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ECON1003 S1 2023

PROBLEM SET 8

1. Consider the Cobb Douglas production function Q = ALα K β . Find the partial elasticities with respect to labor L and capital K.

2. Consider an individual with the Cobb Douglas utility function U1 = x1x2. (a) Find the MRS

(b) Now consider a new utility function U2 = ln(U1 (x1, x2)). Find the MRS for the new utility function. Compare this with the previous part.

3. Find the differentials for the following functions:

(a) f(x) = x2.

(b) f(x, y) = 3ex + y.

(c) f(x, y) = ln(x) + 5ln(y).

(d) F(L, K) = 10L0.2K0.6

4. Consider a national income model with the following equations.

Y = C + I0 + G0  (1)

C = C0 + b(Y − T ) (2)

T = tY (3)

(a) Solve for the equilibrium levels of C and Y .

(b) Use partial derivatives to derive expressions for changes in the equilibrium level of Y with respect to the parameters C0, G0, I0, and t.

(c) Redo the problem using differentials. Do you get similar results for the partial derivatives derived in part (b)