Question

A local farmer raises peaches using land (K) and labor (L), and has an output of (,K) = 0.5 K0.5 bushels of peaches.

A. Find several input combinations that give the farmer 6 bushels of peaches. Sketch the associated isoquant on a graph, with L on the x-axis and K on the y-axis.

B. In the short run, the farmer only has 4 units of land. What is his short-run production function? Graph it for values of L from 0 to 16, with L on the xaxis and output on the y-axis. What is the name of the slope of this curve?

C. Assuming the farmer still only has 4 units of land, how much extra output does he get from adding 1 extra unit of labor if he is already using only 1 unit of labor? How much extra output does he get from adding 1 extra unit of labor if he is already using 4 units of labor?

D. In the long run, the farmer can change both his amount of land and his amount of labor. Suppose he increases the size of his orchard to 16 units of land. Add to the graph you drew in (b) a new curve showing output as a function of labor when land is fixed a

Answer 1

A local farmer raises peaches using land and labor. This answer explains how to find input combinations that give the farmer a certain amount of output, the short-run production function for a specific input, the extra output from adding labor in two scenarios, and the long-run production function when land is fixed.

Step-by-step explanation:

A. To find input combinations that give the farmer 6 bushels of peaches, we can rearrange the production function equation Q = 0.5K^0.5 to solve for K: K = (2Q)^2. Substituting Q = 6, we find K = (2 * 6)^2 = 144. So, one input combination is K = 144 and L = 6. Another input combination could be K = 9 and L = 12. We can sketch the isoquant on a graph by plotting these input combinations as points.

B. In the short run, the farmer only has 4 units of land. The short-run production function can be found by substituting K = 4 in the production function equation: Q = 0.5(4)^0.5 = 1. This means that with 4 units of land, the farmer can produce 1 bushel of peaches. We can graph this production function by plotting the points (0,0), (1,4), (2,8), and so on, with L on the x-axis and Q on the y-axis. The slope of this curve is called the marginal product of labor.

C. Assuming the farmer still only has 4 units of land, to find the extra output from adding 1 extra unit of labor when using 1 unit of labor, we can find the difference between the output when using 2 units of labor and the output when using 1 unit of labor. Similarly, to find the extra output from adding 1 extra unit of labor when using 4 units of labor, we can find the difference between the output when using 5 units of labor and the output when using 4 units of labor.

D. In the long run, the farmer can change both the amount of land and the amount of labor. If the farmer increases the size of his orchard to 16 units of land, we can find the new curve by substituting K = 16 in the short-run production function. The resulting production function will show output as a function of labor when land is fixed at 16 units.