45 25 e-45 75 40 Units of labor 3. Given the above graph, as you move from point
ID: 1166280 • Letter: 4
Question
45 25 e-45 75 40 Units of labor 3. Given the above graph, as you move from point A to point B a. What happens to output and MRTS? Explain in detail. b. If the price of labor is $9 and the price of capital is $3, what is the marginal rate of technical substitution at the optimal input choice? How much money are you using to produce 45 units of output? i. c. If the price of labor is $5 and the price of capital is $8, what is the marginal rate of technical substitution at the optimal input choice? i. How much money are you spending to produce 45 d. Use your own words to explain the relationship of input prices and units of output? MRTSExplanation / Answer
Solution :
a) Points A and B, both lie on the same isoquant. Along an isoquant (as the name suggests : iso meaning equal and quant meaning quantity), the quantity remains same. So, as we move from point A to point B, the level of output remains same, i.e, Q = 45 units
MRTS (Marginal rate of technical substitution) is the rate at which one factor is substituted for another factor, while keeping the level of output same, i.e, rate at which one factor is decreased while othe factor is increased such that the same output level is produced. It is the slope of isoquant.
As we move from point A to B, amount of capital, K decreases and amount of labor, L increases. Initially, we have to give up more capital to increase labor even by a small amount (isoquant is steeper near point A as compared to point B). As we keep moving towards B, this rate decreases, as in lesser and lesser amount of K is to be sacrificed/ given up to increase the amount of labor by a unit (so, isoquant is flatter around point B as compared to point A). Thus, we can conclude, that as we move from A to B, the rate at which we substitute labor, L for capital, K decreases, or the MRTS decreases.
b) Pl = $9 , Pk = $3. We know that at the optimal point, isocost is tangent to isoquant curve, i.e, slope of isoquant = slope of isocost line. (isocost line is the line on which any combination of two factors result in the same amount of total cost).
Slope of isoquant = MRTS
Slope of isocost line = ratio of factor/input prices = Pl/Pk = 9/3 = 3
Thus, at optimal input choice, MRTS = 3.
i) Amount of money used for Q = 45 units: Total cost = L*Pl + K*Pk. ...(1)
With price ratio/slope so high (compared to part c, we'll see in a bit), we know this price combination ($9 for labor and $3 for capital) is for the steeper looking isocost curve. From the graph, it means we refer to the curve with intercepts - (15,0) and (0,45). So, the optimal input choice is point A, where output level, Q = 45 units. Since, we said isocost has same total cost at all points lying on it, on this isocost,
Total cost = 15*9 + 0*3 (at point (15,0)) = 0*9 + 45*3 (at point (0,45)) = $135 (using (1))
c) Following the same procedure as b) part:
Pl = $5 , Pk = $8
Slope of isoquant = MRTS
Slope of isocost line = ratio of factor prices = Pl/Pk = 5/8 = 0.625
Thus, at optimal input choice, MRTS = 0.625.
i) Amount of money used for Q = 45 units
With price ratio/slope so low (compared to part b, 0.625 < 3), we know this price combination ($5 for labor and $8 for capital) is for the flatter looking isocost curve. From the graph, it means we refer to the curve with intercepts - (40,0) and (0,25). So, the optimal input choice is point B, where output level, Q = 45 units. Since, we said isocost has same total cost at all points lying on it, on this isocost,
Total cost = 40*5 + 0*8 (at point (40,0)) = 0*5 + 25*8 (at point (0,25)) = $200
d) As we saw from above, MRTS (= MPl/MPk) is the rate at which we can substitute amounts of one factor for another, to keep the quantity same. Ratio of factor prices is the rate at which amounts of one factor can be traded for another, while keeping the total cost same.
When we find the optimal input choice, we compare these 2 ratios, in the following sense:
If MPl/MPk > Pl/Pk or by cross multiplying, MPl/Pl > MPk/Pk, this means that more output is generated by spending last dollar on labor than on capital or marginal output generated by increasing one unit of labor is less expensive than by increasing a unit of capital. Thus, it is advantageous to increase labor units, and decrease capital units. As, the labor increases, MPl decreases (i.e, additional labor generates lower additional output), and as capital decreases, MPk increases. Labor is increased till MPl decreases to the point where
MPl/MPk = Pl/Pk.
Similarly if MPl/MPk < Pl/Pk or MPl/Pl < MPk/Pk, it means it is advantageous to invest more in capital and less in labor. Capital input increases while labor input decreases. This in turn decreases MPk and increases MPl, till the point where MPl/Pl = MPk/Pk.
So, the efficient and optimal input choice is where MPl/MPk = Pl/Pk, where the firm can produce same units of output more efficiently, i.e, at a lower cost.
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