Production Function Basics Flashcards
What does A, TFP, include?
Technology Availability of raw materials Prices of intermediate and input goods Institutions, culture Labor and capital utilization Should take out labor quality
Express “y” gdp per worker
Y/N = AKaN(1-a)/N Divide by NaN(1-a) Get AKa/Na, –> (K/N)a is little ‘k’ to the alpha so Little y = Aka
What is η(k) and η(n)? How are they related?
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)
Define η(k) and η(n) using factor shares
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
What does the zero pure profits, perfect competition scenario say about output as a function of factor inputs and prices?
P*Y = w*N + r pk*K
What is the mathematical expression for diminishing returns to capital? Diminishing returns to labor?
Fkk 0
What is Okun’s law?
(Ybar-Y) / y = 2*(u-ubar) Where ybar is full employment output and ubar is full employment unemployment
Express the growth rate of output (Y) in terms of A, K and N
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
Express the growth rate of output per capita (Y/N) in terms of A, K and N
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
What is the expression for the standard Solow residual?
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)
What is the expression for the Solow residual assuming varied labor quality? Assuming varied utilization rates?
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
If labor quality varies, how do you express the growth rate of labor?
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)
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 (?)
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.
What is the first key lesson of the Solow TFP?
In the long run, most differences in growth of output per worker across countries is due to differences in TFP.
What are the findings of Alwyyn Young?
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.
Another implication of how utilization rates enter into TFP?
TFP is proyclical because it’s higher in booms and lower in recessions.
What are the implications of the L=Na1*Na2 equation on calculating TFP growth properly?
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.
How do you account for little y properly assuming varied labor input?
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
What are the underlying elements of items 2, 3 and 4 for properly accounting for TFP?
– 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)
How does the Alwyyn Young paper help us understand China and India’s growth?
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
If Y = F(K,K,L,t) what is the skill premium?
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)
Empirically when trying to get at skilled labor demand, what do we observe?
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
What are the policy implications of skill-biased technological change?
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
How do you determine output via factor demand?
Maximize π by P*AF(K,N) - w*N-rpkK
What is the relationship between nk and nnmarginal products/factor prices/
