Calculation of Decreasing Return to Scale - Assignment Example

Calculation of Decreasing Return to Scale a). F (K, L) =K1̸2L1/3 To solve this let us assume the units of capital (K) are 5 and the units of labour (L) are 4, substituting 5and 4 from the production function, the following results are obtained.F (K, L) =K1̸2L1/3 = (5)1̸2× (4)1/3 =2.5+1.33=3.8333 Now, suppose K is reduced by 1 unit and L is increased by 1 unit, such that the new, K, become=6 and the new L become=3. Then the new output will be as follows; F (K, L) = (4)1̸2× (3)1/3 =2+1.442=3.442.

Therefore, this production function indicates decreasing return to scale, because after reducing the units of capital by 1unit and increasing the units of capital by 1, the output decreased by 0.391, obtained by finding the difference between 3.833 and 3.442. b).F (K, L) =K+L=K+L Let us assume that K= 5and L=2, then by substituting the values from the above production function we can be able to know the type of return to scale being exhibited;F (K, L) =K+L=K+L=5+2 =7 Suppose K is reduced by 1unit and L is increased by 1 unit, the new K becomes 4 and the new L become 3.

This indicate that the production function, F (K, L) =K+L=K+L, exhibit a constant return to scale because even after reducing K and increasing L proportionally, the output remain ,7,constantC).F (K, L) =K+L² Assume that K=5 and L=4, therefore, F (K, L) =K+L²=5+ (4)²=21. Supposing that K is increased by one unit and L is reduced by 1 unit, such that the new K=6 and the new L=3. The new output will be; F (K, L) =K+L²=6+ (3)²= 15 units. This indicates that production function above exhibits reducing to scale. d). 𝐹 (𝐾, 𝐿) = 𝐾𝛼𝐿1−𝛼+7 Assuming that K=6 and L=5, while 𝛼=constant=1, the replacing the values from the production function, 𝐹 (𝐾, 𝐿) = 𝐾𝛼𝐿1−𝛼+7=61×5(1-1) +7=6×1+7=13.

Supposing that K is increased by 1 unit while L is reduced by 1 unit, then the new output becomes; 𝐹(𝐾,𝐿)= 𝐾𝛼𝐿1−𝛼+8=51×5(1-1)+7=8×1+7=15.This indicates that the production function, exhibits increasing return to scale. 2. Given Cobb-Douglass production function 𝑌=𝐴𝐾𝛼𝐿1−𝛼. Based on the equation above, the amount of capital per worker may be derived by dividing both side by L follow; 𝑌=𝐴𝐾𝛼𝐿1−𝛼Y =𝐴𝐾𝛼𝐿1−𝛼 = (K) 𝛼=𝐴𝐾𝛼 L The production level per worker is Y = 𝐴𝐾𝛼.

This means the level of capital per worker is decreasing relative to the level of capital per workers as shown in the diagram below. Output and investment per worker 4 Y=AF (K) 3.5 MPK 3 2.

5 I=S (AF) k 2 1.5 10.5 0 50 100 150 200 Capital (k) per worker3.countryCapital per personPer capital GDPCapital relative to U.SGDP relative to U.SPredicted *YImplied TFT to match DataUnited States135,87742,8771.001.001.001.00France 109,02329,6333.681.45429688052.45Mexico33,16811,2040.343.834291132953Kenya2,3792,02511.5521.174286993.974. a).

Slow model may be derived from the cobras model and may be represented as follows; Q = A Ka L b. The model asserts that an increase in either of the three factors namely; capital, k, labour, L, and multifactor productivity, A, leads to an increase in output (Jones, pp.160-167). b).Graph showing how output evolves overtime AInputs(L, k) growth m Output in Units The rate of growth increases upward starting from point M up to a stable state point b where it remains constant and start declining slowly towards the Right (Jones, pp.165).c).

If A grew at a constant rate, GDP will consequently grow because as one of the input is increased, GDP increases. d). Consumption reduces savings and investment, therefore, if TFP increases economy will grow. 5). a). If investment rates double, GDP per capital would also doubles because investment has a proportional impact on GDP per capital. 6). a).At steady state, capital stock and depreciation are offset. Dk=0i = sy = s f (k)Where i=investment, and =depreciation (Jones, pp.175-180)Dk =s* f (k) – dkAt steady state=2648billions-8%×2648billionThe steady state capital stock for the economy=$2436.16b). 2648billions-20%×2648billion=$2118.4c). Dk =s* f (k) – dk a).

=2648billions (0)-8%×2648billion= ($2118.4)=$2118.4 2648billions-20%×2648billion=$2118.4b).2648billions(0)-20%×2648billion=$529.6Works CitedJones, Charles I. Macroeconomics. New York: W.W. Norton, 2011. Print.