameter that affects how much output can be produced from a given set of inputs. Usually, we think of A as reflecting technological change. Increases in A mean that the amount of output obtainable from any given set of labor and capital inputs will increase as well.
returns to scale
A change in the amount of output in response to a proportional increase in all of the inputs.
isoquant 等产量曲线
A curve representing all the combinations of inputs that allow a firm to make a particular quantity of output. 一条线上表示所有可能的搭配L+K，生产出同种产量。 Each isoquant shows the possible combinations of labor (L) and capital (K) that produce the output (Q) levels 1, 2, and 4 units. The negative slope of the isoquant is the marginal rate of technical substitution of labor L for capital K. At point A, the marginal product of labor is high relative to the marginal product of capital, and a relatively small decrease in labor would require a large quantity of capital to hold output constant. At point B, the marginal product of labor is low relative to the marginal product of capital, and a relatively small decrease in capital would require a large quantity of labor to hold output constant. K在y， L在x。 在isoquant 上，越往下走， substitution 的rate 越低。 一开始愿意用很多k换l， 后来不愿意用那么多k换l
expansion path 扩展线
A curve that illustrates how the optimal mix of inputs varies with total output. 一条线连接所有的在isoost 和isoquant 相交的点。
total cost curve
A curve that shows a firm's cost of producing particular quantities.
isocost line
A curve that shows all of the input combinations that yield the same cost. 全部搭配， 一样的成本价格。 C = RK + WL This means that the y-intercept of the isocost line is C/R, while the slope is the (negative of the) inputs' price ratio, -W/R.
diminishing marginal product
A feature of the production function; as a firm hires additional units of a given input, the marginal product of that input falls. 每增加一个input， marginal output下降。
cost minimization
A firm's goal of producing a specific quantity of output at minimum cost.
intermediate good
A good that is used to produce another good.
production function 生产函数；生产功能
A mathematical relationship that describes how much output can be made from different combinations of inputs.
The Impact of Technological Change
An improvement in technology shifts the isoquant image inward to image . The new cost-minimizing input combination (L2, K2) is located at the tangency between Q2 and the isocost C2. (L2, K2) uses fewer inputs and is, therefore, cheaper than the original cost-minimizing input combination (L1, K1) located at the tangency between Q1 and the isocost C1. 一样的Q iscoquant 和 isocost 都往里面shift。 因为只需要更少的k和l 可以产出一样的q
total factor productivity growth／ technological change
An improvement in technology that changes the firm's production function such that more output is obtained from the same amount of inputs. 一样的input ， output 增加 因为科技的改变。 Q = Af(K, L)
Suppose that the wage rate is $10 per hour and the rental rate of capital is $25 per hour. Write an equation for the isocost line for a firm. Draw a graph (with labor on the horizontal axis and capital on the vertical axis) showing the isocost line for C = $800. Indicate the horizontal and vertical intercepts along with the slope. Suppose the price of capital falls to $20 per hour. Show what happens to the C = $800 isocost line including any changes in intercepts and the slope.
An isocost line always shows the total costs for the firm's two inputs in the form of C = RK + WL. Here, the wage rate (W) is $10 and the rental rate of capital (R) is $25, so the isocost line is C = 10L + 25K. We can plot the isocost line for C = $800 = 10L + 25K. One easy way to do this is to compute the horizontal and vertical intercepts. The horizontal intercept tells us the amount of labor the firm could hire for $800 if it only hired labor. Therefore, the horizontal intercept is $800/W = $800/$10 = 80. The vertical intercept tells us how much capital the firm could hire for $800 if it were to use only capital. Thus, it is $800/R = $800/$25 = 32. We can plot these points on the following graph and then draw a line connecting them. This is the C = $800 isocost line labeled C1. We can calculate slope in several different ways. First, we can simply calculate the slope of the isocost line as drawn. Remember that the slope of a line is ΔY/ΔX (i.e., rise over run). Therefore, the slope is image We can also rearrange our isocost line into slope-intercept form by isolating K: 800 = 10L + 25K 25K = 800 - 10L K = (800/25) - (10/25)L = 32 - 0.4L This equation tells us that the vertical intercept is 32 (which we calculated earlier) and -0.4 is the slope. image If R falls to $20, the horizontal intercept is unaffected. If the firm is only using labor, a change in the price of capital will have no impact. However, the vertical intercept rises to $800/R = $800/$20 = 40 and the isocost line becomes steeper (C2). The new slope is -W/R = -$10/$20 = -0.5.
fixed inputs
Inputs that cannot be changed in the short run.
