Answer to Question #165771 in Macroeconomics for John Bates

Question

First, suppose we have a representative individual living for two periods and has utility, U = ln C1 + ln C2. Let his/her labour income in the first period to be Y1 and zero in the second period. Let the rate of return to savings, r, to be influenced by a random shock, find the first-order condition for his/her choice of C1. Explain how, if at all, does consumption respond to the uncertainty in the rate of return?

Second, suppose we extend the time horizon to infinitely lived agents, and let the individual also supply capital to a representative firm in the economy. Formulate and solve the firm’s profit maximization problem. You can assume price level to be fixed and normalized to one, but you should explain the economic intuitions of the first-order conditions.

Answer 1

Denote consumption in the first period C1, and in the second C2, the real interest rate r. In the first period, the individual consumes C1 and saves Y1-C1

$U = ln C1 + ln C2$

where from

lnC1=U-lnC2

$C1=e^{U-lnC2}$

$rate of return=\frac{profit}{cost}\times100$

$profit=I-C=Y-C1=Y-e^{U-lnC2}$

I=Y1=Y - income

С=cost - consumption С1

$rate of return=\frac{profit}{cost}\times100=\frac{Y-e^{U-lnC2}}{e^{U-lnC2}}\times100$

In the second period, the individual consumes all the income of the second period and the savings of the first period, increased by taking into account the percentage of savings:

$Cost=Y2+(Y1-C1)(1+r)$

Y2=0

$C1=e^{U-lnC2}$

$U = ln C1 + ln C2$

$Cost=Y2+(Y1-C1)(1+r)$

$Cost=e^{Y2+(Y1-e^{U-lnc2})(1+r)}$

Y2=0

$Cost=e^{(Y1-e^{U-lnc2})(1+r)}$

Y1=Y

$Cost=e^{(Y-e^{U-lnc2})(1+r)}$

$rate of return=\frac{profit}{cost}\times100=\frac{Y2-e^{(Y-e^{U-lnc2})(1+r)}}{e^{(Y-e^{U-lnc2})(1+r)}}$

Y2=0

$rate of return=\frac{profit}{cost}\times100=\frac{Y2-e^{(Y-e^{U-lnc2})(1+r)}}{e^{(Y-e^{U-lnc2})(1+r)}}\times100=\frac{-e^{(Y-e^{U-lnc2})(1+r)}}{e^{(Y-e^{U-lnc2})(1+r)}}\times100=-100$

The condition for maximizing the profit of the firm is the use of such an amount of resource, in which the equality is satisfied:

$MRC=MRP$

$MRP=MR\times MP$

$MPC=MC\times MP$

$MR\times MP=MC\times MP$

If the firm is not able to influence the prices of resources, then it buys resources in a completely competitive market of factors of production, then the MRC values will be the same for all hired units of the resource and will amount to the price of a unit of the resource Ra. Profit maximization in this case is achieved if Pa=MRP.

So, at any price online Pa, the firm may determine the amount of used resource, i.e. QD resource, wherein the condition. Pa= MRP. Then the firm can find a match between the price of the resource Ra and the QD of the resource or determine the demand for the resource. The resource demand curve is the MRP curve, and the supply curve is the MRC curve.

In the long run, when all resources are variable, by releasing any volume of output using multiple resources, such as pile L and capital K, the firm can minimize the cost per unit of output if the condition is met:

$\frac{MPk}{Pk}=\frac{MPL}{PL}$

where MPKand MPL— are the marginal products of capital and labor; Pk and PL — are the unit prices of capital and labor.

The considered equality allows us to find the ratio of resources that provide the company with the minimum costs for a given volume of output, but it does not guarantee that in this case the company receives the maximum possible profit. It was proved above that using a single resource, say A, the firm maximizes profit when the value of the marginal product in monetary terms is equal to the marginal cost of the resource:

$MRPa=MRCa$

Using only two resources, such as labor and capital, a firm maximizes profit when a given rule is satisfied for each resource, i.e. MRPL = MRCL, and MRPK = MRCK. Then, in a generalized form, the condition for maximizing profit when using two resources can be represented as follows:

$\frac{MRPL}{MRCL}=\frac{MRPK}{MRCK}$

If the firm is not able to influence the prices of resources, then the MRC is equal to the price of the resource and the last equality takes the form:

$\frac{MRPL}{PL}=\frac{MRCK}{PK}=1$