Answer to Question #170374 in Microeconomics for Oliva

Question

2. Given F(K,L) = AKa Lb

, show that a and b determine whether the function is contact, increasing 

or decreasing return to scale (show one example in each case).

3. Explain two major approaches to the analysis of consumer behavior.

4. How is consumer equilibrium reached (explain in words) ? Mathematically, show how it is reached

Answer 1

a) F(K,L) = AK$^{a}$L$^{b}$

Using the properties of homogeneity, multiply each input by a constant, say, $k$, and factor

F($k$K,$k$L) = $A(kK)^{a}(kL)^{b}$

= $Ak^ak^ak^bL^b$

= $k^{a+b}(AK^aL^b)$

= $k^{a+b}(F(K,L))$

For a strict Cobb-Douglas production function, where $a+b=1$, exhibits a constant returns to scale. Where $a+b≠1$ (a generalized production function), if $a+b>1$, the function exhibits an increasing returns to scale and a decreasing returns to scale if $a+b<1$

b) Approaches to consumer behavior analysis include the preference approach and utility approach.

i) The preference approach makes use of the indifference curve to show how indifferent a consumer is to the consumption of different bundles of commodities given his budget constraints.

ii) The utility approach looks at consumer behavior from the aspect of ranking. The approach seeks to measure the satisfaction derived from the consumption of certain goods by consumers.

c) Consumers equilibrium is achieved at the point where the budget constraints is at tangency with the indifference curve. This gives the optimum bundle available to a consumer given his preference and constraint. At this point, the ratio of the marginal utilities, $\frac{{MU}_{x}}{{MU}_{y}}$, must equal the price ratio,$\frac{{P}_{x}}{{P}_{y}}$, of the goods in question.

To derive this mathematically, assume a utility function $x^{0.5}y^{0.5}$ subject to the constraint $P_{x}x+P_{y}=M$, the consumer's equilibrium is derived below:

$U=X^{0.5}Y^{0.5}+\lambda(M-P_{x}-P_{y})$

$U_x=0.5X^{-0.5}Y^{0.5}-\lambda{P_x}=0$

$U_y=0.5X^{0.5}Y^{-0.5}-\lambda{P_y}=0$

$U_\lambda= M-P_x-P_y=0$

where $0.5X^{-0.5}Y^{0.5}=U_x=MU_x$ and $0.5X^{0.5}Y^{-0.5}=U_y=MU_y$

$\lambda=\frac{{0.5X^{-0.5}Y^{0.5}}}{{P_x}}=\frac{MU_x}{P_x}$

$\lambda=\frac{{0.5X^{0.5}Y^{-0.5}}}{{P_y}}=\frac{MU_y}{P_y}$

equating $\lambda$:

$\frac{MU_{x}}{P_x}=\frac{MU_y}{P_y}$

$\frac{MU_x}{MU_y}=\frac{P_x}{P_y}$