Production Function Basics

Question 0

What does A, TFP, include?

Answer 0

Technology Availability of raw materials Prices of intermediate and input goods Institutions, culture Labor and capital utilization Should take out labor quality

Question 1

Express “y” gdp per worker

Answer 1

Y/N = AK^aN^(1-a)/N Divide by N^aN^(1-a) Get AK^a/N^a, –> (K/N)^a is little ‘k’ to the alpha so Little y = Ak^a

Question 2

What is η(k) and η(n)? How are they related?

Answer 2

ηk is share of capital in output–> K*Fk(K,N)/F(K,N) η(n) is share of labor in output–> N*F(K,N)/F(K.N) η(k) = 1-η(n)

Question 3

Define η(k) and η(n) using factor shares

Answer 3

η(k) = [dY/dK]/[Y/K] => r*pk*K/(P*Y) η(n) = [dY/dN]/[Y/N] => r*w/(P*Y) In other words, each factor’s elasticity equals its share in the value of output

Question 4

What does the zero pure profits, perfect competition scenario say about output as a function of factor inputs and prices?

Answer 4

P*Y = w*N + r pk*K

Question 5

What is the mathematical expression for diminishing returns to capital? Diminishing returns to labor?

Answer 5

Fkk 0

Question 6

What is Okun’s law?

Answer 6

(Ybar-Y) / y = 2*(u-ubar) Where ybar is full employment output and ubar is full employment unemployment

Question 7

Express the growth rate of output (Y) in terms of A, K and N

Answer 7

Y’/Y = A’/A +a(K’/K)+b(N’/N) Also equals A’/A + η(k)(K’/K) + η(n)(N’/N) Also equals A’/A +(KFk/F)*K’/K+(NFn/F)N’/N And for short periods ΔY/Y = ΔA/A + aΔK/K + bΔN/N

Question 8

Express the growth rate of output per capita (Y/N) in terms of A, K and N

Answer 8

Y’/Y - N’/N = A’/A +a*(K’/K-N’/N) In other words, growth in GDP per capita equals growth in TFP plus growth of capital per worker

Question 9

What is the expression for the standard Solow residual?

Answer 9

A’/A = Y’/Y - η(k)*[K’/K] - η(n)*[N’/N] A’/A = η(k)*(Y’/Y-K’/K) + η(n)*(Y’/Y-N’/N)

Question 10

What is the expression for the Solow residual assuming varied labor quality? Assuming varied utilization rates?

Answer 10

If L = h*N, then A’/A = η(k)*[Y’/Y-K’/K] + η(l)[Y’/Y-N’/N] - η(l)*[h’/h) and if you fail to adjust for quality, you will overestimate TFP growth A’/A by a factor of η(l)*(h’/h) A = A*μk*μn

Question 11

If labor quality varies, how do you express the growth rate of labor?

Answer 11

If labor quality varies, that means that L = N1^a1*N2^a2 So L’/L = N1/(N1+N2)*[N’1/N1] + N2/(N1+N2)*[N’2/N2] And N’/N = a1*(N’1/N1) + a2*(N’2/N2)

Question 12

If labor quality varies, how do you express the growth rate of the total, unadjusted labor force (L1= N1+N2)? You are looking for L’/L-N’/N, which is the measurement error for N = N1 + N2 (?)

Answer 12

It equals [a1 - N1/(N1+N2)] *(N’1/N1-N’2/N2] Equals w1N1/(w1N1+w2N2) - N1/(N1+N2) * [N’1/N1 - N’2/N2] Share of N1 wages as pct of total (alpha) minus share of N1 quantities as pct of total times growth rate differential of N1 over N2.

Question 13

What is the first key lesson of the Solow TFP?

Answer 13

In the long run, most differences in growth of output per worker across countries is due to differences in TFP.

Question 14

What are the findings of Alwyyn Young?

Answer 14

Korea, Taiwan, HK, Singapore had huge GDP and GDP per capita growth, TFP looked like it was 6-7% versus 1-3% in developed economies. Perfectly explainable by factor accumulation such as: 1) increased labor force participation (esp women) 2) Shift towards more educated labor force 3) Movement away from agricultural sector (low productivity) 4) Capital deepening and changes in capital composition Means that TFP growth is not sustainable indefinitely dt decreasing returns to capital and labor at some point.

Question 15

Another implication of how utilization rates enter into TFP?

Answer 15

TFP is proyclical because it’s higher in booms and lower in recessions.

Question 16

What are the implications of the L=Na1*Na2 equation on calculating TFP growth properly?

Answer 16

If w1>w2 and w1 is the wage of skilled labor, then skilled labor has higher marginal productivity and the workforce shifts to higher skilled labor. In the N’/N-L’/L equation just using raw unadjusted L’/L will underestimate the true quality-adjusted change in labor N’/N and overstates TFP growth.

Question 17

How do you account for little y properly assuming varied labor input?

Answer 17

1) Formulate basic GDP per capita estimate 2) Adjust participation for (L/P) to focus on non agricultural sector to get a non agricultural estimate 3) Then use actual labor shares 4) Capital deepening adjustment (Y’/Y-K’/K) 5) Make a quality adjustment for labor and capital by including hours worked and wages and quantities of labor type, prices and quantity of capital type

Question 18

What are the underlying elements of items 2, 3 and 4 for properly accounting for TFP?

Answer 18

– If mortality rates go down, women enter labor force, population booms, that means P increases so 2) adjustment changes – Higher savings and investment rates imply more capital deepening – Educational reforms and sectoral changes can enhance 3)

Question 19

How does the Alwyyn Young paper help us understand China and India’s growth?

Answer 19

China has combined massive capital deepening with high rates of TFP growth largely due to public-private reforms, tech transfer, etc. India focused on skills upgrading and has lousy TFP growth so far. Both are pretty under capitalized though so room exists for K/N to keep growing

Question 20

If Y = F(K,K,L,t) what is the skill premium?

Answer 20

Wh(t)/Wl(t) = C*[A(t)/B(t)]^(σ-1/σ)*(H/L)^(-1/σ) Technological change is skill biased if A(t) grows faster than B(t) and σ > 1 Δ[(H/L)]/(H/L) / [(Δ(Wh/Wl)]/(Wh/Wl)

Question 21

Empirically when trying to get at skilled labor demand, what do we observe?

Answer 21

Wh(t)/Wl(t) and H(t)/L(t) We don’t observe g(t) or σ directly, have to back out g(t) after picking a σ

Question 22

What are the policy implications of skill-biased technological change?

Answer 22

US and Canada subject to same technological change but wage inequality rose in US and not in Canada because H/L (supply) rose faster in Canada than in US, pushing wages down. Also explains decline in skill premium in the US in 70’s; baby boomer generation entered workforce, more educated so higher H/L pushed wage premium down

Question 23

How do you determine output via factor demand?

Answer 23

Maximize π by P*AF(K,N) - w*N-rp_kK

Question 24

What is the relationship between n_k and n_nmarginal products/factor prices/