Xiaoxi and I decided to go into business together! We want to open a small t-shi
ID: 1218870 • Letter: X
Question
Xiaoxi and I decided to go into business together! We want to open a small t-shirt company. We will import plain t-shirts and design/make them into unique graphic t-shirts (with dyes and a small printer machine). Let Xi="raw" t-shirts and let X_2= everything else (dyes and printing stuff including capital). There are several different type of printing "technology" for us to choose from, and Xiaoxi and 1 disagree about which is best for our business plan. Help us decide by looking at the following three production functions. For each one calculate/determine for us: (i) if the function exhibits increasing, decreasing, or constant returns to scale; (ii) MP_1 and whether it is increasing, decreasing, or stays constant with respect to changes in X_1; (iii) MP2 and whether it is increasing, decreasing, or stays constant with respect to changes in x_2. (a) Technology #1: f(x_1,x_2) = 0.5x_1 +2x_2 (b) Technology #2:/(x_1,x2) = 0.2x_1x^2_2 (c) Technology #3: f(x_1,x_2) = 0.8x_1 + x^0.5_2 (d) Which technology do you think we should choose? Briefly explain why.Explanation / Answer
a) Technology 1: f = 0.5x1+2x2
i) Let x1 = ax1 and x2 = ax2 where a is a positive number,
new production function, f2 = 0.5ax1 +2ax2 = a(0.5x1+2x2) = af = a * old production function.
By increasing both inputs with an integer, the output increases with same integer. Thus this technology possess constant return to scale.
ii) MP1 = df/dx1 = 0.5 it is decreasing because there is no x1 in MP1.
iii) MP2 = df/dx2 = 2 it is decreasing because there is no x2 in MP2.
b) Technology 2: f = 0.2x1x22
i) Let x1 = ax1 and x2 = ax2 where a is a positive number,
new production function, f2 = 0.2ax1(ax2)2 = a3(0.2x1+x22) = a3 * old production function.
By increasing both inputs with an integer, the output increases with more than the integer. Thus this technology possess increasing return to scale.
ii) MP1 = df/dx1 = 0.2x22. it is decreasing because there is no x1 in MP1.
iii) MP2 = df/dx2 = 0.4x1x2. if we put x2 = ax2 in MP2 we will get,
0.4x1ax2 = a*old MP2. Hence constant return.
c) Technology 3: f = 0.8x1 + x20.5
i) Let x1 = ax1 and x2 = ax2 where a is a positive number,
new production function, f2 = 0.8ax1 + (ax2)0.5 = a1.5(0.2x1+x22) = a1.5 * old production function.
By increasing both inputs with an integer, the output increases with more than the integer. Thus this technology possess increasing return to scale.
ii) MP1 = df/dx1 = 0.8. it is decreasing because there is no x1 in MP1.
iii) MP2 = df/dx2 = 0.5x2-0.5. if we put x2 = ax2 in MP2 we will get,
0.5a-0.5x2-0.5 = a-0.5*old MP2. Hence decreasing return.
d) Technology 2 should be used because of its increasing return to scale of production function and constant return of MP2.
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