ΔQ = MPL × ΔL + MPK × ΔK = 0 因为产量不变。
MRTSLK＝ － ΔK/ΔL = MPL/MPK
The short-run production function for a firm that produces pizzas is image , where Q is the number of pizzas produced per hour, image is the number of ovens (which is fixed at 3 in the short run), and L is the number of workers employed. Write an equation for the short-run production function for the firm showing output as a function of labor. Calculate the total output produced per hour for L = 0, 1, 2, 3, 4, and 5. Calculate the MPL for L = 1 to L = 5. Is MPL diminishing? Calculate the APL for L = 1 to L = 5.
Solution: To write the production function for the short run, we plug image into the production function to create an equation that shows output as a function of labor: image To calculate total output, we plug in the different values of L and solve for Q: image The marginal product of labor is the additional output generated by an additional unit of labor, holding capital constant. We can use our answer from (b) to calculate the marginal product of labor for each worker: image Note that, because MPL falls as L rises, there is a diminishing marginal product of labor. This implies that output rises at a decreasing rate when labor is added to the fixed level of capital. The average product of labor is calculated by dividing total output (Q) by the quantity of labor input (L): image
Marginal Product
The additional output that a firm can produce by using an additional unit of an input (holding use of the other input constant). MPL = ΔQ/ΔL
A firm is employing 100 workers (W = $15/hour) and 50 units of capital (R = $30/hour). At the firm's current input use, the marginal product of labor is 45 and the marginal product of capital is 60. Is the firm producing its current level of output at the minimum cost or is there a way for the firm to do better? Explain.
The cost-minimizing input choice occurs when MPL/W = MPK/R. We need to determine if this is the case for this firm: MPL = 45 and W = 15, so MPL/W = 45/15 = 3 MPK = 60 and R = 30, so MPK/R = 60/30 = 2 Therefore, MPL/W > MPK/R. The firm is not currently minimizing its cost. 对比MPL/W ， MPK/R 哪一个数字大就增多哪个， 减少数字少的。 Because MPL/W > MPK/R, $1 spent on labor yields a greater marginal product (i.e., more output) than $1 spent on capital. The firm would do better by reducing its use of capital and increasing its use of labor. Note that as the firm reduces capital, the marginal product of capital will rise. Likewise, as the firm hires additional labor, the marginal product of labor will fall. Ultimately, the firm will reach its cost-minimizing input choice where MPL/W = MPK/R.
slope of the production function
The marginal product of labor， as production q goes up, slope flattens.
learning by doing
The process by which a firm becomes more efficient at production as it produces more output.
production
The process by which a person, company, government, or non-profit agency creates a good or service that others are willing to pay for.
average product
The quantity of output produced per unit of input.
marginal rate of technical substitution (MRTSXY)
The rate at which the firm can trade input X for input Y, holding output constant. The negative of the slope of the isoquant is called the marginal rate of technical substitution of one input (on the x-axis) for another (on the y-axis), It is the quantity change in input Y necessary to keep output constant if the quantity of input X changes by 1 unit. 如果x change了1个单位， y 要change 多少才能保持一样的产量Q.
Figure 6.11 A Change in the Relative Price of Labor Leads to a New Cost-Minimizing Input Choice
When labor becomes relatively more expensive, the isocost line shifts from C1 to C2. With the steeper isocost line, the cost-minimizing input choice shifts from point A, with a high ratio of labor to capital, to point B, with a low ratio of labor to capital. 还在同一条iscoquant上，但是点会向上移动，更多的k，更少的l isocost 会变， steeper。 l 往里。k往上。
When Capital Becomes More Expensive, the Isocost Line Becomes Flatter
When the price of capital increases from R = $20 to R = $40 and the price of labor stays constant at W = $10, the slope of the isocost changes from image to image . The quantity of inputs the firm can buy for $100 decreases and the isocost line becomes flatter.
When Labor Becomes More Expensive, the Isocost Line Becomes Steeper
When the price of labor increases from W = $10 to W = $20 and the price of capital stays constant at R = $20, the slope of the isocost changes from image , or -1. The isocost line, therefore, becomes steeper, and the quantity of inputs the firm can buy for $100 decreases.