Chapter 17. Chapter 17

Work It Out
Chapter 17
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Question 17.1

Assume that Brazil and the United States have different production functions [MATH: q=f(k) ](output per worker is a function of capital per worker), where [MATH: q ]() is output per worker and [MATH: k ]() is capital per worker. Let [MATH: q=Ak^{\frac{1}{3}} ](output per worker is proportional to capital per worker to one third). You are told that relative to the United States [MATH: = 1 ](), Brazil has an output per worker of 0.40 and capital per worker of 0.33. Can [MATH: A ]() be the same in Brazil as in the United States? If not, compute the level of [MATH: A ]() for Brazil. What is Brazil’s MPK relative to the United States?
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Video transcript

Work It Out, Chapter 17

(Transcript of audio with descriptions. Transcript includes narrator headings and description headings of the visual content)

(Speaker)
We are told that Brazil and the United States have different production functions q equals f times k, where q is output per worker and k is capital per worker. Let q equal Ak to one-third. You are told that relative to the United States equals 1, Brazil has an output per worker of 0.40 and capital per worker of 0.33. We are asked if the productivity level A can be the same in Brazil as in the United States. If the answer is no, we are asked to compute the level of A for Brazil relative to the United States. We are also asked what is Brazil’s marginal product of capital relative to the U.S. marginal product of capital. From the problem we know that: Brazil and the United States have different production functions.

(Description)
The following text is written: Setting the problem: q subscript B does not equal q subscript US.

(Speaker)
The general production function is q equals Ak to one-third.

(Description)
The following relation is written below the previous one: q equals A times k to the power of one-third.

(Speaker)
Brazil’s output per worker relative to the U.S. output per worker is q B over q US equals 0.4.

(Description)
The following relation is written below the previous one: q subscript B over q subscript US equals 0.4.

(Speaker)
Brazil’s capital per worker relative to the U.S. capital per worker is k B over k US equals 0.33.

(Description)
The following relation is written below the previous one: k subscript B over k subscript US equals 0.33.

(Speaker)
The question will ask if A can be the same in Brazil as in the United States. We know that the assumption of equal productivity levels across countries is wrong, because capital does not flow to poor countries as predicted by this assumption. Therefore, AB does not equal A US.

(Description)
The following relation is written: A subscript B does note equal A subscript US.

(Speaker)
Since A is not equal, we have to compute it. However, in order to compute A, we need to first compute Brazil’s marginal product of capital relative to the U.S. marginal product of capital. We start with the formula MPK B over MPK US. Plugging in the ratios of output per worker and capital per worker, which are given, we obtain the ratio of marginal product of capital.

(Description)
The following equation is written: MPK subscript B over MPK subscript US equals the fraction with q subscript B over q subscript US in the numerator, and k subscript B over k subscript US in the denominator, equals 0.4 over 0.33 equals 1.21.

(Speaker)
To find the ratio of the productivity levels, we start from the relative marginal product of capital and plug in the formulas for each MPK.

(Description)
The following equation is written: MPK subscript B over MPK subscript US equals the fraction with q subscript B over q subscript US in the numerator, and k subscript B over k subscript US in the denominator, equals the fraction with A subscript B multiplied by k subscript B to the one-third divided by A subscript US multiplied by k subscript US to the one-third in the numerator, and k subscript B over k subscript US in the denominator, equals the fraction with A subscript B over A subscript US multiplied by k subscript B over k subscript US all to the power one-third in the numerator, and k subscript B over k subscript US in the denominator, equals the fraction with A subscript B over A subscript US in the numerator, and k subscript B over k subscript US multiplied by k subscript B over k subscript US all to the negative one-third in the denominator, equals the fraction with A subscript B over A subscript US in the numerator, and k subscript B over k subscript US all to the two-thirds in the denominator, equals the fraction with A subscript B over A subscript US in the numerator, and 0.48 in the denominator, equals 1.21.

(Speaker)
After isolating the productivity level ratio, we can compute it.

(Description)
The following equation is written below the previous one: A subscript B over A subscript US equals 1.21 times 0.48 equals 0.58.

(Speaker)
Brazil’s productivity level is 58 percent of that of the United States.